Pascal Auscher
email: pascal.auscher@universite-paris-saclay.fr

AND Moritz Egert
email: moritz.egert@universite-paris-saclay.fr

Kaj Nystr
email: kaj.nystrom@math.uu.se

Alan Mcintosh

L 2 WELL-POSEDNESS OF BOUNDARY VALUE PROBLEMS FOR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS

Keywords: 2010 Mathematics Subject Classification. Primary: 35K40, 35K46, 42B37. Secondary: 26A33, 42B25, 47A60, 47D06 Second order parabolic systems, parabolic Kato square root estimate, parabolic Dirac operator, Dirichlet and Neumann problems, (non-tangential) maximal functions, square function estimates, a priori representations, boundary layer operators, half-order derivative

We prove the first positive results concerning boundary value problems in the upper halfspace of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, wellposedness results with data in L 2 -Sobolev spaces together with invertibility of layer potentials, and perturbation results. In the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversally independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation. Contents 1. Introduction 1 2. Main results 3 3. Homogeneous fractional Sobolev spaces 29 4. The parabolic Dirac operator 34 5. The correspondence to a first order system: Proof of Theorem 2.2 36 6. Resolvent estimates 38 7. Quadratic estimates for P M : Proof of Theorem 2.3 42 8. Reverse Hölder estimates and non-tangential maximal estimates 50 9. Duality results for λ-independent operators 62 10. The operator sgn(P M ) 64 11. Inverting parabolic operators by layer potentials 65 12. Well-posedness results for λ-independent coefficients 66 13. Uniqueness for Dirichlet problems 73 14. Miscellaneous generalisations and open problems 80

Introduction

In this paper we study boundary value problems for parabolic equations of type

Lu := ∂ t u -div X A(X, t)∇ X u = 0. (1.1)
Here, X = (λ, x) ∈ R n+1 + := (0, ∞) × R n and t ∈ R. We assume that the matrix A is measurable, strongly elliptic, and possibly depends on all variables. Parabolic systems will be included in our considerations as well. The focus on the upper parabolic half-space R n+2 + here is of course arbitrary and we could as well have decided to work in the lower one.

We stress that with mere measurable dependence in time there are no positive results on solvability of boundary value problems in this context up to now. Indeed, most studies concerning second order parabolic boundary value problems have focused on the heat equation in (parabolic) Lipschitz-type domains and some generalisations. For the heat equation, we mention [START_REF] Brown | The method of layer potentials for the heat equation in Lipschitz cylinders[END_REF][START_REF] Brown | The initial-Neumann problem for the heat equation in Lipschitz cylinders[END_REF] devoted to timeindependent Lipschitz cylinders and the important achievements in [START_REF] Hofmann | Parabolic singular integrals of Calderon-type, rough operators and caloric layer potentials[END_REF][START_REF] Hofmann | L 2 solvability and representation by caloric layer potentials in time-varying domains[END_REF][START_REF] Lewis | The method of layer potentials for the heat equation in time-varying domains[END_REF] for time dependent Lipschitz-type domains. As for generalisations, [START_REF] Hofmann | The Dirichlet problem for parabolic operators with singular drift terms[END_REF] studies the Dirichlet problem for parabolic equations with singular drift terms (containing the ones coming from an elaborate change of variables from the heat equation) with coefficients having some smoothness and smallness in the sense of a Carleson condition. See also [START_REF] Dindos | The Dirichlet problem for second order parabolic operators satisfying Carleson condition[END_REF] and [START_REF] Dindos | BMO solvability and the A∞ condition for second order parabolic operators[END_REF].

In a series of recent papers [START_REF] Castro | Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients[END_REF][START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF][START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF], the solvability for Dirichlet, regularity and Neumann problems with L 2 data was established for the class of parabolic equations (1.1) under the assumptions that the elliptic part is independent of both the time variable t and the variable λ transverse to the boundary, and that the elliptic part has either constant (complex) coefficients, real symmetric coefficients, or small perturbations thereof. This is proved by first studying fundamental solutions and proving square function estimates and non-tangential estimates for the single layer potential along the lines of the approach for elliptic equations in [START_REF] Alfonseca | Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L ∞ coefficients[END_REF]. In the case of real symmetric coefficients, solvability is then established using a Rellich-type argument.

An important step in [START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF] -revisiting an earlier idea of [START_REF] Hofmann | L 2 solvability and representation by caloric layer potentials in time-varying domains[END_REF][START_REF] Kaplan | Abstract boundary value problems for linear parabolic equations[END_REF] -is the introduction of a reinforced notion of weak solution by means of a sesquilinear form associated with (1.1), showing its coercivity and proving a fundamental square function estimate. In fact, what is implicitly shown in [START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF] by adapting the proof of the Kato conjecture for elliptic operators [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF], is that the operator ∂ t -div x A(x)∇ x is maximal-accretive in L 2 (R n+1 ) and that the domain of its square root is that of the associated form. Note that this is for one dimension lower than (1.1). With this at hand, [START_REF] Castro | Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients[END_REF] proves a number of technical estimates for the single layer potential for L based on the DeGiorgi-Nash-Moser condition. Finally, [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF] addresses solvability and uniqueness issues by proving invertibility of the layer potentials. However, the methods there are limited to time independent coefficients and removing this constraint requires new ideas.

We owe to Alan McIntosh the insight that boundary value problems involving accretive sesquilinear forms could be addressed via a first order system of Cauchy-Riemann type. For elliptic equations this approach was pioneered in [START_REF] Auscher | Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems[END_REF] and carried out systematically in [START_REF] Auscher | Solvability of elliptic systems with square integrable boundary data[END_REF] using a simpler setup. It relies on three main properties of the corresponding perturbed Dirac operator at the boundary: bisectoriality, off-diagonal estimates for its resolvents and a bounded holomorphic functional calculus on L 2 , which in this case has been obtained by a T (b) argument based on a remarkable elaboration on the solution of the Kato conjecture in [START_REF] Axelsson | Quadratic estimates and functional calculi of perturbed Dirac operators[END_REF]. Non-tangential maximal estimates to precise the boundary behaviour were also obtained. Uniqueness via a semigroup representation, later identified to layer potential integrals in the case of real elliptic equations in [START_REF] Rosén | Layer potentials beyond singular integral operators[END_REF], are obtained in [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF].

In this paper we implement this strategy for boundary value problems of parabolic equations. The outcome of our efforts is the possibility to address arbitrary parabolic equations (and systems) as in (1.1) with coefficients depending measurably on time and on the transverse variable with additional transversal regularity, the removal of time independence being the substantial step.

In the spirit of a first order approach for parabolic operators, we mention two articles [START_REF] Cerejeiras | Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains[END_REF][START_REF] Cerejeiras | Fischer decomposition and special solutions for the parabolic Dirac operator[END_REF] proposing a factorisation of the heat operator via a (non-homogeneous) Dirac operator valued in some Clifford algebra that involves only first and zero order partial derivatives. But this does not seem to meet our needs here: First, we have non-smooth coefficients and, second, half-order derivatives in time turn out to be part of the data for the boundary value problems. Hence, it is natural that our Dirac operator comes with such derivatives, too.

To proceed, we have to overcome serious difficulties compared with the elliptic counterpart since half-order time-derivatives appearing in the Dirac operator are non-local and have poor decay properties. However, we are still able to obtain just enough decay in the off-diagonal estimates of its resolvents to obtain through a T (b) argument the square function estimate that opens the door to all our further results concerning boundary value problems with L 2 data. Additionally, we prove important non-tangential maximal estimates via some new reverse Hölder inequalities for reinforced weak solutions.

Concerning the Dirichlet problem, the first order approach only comprises uniqueness in the sense of L 2 convergence with square function control. However, in the cases where we can solve, we also prove uniqueness in the classical sense of non-tangential convergence at the boundary with non-tangential maximal control, which is the largest possible class. This includes that the "unique" solution of the Dirichlet problem with non-tangential maximal control in L 2 also enjoys the square function estimates. Proof of uniqueness is done via new arguments, not requiring any form of regularity of solutions such as the DeGiorgi-Nash-Moser estimates, and hence it does apply to parabolic systems in particular.

As a particular outcome of independent interest, we obtain that the parabolic operator ∂ tdiv x A(x, t)∇ x can be defined as a maximal-accretive operator in L 2 (R n+1 ) and that it satisfies a Kato square root estimate -its square root has domain equal to that of the defining form. Note that we allow measurable t-dependence on the coefficients in contrast to [START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF], which makes a huge difference. This has a worth-mentioning consequence. In [START_REF] Auscher | The Dirichlet problem for second order parabolic operators in divergence form[END_REF], we use the solution of the parabolic Kato problem to prove that the L p Dirichlet problem corresponding to (1.1) is well-posed if A = A(x, t) is real valued but possibly non-symmetric, measurable, and p ∈ (1, ∞) is sufficiently large, which is the best possible result in general.

In the next section we will give a comprehensive presentation of our results, postponing details for later. It will come with the introduction of some necessary notation, but we try to make the presentation as fluent as possible to give a general overview. Also the progression does not necessarily follow the order in which things are proved. The subsequent Sections 3 -13 are devoted to the proofs of our results. In the final Section 14 we formulate further generalisations of our results and state additional open problems. Let us mention two of them here. One concerns boundary value problems for (1.1) in time dependent Lipschitz-type domains λ > ϕ(x, t). While Lipschitz changes of variables preserve the class of elliptic equations -div X A∇ X u = 0, this is not the case for equations ∂ t u -div X A∇ X u = 0. Already the simple change of variables (λ, x, t) → (λ -ϕ(x, t), x, t) requires that ϕ Lipschitz in both x and t and brings in a drift term. But the natural regularity on ϕ is some half-order regularity in t and requires more elaborate changes of variables such as the ones used in the works mentioned before. This results in creating drift terms and brings in leading coefficients to which existing results including ours do not apply. Another one would be to allow for data in other spaces which, if one uses the first order approach, requires to develop a Hardy space theory associated with parabolic Dirac operators in the first place.

Main results

2.1. The coefficients. For the time being, A = A(X, t) = {A i,j (X, t)} n i,j=0 is assumed to be an (n + 1) × (n + 1)-dimensional matrix with complex coefficients satisfying the uniform ellipticity condition (2.1)

κ|ξ| 2 ≤ Re(A(X, t)ξ • ξ), |A(X, t)ξ • ζ| ≤ C|ξ||ζ|,
for some κ, C ∈ (0, ∞), which we refer to as the ellipticity constants of A, and for all ξ, ζ ∈ C n+1 , (X, t) ∈ R n+2 + . Here u • v = u 0 v 0 + ... + u n v n , ū denotes the complex conjugate of u and u • v is the standard inner product on C n+1 . We use X = (x 0 , x) and x = (x 1 , . . . , x n ). Most often, we specialize the variable transverse to the boundary by setting λ = x 0 .

Reinforced weak solutions.

If Ω is an open subset of R n+1 , we let H 1 (Ω) = W 1,2 (Ω) = W 1,2 (Ω; C) be the standard Sobolev space of complex valued functions v defined on Ω, such that v and ∇v are in L 2 (Ω; C) and L 2 (Ω; C n ), respectively. A subscripted 'loc' will indicate that these conditions hold locally. We shall say that u is a reinforced weak solution of (1.1) on R n+1

+ × R if u ∈ Ėloc := Ḣ1/2 (R; L 2 loc (R n+1 + )) ∩ L 2 loc (R; W 1,2 loc (R n+1 + ))
and if for all φ ∈ C ∞ 0 (R n+2 + ),

∞ 0 R n+1 A∇ λ,x u • ∇ λ,x φ + H t D 1/2 t u • D 1/2
t φ dx dt dλ = 0.

Here, D

1/2 t is the half-order derivative and H t the Hilbert transform with respect to the t variable, designed in such a way that

∂ t = D 1/2 t H t D 1/2
t . The space Ḣ1/2 (R) is the homogeneous Sobolev space of order 1/2. In Section 3 we shall review properties of this space, which we define as a subspace of L 2 loc (R) in order to deal with proper distributions. Still our definition is equivalent to other common ones: it is the completion of C ∞ 0 (R) for the norm D 1/2 t

• 2 and, modulo constants, it embeds into the space S ′ (R)/C of tempered distributions modulo constants.

At this point we remark that for any u ∈ Ḣ1/2 (R) and φ ∈ C ∞ 0 (R) the formula

R H t D 1/2 t u • D 1/2 t φ dt = - R u • ∂ t φ dt
holds, where on the right-hand side we use the duality form between Ḣ1/2 (R) and its dual Ḣ-1/2 (R) extending the complex inner product of L 2 (R). It follows that a reinforced weak solution is a weak solution in the usual sense on R n+1 + ×R: it satisfies u ∈ L 2 loc (R; W 1,2 loc (R n+1 + )) and for all φ

∈ C ∞ 0 (R n+2 + ), R R n+1 + A∇ λ,x u • ∇ λ,x φ dx dλ dt - R R n+1 + u • ∂ t φ dx dλ dt = 0.
This implies ∂ t u ∈ L 2 loc (R; W -1,2 loc (R n+1 + )). Conversely, any weak solution u in Ḣ1/2 (R; L 2 loc (R n+1 + )) is a reinforced weak solution.

Energy solutions.

As a first illustration of this concept of interpreting (1.1), let us quickly explain how to obtain reinforced weak solutions using the form method. This seems to be surprising at first sight and relies on the fundamental revelation of coercivity for the parabolic operator L explored first in [START_REF] Kaplan | Abstract boundary value problems for linear parabolic equations[END_REF] and later on for instance in [START_REF] Castro | Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients[END_REF][START_REF] Hofmann | L 2 solvability and representation by caloric layer potentials in time-varying domains[END_REF][START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF][START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF]. It will also allow us to touch some of the paper's central themes. For the moment, time or even transversal dependency of the coefficients will not be any obstacle.

We are looking for reinforced weak solutions that belong to the energy class Ė of all v ∈ Ėloc for which

v Ė := ∇ λ,x v 2 L 2 (R n+2 + ) + H t D 1/2 t v 2 L 2 (R n+2 + ) 1/2 < ∞.
Consequently, these are called energy solutions. When considered modulo constants, Ė is a Hilbert space and it is in fact the closure of C ∞ 0 R n+2 + for the homogeneous norm • Ė. Any v ∈ Ė can be defined as a uniformly continuous function on [0, ∞) with values in the homogeneous parabolic Sobolev space Ḣ1/4 ∂t-∆x . Here, Ḣs ±∂t-∆x is defined as the closure of Schwartz functions v ∈ S(R n+1 ) with Fourier support away from the origin for the norm F -1 ((|ξ| 2 ± iτ ) s Fv) 2 . This yields a space of tempered distributions if s < 0 and of tempered distributions modulo constants in L 2 loc (R n+1 ) if 0 < s ≤ 1/2. Of course the choice of the sign in front of iτ defines the same space up to equivalent norms. We implicitly assume the branch cut for z s on (-∞, 0] but any other possible choice yields the same space with an isometric norm. Conversely, any g ∈ Ḣ1/4 ∂t-∆x can be extended to a function v ∈ Ė with trace v| λ=0 = g. We will include proofs of these statements in Section 3. [START_REF] Alfonseca | Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L ∞ coefficients[END_REF].

By an energy solution to (1.1) with Neumann boundary data ∂ ν A u| λ=0 = f we mean a function u ∈ Ė such that for all v ∈ Ė,

R n+2 + A∇ λ,x u • ∇ λ,x v + H t D 1/2 t u • D 1/2 t v dλ dx dt = -f, v| λ=0 , (2.2)
where • , • denotes the pairing of Ḣ1/4 ∂t-∆x with its dual Ḣ-1/4 -∂t-∆x extending the inner product on L 2 (R n+1 ) and ∂ ν A u(λ, x, t) := [1, 0, . . . , 0] • (A∇ λ,x u)(λ, x, t) (2.3) is the conormal derivative, inwardly oriented on the upper half-space. In particular, u is a reinforced weak solution to (1.1) and the Neumann boundary data f ∈ Ḣ-1/4 ∂t-∆x is attained in the weak sense prescribed by (2.2). Similarly, by an energy solution to (1.1) with Dirichlet boundary datum u| λ=0 = f ∈ Ḣ1/4 ∂t-∆x we mean a function u ∈ Ė such that for all v ∈ Ė0 , the subspace of Ė with zero boundary trace,

R n+2 + A∇ λ,x u • ∇ λ,x v + H t D 1/2 t u • D 1/2 t v dλ dx dt = 0.
The key to solving these problems is the introduction of the modified sesquilinear form

a δ (u, v) := R n+2 + A∇ λ,x u • ∇ λ,x (1 -δH t )v + H t D 1/2 t u • D 1/2 t (1 -δH t )v dλ dx dt,
where δ is a real number yet to be chosen. The Hilbert transform H t is a skew-symmetric isometric operator with inverse -H t on both Ė and Ḣ1/4 ∂t-∆x . Hence, 1 -δH t is invertible on these spaces for any δ ∈ R. The upshot is that if we fix δ > 0 small enough, then a δ is coercive on Ė since (2.4) Re a δ (u, u)

≥ (κ -Cδ) ∇ λ,x u 2 2 + δ H t D 1/2 t u 2 2
where κ, C are the constants in (2.1). Hence, solving the Neumann problem with data f means finding u ∈ Ė such that

a δ (u, v) = -f, (1 -δH t )v| λ=0 (v ∈ Ė)
and the Lax-Milgram lemma applied to a δ on Ė yields a unique such u. We remark that this exactly means (2.2) upon replacing (1 -δH t )v by v, so that ∂ ν A u| λ=0 = f holds in the respective sense. For the Dirichlet problem we take an extension w ∈ Ė of the data f and apply the Lax-Milgram lemma to a δ on Ė0 to obtain some u ∈ Ė0 such that a δ (u, v) = -a δ (w, v) (v ∈ Ė0 ).

Hence, u + w is an energy solution with data f . Would there exist another solution v, then a δ (u + w -v, u + w -v) = 0 and hence u + w -v Ė = 0 by coercivity. We shall rephrase these observations by saying the the Dirichlet and Neumann problems associated with (1.1) are well-posed for the energy class. In the case of λ-independent coefficients we will rediscover the energy solutions in Section 2.8 within a much broader context.

2.4.

The correspondence to a first order system. Given a reinforced weak solution u as in Section 2.2, we create a vector with n + 2 components, which we call the (parabolic) conormal differential of u,

D A u(λ, x, t) :=    ∂ ν A u(λ, x, t) ∇ x u(λ, x, t) H t D 1/2 t u(λ, x, t)    ,
where the inwardly oriented conormal derivative was defined in (2.3). Note that the last component of D A u contains the half-order time derivative of u, which exists by assumption. Whether or not we choose to include the Hilbert transform in the definition of D A u is not important at this stage: it only affects the algebra of the representation but not its analysis. Note also that in a pointwise fashion,

|D A u| 2 ∼ |∇ λ,x u| 2 + |H t D 1/2 t u| 2 .
Throughout the paper we will represent vectors ξ ∈ C n+2 as

ξ =   ξ ⊥ ξ ξ θ   ,
where the normal (or perpendicular) part ξ ⊥ is scalar valued, the tangential part ξ is valued in C n and the time part ξ θ is again scalar valued.

If u is a reinforced weak solution to (1.1), then the vector valued function F := D A u belongs to the space L 2 loc (R; L 2 loc (R n+1 + ; C n+1 )) × L 2 (R; L 2 loc (R n+1 + ; C)), the global square integrability for the θ-component being with respect to the t variable, and it has the additional structure (2.5) curl x F = 0,

∇ x F θ = H t D 1/2
t F , when computed in the sense of distributions on R n+1 + × R. In fact, the first equation is contained in the second one.

To eventually address boundary value problems, we will assume global square integrability with respect to (x, t) for all components of F . Hence, with the help of Fubini's theorem we put D A u in the space L 2 loc (R + ; L 2 (R n+1 ; C n+2 )), where the local square integrability now is with respect to the λ variable. Throughout the paper we let L 2 be L 2 (R n+1 ; C n+2 ) and H loc be the subspace of L 2 loc (R + ; L 2 ) defined by the compatibility conditions (2.5).

Next, we split the coefficient matrix A as A(λ, x, t) = A ⊥⊥ (λ, x, t) A ⊥ (λ, x, t) A ⊥ (λ, x, t) A (λ, x, t) (2.6) and we recall from [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF] that there is a purely algebraic transformation on the bounded, uniformly elliptic matrix functions that will eventually allow us to write (1.1) as system for the unknown D A u. Proposition 2.1 [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF]Prop. 4.1,p. 68]). The pointwise transformation

A → Â := 1 0 A ⊥ A A ⊥⊥ A ⊥ 0 1 -1 = A -1 ⊥⊥ -A -1 ⊥⊥ A ⊥ A ⊥ A -1 ⊥⊥ A -A ⊥ A -1
⊥⊥ A ⊥ is a self-inverse bijective transformation on the set of bounded matrices which are uniformly elliptic.

We introduce the operators (2.7)

P :=    0 div x -D 1/2 t -∇ x 0 0 -H t D 1/2 t 0 0    , M :=    Â⊥⊥ Â⊥ 0 Â ⊥ Â 0 0 0 1    .
For each fixed λ > 0, the operator M is a multiplication operator on L 2 . The operator P is independent of λ and defined as an unbounded operator in L 2 with maximal domain. The adjoint of P is

P * =    0 div x H t D 1/2 t -∇ x 0 0 -D 1/2 t 0 0    . (2.8)
In Section 5 we will prove the following theorem, establishing the connection between reinforced weak solutions u to (1.1) and a first order differential equation. Theorem 2.2. If u is a reinforced weak solution u to (1.1) and F := D A u ∈ H loc , then (2.9)

R n+2 + F • ∂ λ φ dλ dx dt = R n+2 + M F • P * φ dλ dx dt for all φ ∈ C ∞ 0 (R n+2 + ; C n+2 ).
Conversely, to any F ∈ H loc satisfying (2.9) for all φ, there exists a reinforced weak solution u to (1.1), unique up to a constant, such that F = D A u.

In other words, up to additive constants one can construct all reinforced weak solutions u to (1.1) with the property D A u ∈ L 2 loc (R + ; L 2 (R n+1 ; C n+2 )) by solving the differential equation (2.10)

∂ λ F + P M F = 0 in the distributional sense (2.9) in the space H loc . The equation (2.10) depends on the operator P M , hence on the choice of A to represent L. One can show that any choice of A leads to the same conclusions as far as Dirichlet problems are concerned: solvability and uniqueness, respectively, hold for all choices or none. On the other hand, Neumann problems depend on the choice of A in the formulation and in the conclusion. It could well be that one Neumann problem is solvable or well-posed but not all of them. 2.5. The parabolic Dirac operator for transversally independent equations. The first case to study is that of equations with coefficients independent of the transverse variable λ. In this case also the entries of M are independent of λ. Thus, the differential equation (2.10) becomes autonomous and can be solved via semigroup techniques, provided the semigroup is well-defined. This requires that P M has a bounded holomorphic functional calculus.

Let us quickly recall that an operator T in a Hilbert space is bisectorial of angle ω ∈ (0, π/2) if its spectrum is contained in the closure of the open double sector S ω := {z ∈ C : | arg z| < ω or | arg z -π| < ω} and if for each µ ∈ (ω, π/2) the map z → z(z -T ) -1 is uniformly bounded on C\S µ . Throughout this paper we require basic knowledge of such operators and their functional calculus allowing to define b(T ) for suitable holomorphic functions b on S µ . A reader without a background in this field will find all the necessary results and further information in various comprehensive treatments including [START_REF] Egert | On Kato's conjecture and mixed boundary conditions[END_REF][START_REF] Haase | The Functional Calculus for Sectorial Operators[END_REF][START_REF] Mcintosh | Operators which have an H ∞ functional calculus[END_REF].

As can be seen from (2.7) and (2.8), the operator P contains the non-local operators D

1/2 t and H t D 1/2 t
and it is not self-adjoint. These are the two major differences compared to the corresponding construction of the Dirac operator for elliptic equations. Note that P arises from the heat equation ∂ t u -∆ x u = 0. The non-selfadjointness of P is reflected in the fact that the adjoint equation is not the heat equation itself but the backward heat equation -∂ t u -∆ x u = 0. Hence, we (unfortunately) cannot just quote results from the literature devoted to elliptic equations without investigating the arguments. Nevertheless, we (fortunately) can prove the similar results yielding a harvest of consequences from abstract reasoning regardless of the precise definition of P and M .

We are ready to state a fundamental result of the paper:

Theorem 2.3. The operator P M is a bisectorial operator on L 2 with range R(P M ) = R(P ). It satisfies the quadratic estimate

∞ 0 λP M (1 +λ 2 P M P M ) -1 h 2 2 dλ λ ∼ h 2 2 (h ∈ R(P M )).
The angle ω of bisectoriality and implicit constants in the quadratic estimate depend upon n and the ellipticity constants of A. In particular, P M has a bounded holomorphic functional calculus on R(P M ) = R(P ) on open double sectors S µ for all µ ∈ (ω, π/2). The same holds true for M P on R(M P ) = M R(P ).

The proof is given in Sections 4 and 7. We emphasize that the entries of M are allowed to depend on x and t in a merely measurable fashion. As a consequence of bisectoriality there are topological splittings (2.11)

L 2 = R(P ) ⊕ N(P M ) = M R(P ) ⊕ N(P ).
Existence of a bounded holomorphic functional calculus for P M on R(P ) on sectors S µ in our case means that for any bounded holomorphic function b : S µ → C the functional calculus operator b(P M ) on R(P ) is bounded by b(P M ) R(P )→R(P ) b L ∞ (Sµ) with an implicit constant depending upon µ, n and the ellipticity constants of A. Due to the seminal result of McIntosh [START_REF] Mcintosh | Operators which have an H ∞ functional calculus[END_REF] this property is in fact equivalent to the quadratic estimate stated above. If b is unambiguously defined at the origin, then b(P M ) extends to L 2 by b(0) on N(P M ). This general theory also applies to M P on the closure of its range.

Based on Theorem 2.3 and the observation that H loc identifies with L 2 loc (R + ; R(P )), see Lemma 4.1 below, we can construct solutions to the differential equation (2.10), using functional calculus.

We define the characteristic functions χ + (z) and χ -(z) for the right and left open half planes in C and the exponential functions e -λ [z] , λ > 0. Here [z] := z sgn(z) and sgn(z) := χ + (z) -χ -(z). This gives the generalized spectral projections χ ± (P M ), which are bounded on R(P ) by Theorem 2.3, and the holomorphic semigroup (e -λ[P M ] ) λ>0 generated by [P M ] = P M sgn(P M ). The boundedness of the spectral projections yields a topological splitting into spectral spaces H ± (P M ) := χ ± (P M )R(P ), which can be called generalized Hardy spaces, (2.12)

R(P ) = H + (P M ) ⊕ H -(P M ).
It is convenient to introduce the following generalized Cauchy extension: for h ∈ R(P ) and λ ∈ R\{0},

(2.13) (C 0 h) λ = (C 0 h)(λ, •) := (C + 0 h)(λ, •) = e -λP M χ + (P M )h (if λ > 0), (C - 0 h)(λ, •) = e -λP M χ -(P M )h (if λ < 0). Let us also define the semigroup extension of h for λ > 0 by (Sh) λ = (Sh)(λ, •) := e -λ[P M ] h. (2.14) Hence, (C + 0 h) λ = (Sχ + (P M )h) λ (if λ > 0), (C - 0 h) λ = (Sχ -(P M )h) -λ (if λ < 0)
. The above also applies to M P , which has a bounded holomorphic functional calculus on the closure of its range, too. However, unlike for P M , the range of M P is not independent of M .

Proposition 2.4. The generalized Cauchy extension

F = C + 0 h of h ∈ R(P ) gives a solution to ∂ λ F + P M F = 0 in the strong sense F ∈ C([0, ∞); R(P )) ∩ C ∞ ((0, ∞); D(P M )) with bounds sup λ>0 F λ 2 ∼ χ + (P M )h 2 ∼ sup λ>0 - 2λ λ F µ 2 2 dµ and L 2 limits lim λ→0 F λ = χ + (P M )h, lim λ→∞ F λ = 0.
In addition, the square function estimate

∞ 0 λ∂ λ F 2 2 dλ λ ∼ χ + (P M )h 2 2 (2.15)
holds. Furthermore, F is a solution to ∂ λ F + P M F = 0 also in the weak sense (2.9).

If h belongs to the spectral space H + (P M ), then h = χ + (P M )h is the initial value of its generalized Cauchy extension C + 0 h. The generalized Cauchy extension C - 0 h gives a solution of the same equation with the analogous estimates for λ ∈ R -. Moreover, the counterpart of Theorem 2.2 for the lower half-space (with identical proof) provides the correspondence with parabolic conormal differentials of reinforced weak solutions to (1.1) on R n+2 -. All statements in Proposition 2.4 but the last one are well-known properties in semigroup theory and they apply to any bisectorial operator with a bounded holomorphic functional calculus [START_REF] Haase | The Functional Calculus for Sectorial Operators[END_REF][START_REF] Mcintosh | Operators which have an H ∞ functional calculus[END_REF]. Note that in (2.15) we have ∂ λ F = φ(λP M )h with holomorphic function φ(z) = ze -[z] χ + (z), and thus this square function estimate follows from the analogous one with φ(z) = z(1 + z 2 ) -1 provided by Theorem 2.3. The verification of the equation in the weak sense (2.9) is special to the operator P M and follows by a simple integration by parts.

The converse to Proposition 2.4 is a verbatim modification of the argument originally designed for elliptic systems [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF]Thm. 8.2]. In fact, this relies on an abstract approach that works for any bisectorial operator with a bounded holomorphic functional calculus, see [START_REF] Auscher | Remarks on maximal regularity[END_REF]. Hence, we can directly state the following result. Theorem 2.5. Let F ∈ L 2 loc (R + ; R(P )) be a solution of (2.10) in the weak sense such that

(2.16) sup λ>0 - 2λ λ F µ 2 2 dµ < ∞.
Then F has a L 2 limit h ∈ H + (P M ) at λ = 0 and F is given by the Cauchy extension of h. The analogous result for weak solutions F ∈ L 2 loc (R -; R(P )) of (2.10) also holds upon using the spectral space H -(P M ).

Finally, we mention that the whole functional calculus for P M becomes analytic in A equipped with the L ∞ -norm (or equivalently, in M for the same norm). In particular, this provides us with Lipschitz estimates for the operator norm on R(P ): for all b bounded and holomorphic in a sector

S µ with ω < µ < π/2, b(P M ) -b(P M ′ ) R(P )→R(P ) b ∞ M -M ′ ∞ , (2.17)
where M ′ is associated with an elliptic matrix A ′ in the same class as A and M -M ′ ∞ is small. The implicit constant depends on n, µ and the ellipticity constants of A. This is again an abstract property of the functional calculus for operators of type P M once it is known that implicit constants in the quadratic estimate of Theorem 2.3 depend on the coefficients A only through the ellipticity constants. Details are written out for example in [8, p. 269] or [START_REF] Egert | On Kato's conjecture and mixed boundary conditions[END_REF]Prop. 6.1.25].

2.6. Kato square root estimate. As another important consequence of Theorem 2.3 we obtain the resolution of the Kato problem for parabolic operators in full generality.

In order to set the context, we note that similar to Section 2.3 the use of the energy space

V := H 1/2 (R; L 2 (R n )) ∩ L 2 (R; W 1,2 (R n )) allows us to define the parabolic operator L := ∂ t - div x A (x, t)∇ x as an operator V → V * via a sesquilinear form, Lu, v := R n+1 A ∇ x u • ∇ x v + H t D 1/2 t u • D 1/2 t v dx dt (u, v ∈ V).
Carving out the surprising analogy with elliptic operators even further, [START_REF] Auscher | On non-autonomous maximal regularity for elliptic operators in divergence form[END_REF]Lem. 4] shows that L with maximal domain

D(L) = {u ∈ V : Lu ∈ L 2 (R n+1 )} in L 2 (R n+1
) is maximal accretive: That is to say L is closed, Re Lu, u ≥ 0 holds for all u ∈ D(L), and 1 + L is onto. We refer to [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] for background on this class of operators. Note that all this is for one dimension lower than in Section 2.3 and that we are using an inhomogeneous energy space since we are studying L as an operator in L 2 (R n+1 ). The reader will notice that the following result reads almost identically to the famous Kato problem for elliptic operators solved in [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF]. We shall give its proof in Section 12.1. We emphasize that no assumptions on A = A (x, t) besides measurability and uniform ellipticity have been imposed.

Theorem 2.6. The operator L = ∂ t -div x A (x, t)∇ x arises from an accretive form, it is maximalaccretive in L 2 (R n+1 ), the domain of its square root is that of the accretive form, that is,

D( √ L) = V = H 1/2 (R; L 2 (R n )) ∩ L 2 (R; W 1,2 (R n )). The two-sided estimate √ L u 2 ∼ ∇ x u 2 + D 1/2 t u 2 (u ∈ V)
holds with implicit constants depending only upon n and ellipticity constants of A .

The reader should remark that not even the case A * = A can be treated by abstract functional analysis (while this is the case for the elliptic Kato problem) because L is never self-adjoint. 2.7. Sobolev spaces. The previously obtained characterisation of solutions to the differential equation in (2.10) extends to other topologies.

For s ∈ R we can define the fractional powers [P M ] s as closed and injective operators in R(P M ) = R(P ), using the functional calculus for P M . The homogeneous Sobolev space Ḣs P M based on P M is the completion of D([P M ] s ) ∩ R(P M ) under the norm [P M ] s • 2 . Another option taken in [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF] is to use the fractional powers of (P M ) s with branch cut on i(-∞, 0], which gives the same spaces with isometric norms. This yields an abstract scale of Hilbert spaces that can all be realised within the same linear Hausdorff space and the intersection of any two of them is dense in both of them.

Of course, Ḣ0

P M = R(P ). As a matter of fact, the bounded holomorphic functional calculus extends to Ḣs P M , yielding in particular the spectral spaces Ḣs,± P M , and P M extends to an isomorphism from Ḣs P M onto Ḣs-1 P M . The same abstract construction can be done with M P starting from the base space R(M P ). Further background on these operator-adapted Sobolev spaces is provided in [START_REF] Auscher | Holomorphic functional calculi of operators, quadratic estimates and interpolation[END_REF] or [START_REF] Haase | The Functional Calculus for Sectorial Operators[END_REF]Ch. 6].

In particular, this construction applies to P and in this case we re-obtain concrete homogeneous parabolic Sobolev spaces as defined in Section 2.3. To this end, we note that if h ∈ D(P 2 ) ∩ R(P ), then

P 2 h =    ∂ t -∆ x 0 0 0 -∇ x div x ∇ x D 1/2 t 0 -H t D 1/2 t div x ∂ t    h =    ∂ t -∆ x 0 0 0 ∂ t -∆ x 0 0 0 ∂ t -∆ x    h. (2.18)
Taking the Fourier transform, we easily see that Ḣs P is a closed subspace of ( Ḣs/2 ∂t-∆x ) n+2 and that their norms are equal. Moreover, (2.18) reveals the bounded projection π P onto R(P ) along N(P ), which is well-defined in virtue of the decomposition (2.11) for M = 1, as the smooth parabolic singular integral of convolution π P = P 2 (∂ t -∆ x ) -1 . Thus, Ḣs P is precisely the image of ( Ḣs/2 ∂t-∆x ) n+2 under the bounded extension of π P .

The following lemma allows one to work more concretely with the Sobolev spaces associated with P M . It is a consequence of the bounded holomorphic functional calculus for P M (and P ) and the fact that R(P M ) = R(P ) holds, see [START_REF] Auscher | On L 2 solvability of BVPs for elliptic systems[END_REF]Prop. 4.5] With this notation set up, we can give the alluded a priori characterisations of weak solutions to (2.10) with data in parabolic homogeneous Sobolev spaces. Note that all appearing abstract Sobolev spaces identify with concrete parabolic Sobolev spaces only in the range of the above lemma.

Theorem 2.8. Let -1 ≤ s < 0. The generalized Cauchy extension F = C + 0 h of h ∈ Ḣs P gives a solution to ∂ λ F + P M F = 0, in the strong sense F ∈ C([0, ∞); Ḣs P ) ∩ C ∞ ((0,
F λ = χ + (P M )h, lim λ→∞ F λ = 0.
In addition, the square function estimate

∞ 0 λ -s F 2 2 dλ λ ∼ χ + (P M )h 2 Ḣs P (2.19)
holds. Furthermore, let F ∈ L 2 loc (R + ; R(P )) be a solution of (2.10) in the weak sense (2.9) such that

∞ 0 λ -s F 2 2 dλ λ < ∞.
Then F has a Ḣs P limit h ∈ Ḣs,+ P M at λ = 0 and F is given by the Cauchy extension of h. The analogous results for the same equation on R -also hold upon replacing the positive spectral spaces and projections with their negative counterparts.

As in the case of Proposition 2.4 and Theorem 2.5, also Theorem 2.8 was first obtained in the context of elliptic equations but by an abstract argument which literally applies to any bisectorial operator with bounded holomorphic functional calculus and to P M in particular. References in the context of elliptic equations are [5, Thm. 9.2] for s = -1 and [55, Thm. 1.3] for s ∈ (-1, 0). These results have been put into a much broader context in [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF]Sec. 11]. In particular, in their proof it is not important whether or not P is self-adjoint as long as P itself is bisectorial with bounded holomorphic functional calculus. However, expressions involving B * D in these references should be replaced with M * P * in our context. 2.8. Three special cases. In order to make things more explicit, we single out three special cases of Theorems 2.5 and 2.8. Throughout, we let u be a reinforced weak solution to (1.1) and we write F = D A u, where we recall

|D A u| 2 ∼ |∇ λ,x u| 2 + |H t D 1/2 t u| 2 .
Let us begin by revisiting the case s = -1/2, which already appeared in our discussion of energy solutions in Section 2.3. Indeed, in this case (2.19) becomes

R n+2 + |∇ λ,x u| 2 + |H t D 1/2 t u| 2 dλ dx dt ∼ h 2 Ḣ-1/2 P ∼ ∂ ν A u| λ=0 2 Ḣ-1/4 ∂ t -∆x + u| λ=0 2 Ḣ1/4 ∂ t -∆x , (2.20)
which is the energy estimate for reinforced weak solutions, and finiteness of the left-hand side exactly means that u belongs to the energy class Ė introduced in Section 2.3. The a priori information that

F ∈ C 0 ([0, ∞); Ḣ-1/2 P ) ∩ C ∞ ((0, ∞); Ḣ-1/2 P ∩ Ḣ1/2 P M
) satisfies ∂ λ F + P M F = 0 in the strong sense, allows to integrate by parts in λ for every v ∈ Ė, so to obtain

R n+2 + A∇ λ,x u • ∇ λ,x v + H t D 1/2 t u • D 1/2 t v dλ dx dt = -∂ ν A u| λ=0 , v| λ=0 . (2.21)
Here, the trace of ∂ ν A u is defined in Ḣ-1/4 -∂t-∆x by continuity of F = D A u and • , • denotes the duality pairing with Ḣ1/4 ∂t-∆x extending the inner product on L 2 (R n+1 ). Strictly speaking, we would carry out this integration for v smooth and compactly supported in all variables first and then extend by density. A comparison with (2.2) reveals that this understanding of the Neumann boundary data is the same as in the abstract variational setup but the restriction to transversally independent coefficients also allows for a more concrete understanding in the sense of a L 2 limit. In particular, we may plug in v = (1 -δH t )u with δ > 0 small enough as in Section 2.3 to conclude by coercivity

R n+2 + |∇ λ,x u| 2 + |H t D 1/2 t u| 2 dλ dx dt ∂ ν A u| λ=0 Ḣ-1/4 ∂ t -∆x u| λ=0 Ḣ1/4 ∂ t -∆x . (2.22)
Hence, we obtain the boundedness and invertibility of the Dirichlet to Neumann map at the boundary expressed in the comparability

∂ ν A u| λ=0 Ḣ-1/4 ∂ t -∆x ∼ u| λ=0 Ḣ1/4 ∂ t -∆x ∼ h Ḣ-1/2 P . (2.23)
We shall come back to this point later on in Section 2. [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF].

When s = 0, (2.16) rewrites

sup λ>0 - 2λ λ R n+1 |∇ λ,x u| 2 + |H t D 1/2 t u| 2 dx dt dµ ∼ h 2 2 ∼ ∂ ν A u| λ=0 2 2 + u| λ=0 2 Ḣ1/2 ∂ t -∆x
and, as we will see in Section 2.10, the left hand side is also comparable to a non-tangential maximal norm on D A u. The comparability

∂ ν A u| λ=0 2 ∼ u| λ=0 Ḣ1/2 ∂ t -∆x ∼ ∇ x u| λ=0 2 + H t D 1/2 t u| λ=0 2
for this class of u would be the Rellich estimates for proving solvability of regularity problems and Neumann problems with

L 2 data. When s = -1, (2.19) becomes ∞ o R n+1 |λ∇ λ,x u| 2 + |λH t D 1/2 t u| 2 dx dt dλ λ ∼ h 2 Ḣ-1 P ∼ ∂ ν A u| λ=0 2 Ḣ-1/2 ∂ t -∆x + u| λ=0 2 2 .
In this case the inequality Theorem 2.9. Let u be a reinforced weak solution of (1.1) with

∂ ν A u| λ=0 Ḣ-1/2
∞ 0 R n+1 |λ∇ λ,x u| 2 + |λH t D 1/2 t u| 2 dx dt dλ λ < ∞.
Then there exist unique c ∈ C and h ∈ H

+ (M P ) such that u = c -(e -λ[M P ] h) ⊥ . In particular, u ∈ C 0 ([0, ∞); L 2 (R n+1 )) + C
, where the subscripted 0 means that u vanishes at ∞. Moreover, P h is the element h ∈ Ḣ-1,+ P M appearing in Theorem 2.8. 2.9. Resolvent estimates. An important ingredient in the proof of Theorem 2.3 are off-diagonal estimates for the resolvents of P M . For anyone who knows how these estimates are obtained in the elliptic case, the non-locality of half-order time derivatives may seem as a serious obstacle. A key observation is the following. Lemma 2.10. There exists δ 0 > 0, depending only on dimension and the ellipticity constants of the matrix A, such that for any real p, q with | 1 p -1 2 | < δ 0 and | 1 q -1 2 | < δ 0 , and any λ ∈ R, the resolvent

(1 +iλP M ) -1 is bounded on L p (R; L q (R n ; C n+2 ))
with uniform bounds with respect to λ. The same result holds with M P , P * M or M P * in place of P M . This result will be proved in Section 6 and it allows us to obtain enough decay for our arguments. We prove the following off-diagonal estimates.

Proposition 2.11. There exists

ε 0 > 0 and N 0 > 1 such that if | 1 q -1 2 | < ε 0 , then one can find ε = ε(n, q, ε 0 ) > 0 with the following property. Whenever N ≥ N 0 , then there exists a constant C = C(ε, N, q) < ∞ such that for all balls Q = B(x, r) in R n , all intervals I = (t -r 2 , t + r 2 )
, and all parameters j ∈ N, k ∈ N * and λ ∈ R with |λ| ∼ r:

--

Q×4 j I |(1 +iλP M ) -1 h| q dy ds ≤ CN -qεk -- C k (Q×4 j I)
|h| q dy ds, (2.24)

provided h ∈ (L 2 ∩ L q )(R n+1 ; C n+2 ) has support in C k (Q × 4 j I). Here, C k (Q × J) := (2 k+1 Q × N k+1 J) \ (2 k Q × N k J)
and cQ and cJ denote dilates of balls and intervals, respectively, keeping the center fixed and dilating the radius by c. Analogous estimates hold with P M replaced by M P , P * M or M P * .

Some remarks explaining the nature of this result are necessary. First, Q×I is a standard parabolic cube in R n × R. Some estimates will require that we stretch I leaving Q fixed. Hence the use of 4 j I. Next, since |λ| is on the order of the radius of the parabolic cube there is no power of r/|λ| involved nor is there dependence on j. Furthermore, in the definition of C k (Q × 4 j I) we take the usual double stretching in the x-direction, but we stretch by a factor N in the t-direction. In principle N will be chosen larger than any power of 2 n that we may need to match the decay in the x-direction. In the end, we still have a small power of N -k in front of the integral on the right hand side but note that we have normalized by taking averages (materialized by the dashed integrals) and observe that the Lebesgue measure of C k (Q × 4 j I) is on the order of 2 kn N k times that of Q × 4 j I. Hence, this is much more decay than it looks like.

We do not know whether the resolvent has the classical off-diagonal decay as an operator acting on a space of homogeneous type (here, R n+1 with parabolic distance and Lebesgue measure) in the sense of [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type[END_REF]. So, it is kind of a novelty in this topic that a weaker form of off-diagonal estimates apply, using that our space has identified directions, which is not the case in general. In contrast, all decay estimates obtained in [START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF] for operators with t-independent coefficients have the parabolic homogeneity.

2.10. Non-tangential maximal function estimates. The off-diagonal estimates in Proposition 2.11 are strong enough to prove non-tangential maximal function estimates in Section 8. For (x, t) ∈ R n+1 we define the non-tangential maximal function

N * F (x, t) = sup λ>0 --- Λ×Q×I |F (µ, y, s)| 2 dµ dy ds 1/2
, where Λ = (c 0 λ, c 1 λ), Q = B(x, c 2 λ) and I = (t -c 3 λ 2 , t + c 3 λ 2 ) with positive constants c i and c 0 < c 1 . The numerical values of the constants are irrelevant since by a covering argument any change gives equivalent norms. Theorem 2.12. Let h ∈ R(P M ) and let F = Sh be its semigroup extension as in (2.14). Then

N * F 2 ∼ h 2 ,
where the implicit constants depend only on dimension and the ellipticity constants of A. Furthermore, there is almost everywhere convergence of Whitney averages

lim λ→0 --- Λ×Q×I |F (µ, y, s) -h(x, t)| 2 dµ dy ds = 0 for almost every (x, t) ∈ R n+1 .
Recall that if h ∈ H + (P M ), then the semigroup extension is the same as the Cauchy extension of h, so that this theorem provides us with estimates for reinforced weak solutions to (1.1). As the estimate

- c 1 λ c 0 λ F µ 2 2 dµ N * F 2 2
holds for arbitrary functions F , see Lemma 8.11 below for convenience, the non-tangential maximal estimate can be seen as a further regularity estimate for solutions in the uniqueness class of Theorem 2.5. Of course, the class of solutions to (2.10) with N * F 2 < ∞ is also a uniqueness class by restriction. In addition, the almost everywhere convergence of Whitney averages allows us to give a pointwise meaning to the boundary trace of such solutions. A similar remark applies to data in H -(P M ) for solutions with λ ∈ R -.

There is also a companion result that applies to Dirichlet problems with L 2 data.

Theorem 2.13. Let h ∈ R(M P ) and let F (λ, •) = e -λ[M P ] h be its semigroup extension with respect to M P . Then

N * F 2 ∼ h 2 ,
where the implicit constants depend only on dimension and the ellipticity constants of A. Furthermore, there is almost everywhere convergence of Whitney averages,

lim λ→0 --- Λ×Q×I | F (µ, y, s) -h(x, t)| 2 dµ dy ds = 0 for almost every (x, t) ∈ R n+1 .
Indeed, we recall from Theorem 2.9 that for F as above and h ∈ H + (M P ) the perpendicular part F ⊥ is a solution of the Dirichlet problem with data h ⊥ .

The non-tangential maximal estimates in Theorem 2.12 and Theorem 2.13 depend on apparently new reverse Hölder estimates for reinforced weak solutions of (1.1) that we also prove in Section 8. These estimates are valid in full generality for λ-dependent equations and also for solutions in the lower half-space up to the obvious changes of notation.

Theorem 2.14.

There is a constant C, depending only on the ellipticity constants of A and the dimension n, such that any reinforced weak solution of (1.1) satisfies the reverse Hölder estimate

--- Λ×Q×I |∇ λ,x u| 2 + |H t D 1/2 t u| 2 + |D 1/2 t u| 2 dµ dy ds 1/2 ≤ C k∈Z 1 1 + |k| 3/2 --- 8Λ×8Q×I k |∇ λ,x u| + |H t D 1/2 t u| + |D 1/2
t u| dµ dy ds.

(2.25)

Here, Λ = (λ -r, λ + r), Q = B(x, r), I = (t -r 2 , t + r 2 ] define a parabolic cylinder of radius r < λ/8 and I k := kℓ(I) + I are the disjoint translates of I covering the real line.

Note that the estimates in (2.25) are non-local in the time variable, reflecting the non-locality of half-order time derivatives. The estimates involves both H t D 1/2 t u and D 1/2 t u, which is reminiscent of the arbitrary choice that we made when setting up the first order approach. In fact, the proof of Theorem 2.14 will suggest that control for one of these functions on the left requires both of them on the right. The factor of the parabolic enlargement of Λ and Q on the right-hand side and the corresponding relation between r and λ can be changed from 8 to any factor c > 1 at the expense of C then depending on c as well. Again, this is best done by a covering argument.

2.11. Characterisation of well-posedness of the BVPs (λ-independent). Let us eventually address boundary value problems for parabolic equations. We write Lu = 0 to mean that u is a reinforced weak solution of (1.1). For -1 ≤ s ≤ 0 we let

F Es := N * (F ) 2 (if s = 0), ( ∞ 0 λ -s F 2 2 dλ
λ ) 1/2 (otherwise) and define the solution classes

E s := {F ∈ L 2 loc (R n+2 + ; C n+2 ); F Es < ∞}. Given s ∈ [-1, 0]
, the regularity of the data, we consider the problems (R) L Es :

Lu = 0, D A u ∈ E s , u| λ=0 = f ∈ Ḣs/2+1/2 ∂t-∆x , (N ) L Es : Lu = 0, D A u ∈ E s , ∂ ν A u| λ=0 = f ∈ Ḣs/2 ∂t-∆x . The problem (R) L
Es is formulated as a Dirichlet problem with regularity s/2 + 1/2 on the data but due to

f Ḣs/2+1/2 ∂ t -∆x ∼ [∇ x f, H t D 1/2 t f ] ( Ḣs/2 ∂ t -∆x ) n+1
, it is equivalent to a regularity problem with (parabolic) regularity s on the quantities ∇ x u and H t D 1/2 t u at the boundary. We shall adopt this second point of view because these quantities naturally appear in D A u. On the other hand, (N ) L Es is a Neumann problem. Note that we require the full knowledge D A u ∈ E s , whereas in [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF] the Neumann problem for s = 0 is posed with the non-tangential maximal control of ∇ λ,x u only. However, therein additional information, such as DeGiorgi-Nash-Moser estimates and invertibility of layer potentials, are used to prove uniqueness under this weaker assumption.

Solving the problems (R) .

The implicit constant must be independent of the data. The behaviour at the boundary is the strong convergence specified by our results above. Existence means that this holds for all data in a chosen space. Uniqueness means that there is at most one solution. Well-posedness is the conjunction of both existence of a solution for all data and uniqueness.

It is instructive to write out the general theory for the three most prominent cases that we considered in Section 2.8: In the case s = 0, the boundary value problems are the regularity and Neumann problems with data in L 2 (R n+1 ). For the regularity problem, the data takes the form of an array [∇ x f, H t D 1/2 t f ] for some f ∈ Ḣ1/2 ∂t-∆x . Due to Theorem 2.12 the boundary behaviour can also be interpreted in the sense of almost everywhere convergence of Whitney averages. In the case s = -1, as said, (R) L E -1 is nothing but a Dirichlet problem with L 2 data f and thanks to Theorem 2.13 almost everywhere convergence of Whitney averages of u to its boundary data comes again as a bonus. In the case s = -1/2 we are back to the class of energy solutions, within which we have already obtained well-posedness in Section 2.3. In fact, we have seen in Section 2.8 that the class of energy solutions coincides with the class of reinforced weak solutions such that D A u ∈ E -1/2 .

We will say that the boundary value problem (BVP ) L Es , where BVP designates either N or R, is compatibly well-posed if it is well-posed and if the solution agrees with the energy solution of the problem (BVP ) L E -1/2 when the data belongs to both data spaces. In order to formulate characterisations of well-posedness and compatible well-posedness, we set some new notation. Recall that the trace h of a conormal differential contains three terms: the first one is called the scalar component and denoted by h ⊥ , and the two others are the tangential component and the time component, which we concatenate by h r = [h , h θ ]. Since the trace space of all conormal differentials of reinforced weak solutions has been identified as a generalized Hardy space in Theorems 2.5 and 2.8, the conclusion is that the two maps N ⊥ : h → h ⊥ and N r : h → h r carry the well-posedness. We record this easy but important observation in the following theorem.

Theorem 2.15. The following assertions hold for -1 ≤ s ≤ 0.

(i) (R) Let us recall that Theorems 2.5 and 2.8 also provide a priori representations for the conormal gradients of solutions to (1.1) on the lower half-space R n+2 -, leading to similar characterisations of well-posedness upon replacing positive with negative spectral spaces. Given these characterisations, (compatible) well-posedness extrapolates and compatible well-posedness interpolates. The full details for elliptic Dirac operators can be found in [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF]Thm. 7.8 & 7.7]. They remain unchanged in our setup.

Theorem 2.17. The set of s ∈ (-1, 0) for which (BVP ) L Es is well-posed (compatibly well-posed) is open.

Note that in general we cannot extrapolate from the endpoint cases s = 0 and s = -1.

Theorem 2.18. If (BVP ) L Es i is compatibly well-posed when -1 ≤ s 0 , s 1 ≤ 0, then (BVP ) L
Es is compatibly well-posed for any s between s 0 and s 1 .

We mention that all results are analytic with respect to L ∞ -perturbations of the coefficients A(x, t). Again this is an abstract result proved in [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF]Thm. 7.16], compare also with [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF]].

The assumption of this theorem is that a local Lipschitz estimate holds for the functional calculus of P M on Ḣs P . This one has been described in (2.17) when s = 0 and it extends to any s ∈ [-1, 0] using Lemma 2.7.

Theorem 2.19. Let s ∈ [-1, 0]. If (BVP ) L
Es is (compatibly) well-posed, then so is (BVP ) L ′ Es for any L ′ with coefficients A ′ such that A -A ′ ∞ is sufficiently small, depending on s and the ellipticity constants of A. Moreover, the operator norm of the inverse of the respective projection N • is Lipschitz continuous as a function of A ′ in a neighbourhood of A.

Finally, the Lipschitz estimates (2.17), extended to Ḣs P as explained above, allow us to use the method of continuity to perturb from any equation to any other within arcwise connected components. Usually, we perturb from the heat equation. We record this observation in the following result to be proved in Section 12.9. We say a bounded operator has lower bounds if T f f holds for all f .

L * v = -∂ t v -div X A * (X, t)∇ X v = 0, (2.26)
where A * is the (complex) adjoint of A. Note the sign change in front of the time derivative. This equation can also be rephrased as a first order system. However, in this case it will be convenient to use a different representation to simplify the presentation of the duality results.

We use the backward (parabolic) conormal differential

D A * v(λ, x, t) :=    ∂ ν A * v(λ, x, t) ∇ x v(λ, x, t) D 1/2 t v(λ, x, t)    ,
where the difference compared to D A * v is in the third component. With this choice of conormal differential, the analogue of Theorem 2.2 is that there is, up to constants, a correspondence between reinforced weak solutions v to (2.26) and weak solutions G ∈ H loc := L 2 loc (R + ; R(P * )) to the first order system (2.27)

∂ λ G + P * M G = 0.
Here M = N M * N , where M * is the complex adjoint of M in the standard duality of C n+2 and

(2.28)

N =    -1 0 0 0 1 0 0 0 1   .
Note that this is an operator in the same class as M . In the case of λ-independent coefficients, the identity P * M = (( M ) * P ) * shows that bisectoriality of P * M on L 2 and the bounded holomorphic functional calculus inherit from ( M ) * P . The same analysis as developed above applies to (2.27) The proof of these two results will be given in Section 9.

2.13. The role of sgn(P M ). In Section 2.11 we have characterised well-posedness separately for each BVP on each parabolic half-space by means of the spectral projections χ ± (P M ) and the coordinate projections N ⊥ , N r . In contrast, most classical methods to solving BVPs, such as the method of layer potentials [START_REF] Castro | Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients[END_REF][START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF][START_REF] Verchota | Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains[END_REF] (iv) In particular, well-posedness of (R) L Es and (N ) L Es on both half-spaces is equivalent to simultaneous invertibility of the six operators 1 ± s ⊥⊥ (P M ), s ⊥r (P M ), s r⊥ (P M ), 1 ± s rr (P M ) acting between the respective components of Ḣs P . The experienced reader in parabolic boundary value problems may wonder about the relation between the spectral projections and the so-called Calderón projections. We shall come back to this in Section 2.15.

The proof of Theorem 2.23 will be provided in Section 10. Let us stress that this theorem cannot be improved by purely algebraic reasoning on complementary projections.

First, we could have invertibility of solely s r⊥ (P M ) and failure of invertibility of one 1± s ⊥⊥ (P M ) as well as one of 1 ± s rr (P M ). Indeed, using a representation as in (2.31) for general pairs of complementary projections χ ± , we already see from the simple example in R 2 ,

χ + := 1 0 1 0 , χ -:= 0 0 -1 1 ,
that s r⊥ can be invertible, while 1 -s ⊥⊥ and 1 + s rr are not. In the opposite direction, let ℓ 2 (N) be the space of complex square-summable sequences, let S : (a n ) → (0, a 1 , a

Γ L,+ N D = s ⊥r (P M ) -1 (1 -s ⊥⊥ (P M )) = (1 -s rr (P M )) -1 s r⊥ (P M ) (2.33) and Γ L,+ DN = (Γ L,+ N D ) -1 = s r⊥ (P M ) -1 (1 -s rr (P M )) = (1 -s ⊥⊥ (P M )) -1 s ⊥r (P M ). (2.34)
Similarly, akin operators for the lower half-space can be rediscovered within the functional calculus of P M . In Section 10 we shall use the operators Γ L,± N D , Γ L,± DN to prove the following. It is instructive to compare Proposition 2.25 with Theorem 2.23 and the succeeding remark. Indeed, the additional topological restriction that inverses should be compatible with the ones on energy solutions allows us to overcome the general algebraic obstructions to proving that invertibility of 1 ± s ⊥⊥ (P M ) (or 1 ± s rr (P M )) implies invertibility of s r⊥ (P M ) (or s ⊥r (P M ), respectively). 2.14. Layer potentials. We continue to carve out the link with classical theory of BVPs by introducing abstract layer potentials in the spirit of [START_REF] Rosén | Layer potentials beyond singular integral operators[END_REF], following the presentation in [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF]Section 12.3]. We also go further by constructing explicitly the inverse of ∂ t -div λ,x A(x, t)∇ λ,x on R n+2 from the layer potentials and without local DeGiorgi-Nash-Moser regularity assumptions.

In the following we designate by π M P the projection onto R(M P ) along N(M P ) = N(P ) in (2.11), noting that the notation π P is consistent with Section 2.5. We collect two further properties of the functional calculi that are required for defining the layer potentials rigorously. The first one follows since π M P and π P share the same nullspace and the second one comes from general functional calculus of these operators, see [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF]Sec. 11].

Lemma 2.26. The restrictions π M P : R(P ) → R(M P ) and π P : R(M P ) → R(P ) are isomorphisms and mutual inverses of one another. For every -1 ≤ s ≤ 0, they extend to isomorphisms π M P : Ḣs+1 P → Ḣs+1

M P and π P : Ḣs+1 M P → Ḣs+1 P , respectively. Moreover, the intertwining property Note that the holomorphic functions z → e -λz χ sgn λ (z) are bounded on Re z = 0 when λ = 0. Thus, Lemma 2.7 shows that if -1 ≤ s ≤ 0, then e -λP M χ sgn λ (P M ) is bounded on Ḣs P , while P is an isomorphism from Ḣs+1 P onto Ḣs P . We regroup again -and θ-components of vectors as a single component denoted r. For instance, for a scalar function f ∈ Ḣs/2 ∂t-∆x , we let h = [f, 0] be the vector function in Ḣs P with h ⊥ = f and h r = 0. Throughout, we assume -1 ≤ s ≤ 0. For λ ∈ R \ {0} and f ∈ Ḣs/2+1/2 ∂t-∆x , we define the single layer operator by (2.35)

P π P b(M P )π M P h = b(P M )P h is valid for all h ∈ Ḣs+1
S λ f :=            -P -1 e -λP M χ + (P M ) f 0 ⊥ (if λ > 0), + P -1 e -λP M χ -(P M ) f 0 ⊥ (if λ < 0)
and for f ∈ Ḣs/2 ∂t-∆x we define the double layer operator (2.36)

D λ f :=            -π P e -λM P χ + (M P )π M P f 0 ⊥ (if λ > 0), + π P e -λM P χ -(M P )π M P f 0 ⊥ (if λ < 0).
Note that these definitions differ from the ones in [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF] in that we have introduced some additional projectors in (2.36) to yield rigorous formulae for all Sobolev functions while usage of the projectors is not necessary for f in dense spaces, see [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF]Sec. 7.4] for details. To avoid further confusion, let us notice an unfortunate typo in [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF]: the four formulae (81) -( 85) come with e +t... instead of e -t. We can relate the conormal differentials of the layer operators to the Cauchy extension operators of (2.13). By a direct calculation from the intertwining property in Lemma 2.26,

D A S λ f =            +e -λP M χ + (P M ) f 0 (if λ > 0), -e -λP M χ -(P M ) f 0 (if λ < 0)
and

D A D λ f =            +e -λP M χ + (P M )P f 0 (if λ > 0), -e -λP M χ -(P M )P f 0 (if λ < 0). Note that P f 0 has zero ⊥-component and r-component equal to -∇ x f -H t D 1/2 t f
. In other words, (2.37)

D A S λ f = sgn(λ) C 0 f 0 λ , D A D λ f = sgn(λ) C 0 P f 0 λ , (λ = 0),
where, as we recall, C 0 is the generalized Cauchy extension operator. These expressions, together with the correspondence stated in Theorem 2.2 and the boundedness of the Cauchy extensions, show that D λ f and S λ f are reinforced weak solutions to (1.1). Moreover, the link with the functional calculus of P M provides us with existence of limits at λ = 0 ± including the jump relations and the duality relations between layer potentials.

Proposition 2.27. Let -1 ≤ s ≤ 0. Given f ∈ Ḣs/2 ∂t-∆x , the limits

D A S 0 + f = χ + (P M ) f 0 , D A S 0 -f = -χ -(P M ) f 0

exist in the strong topology of Ḣs P and hence the jump relation for the conormal derivatives and the no-jump relation for tangential derivatives are encoded as

D A S 0 + f -D A S 0 -f = f 0 .
Given f ∈ Ḣs/2+1/2 ∂t-∆x , the limits

D 0 + f = -π P χ + (M P )π M P f 0 ⊥ , D 0 -f = + π P χ -(M P )π M P f 0 ⊥
exist in the strong topology and hence

D 0 + f -D 0 -f = -f, D 0 + f + D 0 -f = -π P sgn(M P )π M P f 0 ⊥ =: 2Kf,
with K being the double layer potential at the boundary.

In addition, there are the duality formulae for

f ∈ Ḣs/2 ∂t-∆x and g ∈ Ḣ-s/2-1/2 -∂t-∆x given by g, S L λ f = S L * -λ g, f ,
and for f ∈ Ḣs/2+1/2 ∂t-∆x and g ∈ Ḣ-s/2-1/2 -∂t-∆x given by g, D L λ f = ∂ ν A * S L * -λ g, f .
Here, S L λ , D L λ are the layer potential operators defined above and associated with the parabolic equation (1.1), while S L * λ is the single layer operator associated with the backward parabolic equation (2.26) defined using the correspondence with the operator P * M of Section 2.12.

Due to the relation (2.37) between layer potentials and the Cauchy extension, there is always a representation by (abstract) layer potential operators,

D A u(λ, •) = D A S λ (∂ ν A u| λ=0 ) -D A D λ (u| λ=0 )
provided all terms are defined. More precisely, the integrated version of this is as follows.

Theorem 2.28 (Green's representation). Let -1 ≤ s ≤ 0. Any solution u of (1.1) with λindependent coefficients such that D A u belongs to the class E s can be represented as

u(λ, •) = S λ (∂ ν A u| λ=0 ) -D λ (u| λ=0 ) + c, (λ > 0),
where c is a constant. The representation holds in C([0, ∞); L 2 loc (R n+1 )) and, moding out the constant, in C([0, ∞);

Ḣs/2+1/2 ∂t-∆x ). In the lower half-space, with our convention of the conormal derivative, the formula reads

u(λ, •) = -S λ (∂ ν A u| λ=0 ) + D λ (u| λ=0 ) + c, (λ < 0).
Again, Proposition 2.27 and Theorem 2.28 hold as abstract results for operators of type P M with a bounded holomorphic functional calculus, see [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF]Sec. 12.3] for the proof in the context of elliptic systems and also [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF]Section 7.4] for a cleaner presentation.

The statement in Theorem 2.28 looks like the representation by the familiar Green's formula although we neither have defined a fundamental solution nor can perform integration by parts. In fact, at this level of generality, one can only construct an inverse, whose kernel is to become the fundamental solution when one has regularity assumptions on solutions. Nevertheless, our method says that all solutions of the BVPs must be obtained by Green's representation, once the abstract operators have integral representations.

We present two different constructions of the inverse. Both inversion formulae will be proved in Section 11. For the first one, we consider the operator L of (1.1) with all variables (λ, x, t) ∈ R n+2 and coefficients independent of λ. The discussion of Section 2.3 adapts and in virtue of the Lax-Milgram lemma L becomes an isomorphism from

Ḣ1/2 ∂t-∆ λ,x onto Ḣ-1/2 ∂t-∆ λ,x .
Theorem 2.29. Assume that the coefficients do not depend on λ. The inverse of L is formally given by the convolution with the operator valued kernel S λ . More precisely, let

E = L 2 (R; Ḣ-1/2 -∂t-∆x ). Given f ∈ Ḣ-1/2 ∂t-∆ λ,x , the integrals (Γ ε,R f ) λ = ε<|λ-µ|<R S λ-µ f µ dµ,
where

f µ := f (µ, •), exists for each 0 < ε < R < ∞ in E ′ . They define uniformly bounded operators Γ ε,R : Ḣ-1/2 ∂t-∆ λ,x → E ′ and lim ε→0,R→∞ Γ ε,R f = L -1 f holds in the sense of strong convergence in E ′ .
Note that Ḣ1/2 ∂t-∆ λ,x continuously embeds into E ′ and so the last equality makes sense. In other words, we have approximated L -1 f not within Ḣ1/2 ∂t-∆ λ,x but within the larger space E ′ . It does not seem as if Γ ε,R or any modification of this convolution maps into Ḣ1/2 ∂t-∆ λ,x except if f ∈ E. Nevertheless, this representation is in agreement with the natural interpretation of the single layer potential being the fundamental solution computed with pole at µ = 0.

One can also consider a parabolic operator with coefficients depending on all variables and construct its inverse with our methods. This gives us a definition in much greater generality than in the literature. Let us do that for

L = ∂ t -div x A (x, t)∇ x with (x, t) ∈ R n+1 . Consider L = ∂ t -div x A (x, t)∇ x -∂ 2
λ in n + 2 variables and let S λ be the single layer potential operator associated with L.

Theorem 2.30. The inverse of L

: Ḣ1/2 ∂t-∆x → Ḣ-1/2 ∂t-∆x is given by L -1 = -∞ -∞ S λ dλ. More precisely, given f ∈ Ḣ-1/2 ∂t-∆x , the integrals ε≤|λ|≤R S λ f dλ (0 < ε < R < ∞)
S λ f dλ = -L -1 f.
In this result, the approximation is within Ḣ1/2 ∂t-∆x , but is not directly related to the layer potentials of L.

2.15. Invertibility of layer potentials. Let us denote the (abstract) layer potentials from the previous section with superscripts L or L * to distinguish them for the forward and backward equation. So far, we have obtained their boundedness. Provided that the boundary layers are invertible, we can solve the Dirichlet/regularity problem for Lu = 0 with boundary data f (recall that these can be thought as the same problem in different topologies) by

(2.38) u = S L λ (S L 0 ) -1 f or (2.39) u = D L λ (D L 0 + ) -1 f, while the Neumann problem Lu = 0 with Neumann data g can be solved via (2.40) u = D L λ (∂ ν A D L 0 ) -1 g or (2.41) u = S L λ (∂ ν A S L 0 + ) -1 g = S L λ (D L * 0 -) * -1 g.
Here we consider λ > 0. Since S L λ and ∂ ν A D L λ do not jump across the boundary λ = 0, we do not need to use the 0 ± symbol. On the other hand, the double layer potentials D L λ and

D L * λ = (∂ ν A S L -λ ) * jump.
In the lower half-space, we would assume λ < 0 and change 0 + to 0 -and vice versa. Hence, including the lower half-space in the discussion, we have eight possible formulae, depending on the invertibility of six boundary operators, which is reminiscent of the connection between simultaneous well-posedness of BVPs in both half-spaces and invertibility of the entries of sgn(P M ) discussed in Section 2.13. This is of course no coincidence -there is a direct translation of the results on sgn(P M ) into the language of layer potentials. Indeed, setting

D r f = [∇ x f, H t D 1/2
t f ] and using the formulae for D A S L λ and D A D L λ as well as the jump relations from Proposition 2.27, the reader may readily check that

χ + (P M ) = ∂ ν A S L 0 + -∂ ν A D L 0 D -1 r D r S L 0 -D r D L 0 + D -1 r , χ -(P M ) = -∂ ν A S L 0 -+∂ ν A D L 0 D -1 r -D r S L 0 D r D L 0 -D -1 r , sgn(P M ) = ∂ ν A S L 0 + + ∂ ν A S L 0 - -2∂ ν A D L 0 D -1 r 2D r S L 0 -D r (D L 0 + + D L 0 -)D -1 r . (2.42) Comparison with (2.31) reveals ±2∂ ν A S L 0 ± = 1 ± s ⊥⊥ (P M ), -2∂ ν A D L 0 = s ⊥r (P M ), 2D r S L 0 = s r⊥ (P M ) and ∓2D r D L 0 + D -1 r = 1 ± s rr (P M ). Hence, Theorem 2.

(i) (R) L

Es is well-posed on both half-spaces if and only if

S L 0 : Ḣs/2 ∂t-∆x → Ḣs/2+1/2 ∂t-∆x is invertible. (ii) (N ) L
Es is well-posed on both half-spaces if and only if

∂ ν A D L 0 : Ḣs/2+1/2 ∂t-∆x → Ḣs/2 ∂t-∆x is invert- ible. (iii) The two operators S L 0 : Ḣs/2 ∂t-∆x → Ḣs/2+1/2 ∂t-∆x , ∂ ν A D L 0 : Ḣs/2+1/2 ∂t-∆x → Ḣs/2 ∂t-∆x

are invertible if and only if the four operators

∂ ν A S L 0 ± : Ḣs/2 ∂t-∆x → Ḣs/2 ∂t-∆x , D L 0 ± : Ḣs/2+1/2 ∂t-∆x → Ḣs/2+1/2 ∂t-∆x , are invertible. (iv) In particular, well-posedness of (R) L
Es and (N ) L Es on both half-spaces is equivalent to simultaneous invertibility of the six boundary layer operators appearing in (iii). (v) The six operators are always invertible when s = -1/2.

The direction from invertibility to solvability is well-known in this field. Our characterisations of well-posedness allowed us to include uniqueness and the full converse. This statement was implicit in the elliptic counterpart in [START_REF] Auscher | Representation and uniqueness for boundary value elliptic problems via first order systems[END_REF].

The conormal derivative of the double layer potential appearing in (ii) is hardly used in the literature and indeed, (iii) makes clear that it needs not be considered for a simultaneous treatment of regularity and Neumann problems. This is in accordance with the case of a real, symmetric and t-independent matrix A, where invertibility of the boundary layers for s = 0 can be achieved through a Rellich-type estimate

∂ ν A u| λ=0 2 ∼ ∇ x u| λ=0 2 + H t D 1/2 t u| λ=0 2 ,
at the boundary for u = S λ f or u = D λ f . Its proof is typically carried out by integration by parts and thus applies equally well in both half-spaces. As a consequence of this, there are lower bounds for all six operators in Theorem 2.31 and invertibility as well as compatibility with the inverses for the energy class can be obtained from the method of continuity, perturbing from the heat equation to L as in Proposition 2.20. This method has been pioneered for elliptic equations in [START_REF] Verchota | Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains[END_REF]. The parabolic version was recently obtained in [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF]Sec. 11.4].

In the general situation, however, Neumann and regularity problems on both half-spaces might not be simultaneously well-posed and then invertibility of one boundary layer operator needs not be necessary for solving the corresponding problem. Indeed, invertibility of

D L 0 ± : Ḣs/2+1/2 ∂t-∆x → Ḣs/2+1/2 ∂t-∆x is sufficient for solving (R) L Es on R n+2
± via (2.39) but it does not follow from a purely algebraic reasoning that invertibility, or even lower bounds, are also necessary for having well-posedness of (R) L Es solely in both half-spaces, see the remarks after Theorem 2.23. The same discussion applies to (N ) L Es and D L * 0 ∓ .

To finish this section, we comment on the connection between the spectral projections and the Calderón projections. We have seen that the entries of χ + (P M ) are given in terms of layer potentials but using the representation at the level of Ḣs P , -1 ≤ s ≤ 0. However, if we use the boundary space

H s := Ḣs/2 ∂t-∆x × Ḣs/2+1/2 ∂t-∆x , then through the isometry [ψ, ω] → [ψ, -D r ω] onto Ḣs P , the projection χ + (P M ) is similar to ∂ ν A S L 0 + ∂ ν A D L 0 -S L 0 -D L 0 +
, defined on H s , which is the celebrated Calderón projection for our parabolic equation on the upper half-space. The same can be done for the lower half-space. As for sgn(P M ), it is similar to the involution on H s given by

2A L := ∂ ν A S L 0 + + ∂ ν A S L 0 - 2∂ ν A D L 0 -2S L 0 -D L 0 + -D L 0 - .
In [32, Section 3], these calculations are carried out for the layer potentials in the case of the heat equation on a bounded Lipschitz cylinder when s = -1/2. The built-in algebra provided by the Dirac operator P M hence gives us immediate access to this technology. This reference shows also a certain coercivity of the involution, which is not available here since t lives in a non-compact interval. In fact, using the natural duality between H -1/2 and its dual, one can only show that

Re A L ψ -ω , -ω ψ = Re R n+2 + A∇u + • ∇u + + Re R n+2 - A∇u -• ∇u -,
where u ± are the energy solutions to (1.1) for λ-independent coefficients in R n+2

± with data corre- sponding to the splitting of [ψ, D r ω] ∈ Ḣ-1/2 P in virtue of the spectral spaces Ḣ-1/2,± P M
. So, we are missing the terms with half-order derivatives in time and there does not seem to be a hidden coercivity either, as already the example of the heat equation and calculations via Fourier transform show. We leave further details to the reader. [For the interested reader, the very same calculations work with the DB formalism of [START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF] for the elliptic equation -div λ,x A(x)∇ λ,x u = 0 on both half-spaces showing that the coercivity holds because the right hand side (integrals on R n+1 + ) would be equivalent to the square of the norm in

Ḣ-1/4 ∆x × Ḣ1/4 ∆x = Ḣ-1/2 (R n ) × Ḣ1/2 (R n
) of the data using the spectral splitting.] 2.16. Well-posedness results for equations with λ-independent coefficients. Finally, we shall address well-posedness of the BVPs. All structural properties of the coefficients A refer to the schematic representation (2.6).

Theorem 2.32.

(i) (R) L Es and (N ) L Es are compatibly well-posed when -1 ≤ s ≤ 0 and A(x, t) has block structure.

(ii) (R) L Es is compatibly well-posed when -1/2 ≤ s ≤ 0 and A(x, t) is upper triangular. (iii) (R) L Es is compatibly well-posed when -1 ≤ s ≤ -1/2 and A(x, t) is lower triangular. (iv) (N ) L Es is compatibly well-posed when -1/2 ≤ s ≤ 0 and A(x, t) is lower triangular. (v) (N ) L Es is compatibly well-posed when -1 ≤ s ≤ -1/2 and A(x, t) is upper triangular. (vi) (R) L Es and (N ) L Es are compatibly well-posed when -1 ≤ s ≤ 0 and A(x, t) = A(x) is Hermit- ian. (vii) (R) L
Es and (N ) L Es are compatibly well-posed when -1 ≤ s ≤ 0 and A(x, t) = A is constant. We remark that additionally all ranges for compatible well-posedness are relatively open in (-1, 0) by Theorem 2.17 and all results are stable under L ∞ -perturbations of the coefficients as in Theorem 2.19. The proof of this theorem will be given in Section 12.

Let us mention that (vi) was proved in [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF] in the case when L is a real equation and s = -1, 0. Here, we may even allow systems of complex parabolic equations, see the discussion in Section 14.1 at the end of this paper. Similarly, (vii) is done in [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF] when s = -1, 0. Item (i) is also proved as a corollary of the estimates in [START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF] when L has additionally t-independent coefficients. In (ii) and (iii) the triangular condition can be improved to the effect that the lower left and upper right coefficient of A can be real-valued with vanishing x-divergence, respectively, since such a change of the representing matrix results in a new elliptic operator that could also be represented by a triangular matrix [START_REF] Auscher | Solvability of elliptic systems with square integrable boundary data[END_REF]Rem. 6.7].

The results on well-posedness are identical for the backward equation with coefficients A(x, t): In fact, it suffices to revert time and to observe that A(x, -t) satisfies the same requirements as A(x, t) in the list above. Likewise, the results remain valid for BVPs on the lower half-space: Here, it suffices to observe that by even reflection at the boundary we obtain a one-to-one correspondence with weak solutions on the upper half-space for an elliptic equation with coefficients

-1 0 0 1 A ⊥⊥ A ⊥ A ⊥ A -1 0 0 1 ,
sharing the same structural properties.

In combination with Theorem 2.31 we obtain a posteriori all possible representation formulae for the unique solution proposed by layer potential theory.

41) are valid if A is either of block form, or constant, or Hermitian and independent of t, or if A is arbitrary and s is sufficiently close to -1/2 (depending on A). Moreover, (2.38) remains valid for the setup of Theorem 2.32(ii)-(iii) and (2.40) for that of Theorem 2.32(iv)-(v).

The Hermitian case (vi) will be treated in Section 12 by a Rellich-type argument. It will then become clear that the assumption of t-independence is necessary in order to compensate some lack of compactness, stemming from the fact that t varies over the whole real line. If we drop this additional assumption, then the natural solution class changes. The following result, also proved in Section 12, captures this effect.

Proposition 2.34. Assume A(x, t) is Hermitian with D 1/2 t A(x, t) ∈ L ∞ (R n ; BMO(R)). (i) Let f ∈ Ḣ1/2 ∂t-∆x ∩ Ḣ1/4 ∂t-∆x .
Then there is a unique reinforced weak solution u to (1.1) such that u| λ=0 = f and

D A u ∈ E 0 ∩ E -1/2 . (ii) Let f ∈ L 2 (R n+1 ) + Ḣ1/4 ∂t-∆x .
Then there is a unique reinforced weak solution u to (1.1) with u| λ=0 = f and the additional property that u can be written as the sum of two solutions

u 1 + u 2 with D A u 1 ∈ E -1 and D A u 2 ∈ E -1/2 .
(iii) Well-posedness in the same classes holds for the Neumann problem:

For (i), ∂ ν A u| λ=0 is taken in L 2 (R n+1 ) ∩ Ḣ-1/4 ∂t-∆x and for (ii) it is taken in Ḣ-1/2 ∂t-∆x + Ḣ-1/4 ∂t-∆x .
Note that (i) is a regularity problem with restricted data. The dual problem (ii) looks like a curious Dirichlet problem but if the data is restricted to f ∈ L 2 (R n+1 ), it will have a strong implication on the (classical) L 2 Dirichlet problem discussed in the next section.

As before, also Proposition 2.34 remains valid for BVPs on the lower half space. The interested reader may verify that the transposition into the language of layer potentials yields the following.

Proposition 2.35. Assume A(x, t) is Hermitian with

D 1/2 t A(x, t) ∈ L ∞ (R n ; BMO(R)). (i) S L 0 : Ḣ0 ∂t-∆x ∩ Ḣ-1/4 ∂t-∆x → Ḣ1/2 ∂t-∆x ∩ Ḣ1/4 ∂t-∆x is invertible and (2.38) holds for the solution (modulo constants) of Lu = 0 with D A u ∈ E 0 ∩E -1/2 and boundary data f ∈ Ḣ1/2 ∂t-∆x ∩ Ḣ1/4 ∂t-∆x . (ii) D L 0 + : L 2 (R n+1 )+ Ḣ1/4 ∂t-∆x → L 2 (R n+1 )+ Ḣ1/4 ∂t-∆x is invertible and (2.39) holds for the solution of Lu = 0 with D A u ∈ E -1 + E -1/2 and boundary data f ∈ L 2 (R n+1 ) + Ḣ1/2 ∂t-∆x .
2.17. Uniqueness for Dirichlet problems. So far, the Dirichlet problem with L 2 data is only formulated as (R) L E -1 , that is, within the class of reinforced weak solutions having controlled square function norm

D A u 2 E -1 = R n+2 + |λ∇ λ,x u| 2 + |λH t D 1/2 t u| 2 dλ dx dt λ < ∞
and the boundary behaviour can understood in the sense of L 2 convergence of u(λ, •) when λ → 0 since u is continuous with values in L 2 (R n+1 ) as a consequence of Theorem 2.9. As we have seen in Theorem 2.13, such solutions also satisfy the non-tangential maximal bound N * (u) 2 D A u E -1 and enjoy a form of non-tangential convergence for Whitney averages. Thus it makes sense to pose the question of uniqueness in the larger and more classical class involving non-tangential approach in the spirit of earlier references such as [START_REF] Hofmann | L 2 solvability and representation by caloric layer potentials in time-varying domains[END_REF][START_REF] Hofmann | The Dirichlet problem for parabolic operators with singular drift terms[END_REF][START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF].

Definition 2.36. We say that (D) L 2 is well-posed if given any f ∈ L 2 (R n+1 ) there is a unique weak solution Lu = 0 which satisfies N * (u) 2 < ∞ and --- Λ×Q×I |u(µ, y, s) -f (x, t)| dµ dy ds → 0
almost everywhere when λ → 0, where Λ × Q × I and (x, t) are related as in the definition of N * . Uniqueness of (D) L 2 means that if a weak solution Lu = 0 satisfies N * (u) 2 < ∞ and tends to zero in the sense above when λ → 0, then u = 0 almost everywhere.

Note that this problem is concerned with general weak solutions but we will see in Lemma 13.7 below that the non-tangential control a priori forces u to be reinforced. By reverse Hölder estimates on weak solutions (see Lemma 8.2 below), we may replace L 2 -averages of u by L 1 -averages up to taking slightly bigger Whitney regions. Hence, this is the largest possible class in which uniqueness may be asked for. When DeGiorgi-Nash-Moser regularity on solutions is assumed, the conditions in our definition amount to the usual pointwise non-tangential control and non-tangential convergence to the boundary almost everywhere. As our main result on (D) L 2 we will prove in Section 13 the following Theorem 2.37. Existence for the adjoint regularity problem (R) L * E 0 (that is, with regularity data in L 2 (R n+1 ; C n+1 )) implies uniqueness for (D) L 2 . This result is in spirit of the earlier results in [START_REF] Kenig | The Neumann problem for elliptic equations with nonsmooth coefficients[END_REF] on elliptic equations, see also [START_REF] Alfonseca | Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L ∞ coefficients[END_REF]Lem. 4.31]. For parabolic equations, compare with [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF]Lem. 6.1], where much more is assumed to get the same conclusion, including in particular DeGiorgi-Nash-Moser regularity as in all earlier references.

Corollary 2.38. Assume the adjoint regularity problem

(R) L * E 0 is (compatibly) well-posed. Then so is (D) L 2 . Moreover, given f ∈ L 2 (R n+1 ), the weak solution u with data f is reinforced, continuous in λ valued in L 2 (R n+1 ) and satisfies N * (u) 2 ∼ D A u E -1 ∼ sup λ>0 u(λ, •) 2 ∼ f 2 .
Indeed, uniqueness in the corollary above follows from the previous theorem, while existence follows from the previous discussion and Theorem 2.21 on duality of boundary value problems. The last two equivalences are consequences of Theorem 2.2 and Theorem 2.8 (or Theorem 2.9). For the first equivalence, the comparison " " was already mentioned in Theorem 2. [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF] Finally, it turns out that there also is a natural solution class defined by the non-tangential maximal function within which we have uniqueness in the setup of Proposition 2.34. By this means, we can complete the discussion of the previous section by adding the following result for the Dirichlet problem with Hermitian time dependent coefficients.

Theorem 2.40. Assume A(x, t) is Hermitian with

D 1/2 t A(x, t) ∈ L ∞ (R n ; BMO(R)). Let f ∈ L 2 (R n+1 ).
There is a unique weak solution Lu = 0 which satisfies N * (

u max{1, √ λ} ) 2 < ∞ and --- Λ×Q×I |u(µ, y, s) -f (x, t)| dµ dy ds → 0
almost everywhere when λ → 0. This solution is the same as in Proposition 2.34(ii).

We note that the growth in λ in the non-tangential control is reflecting the regularity of A in the assumption. The proof will also be given in Section 13.

Consequences for λ-dependent coefficients.

There is an automatic improvement of our results to a situation where A depends on the transversal variable λ by treating A in (1.1) as a perturbation of λ-independent coefficients A 0 . This is done at the level of the first order equation, reformulating

∂ λ F + P M F = 0 as (2.43) ∂ λ F + P M 0 F = P (M 0 -M )F.
This inhomogeneous problem can then be treated by maximal regularity results obtained in [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF] in the cases s = 0 and s = -1 and in [START_REF] Rosén | Layer potentials beyond singular integral operators[END_REF] for the intermediate cases -1 < s < 0 but changing DB to P M therein. We could have just stopped here but we felt that it is worth helping the reader making it through the transposition to our setup. Generally, the estimates for -1 < s < 0 transpose immediately since they only use semigroup theory. The estimates for s = -1 and s = 0 require more harmonic analysis. They are obtained using an abstract framework that consists of the following:

(i) A Hilbert space H, vector valued Lebesgue spaces

Y := L 2 (R + , λ dλ; H), Y * := L 2 (R + , dλ/λ; H) and a space X with continuous embedding Y * ⊂ X ⊂ L 2 loc (R + , dλ; H).
(ii) A bisectorial operator DB 0 with a bounded holomorphic functional calculus on H such that h → (e -λ[DB 0 ] h) λ>0 and h → (e -λ[D * B * 0 ] h) λ>0 are bounded from H into X . As for our setup, we let H = R(P ), X = {F ∈ L 2 loc (R + , dλ; H) : N * (F ) 2 < ∞} and use P M 0 instead of DB 0 . Then the embedding Y * ⊂ X is classical, see also Lemma 8.11 below. The H → X boundedness of the P M -semigroup follows from the non-tangential estimate stated in Theorem 2.12 and for the P * M -semigroup we use the analogue for the backward equation, see also Section 2.12. The fact that [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF] works within Euclidean space and Lebesgue measure is of no harm either since only the doubling character of the measure is used. This is also the case for R n+1 equipped with parabolic distance and Lebesgue measure. The needed theory on tent spaces above such spaces of homogeneous type has been written in detail in [START_REF] Amenta | Tent spaces over metric measure spaces under doubling and related assumptions[END_REF]. The needed estimates for tent spaces with Whitney averages, proved in [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF] and next in [START_REF] Huang | Weighted tent spaces with Whitney averages: factorization, interpolation and duality[END_REF][START_REF] Hytönen | On the Carleson duality[END_REF], transpose to such a setting, see the final remark in [START_REF] Huang | Weighted tent spaces with Whitney averages: factorization, interpolation and duality[END_REF]. Below, we will summarize the main results informally and give an outline of the general strategy, leaving the care of checking details to the interested readers.

The condition imposed on A(λ, x, t) is that there exists some λ-independent coefficients A 0 (x, t), uniformly elliptic and bounded, such that the discrepancy

A -A 0 s = A -A 0 ∞ (if -1 < s < 0), A -A 0 * (if s = -1, 0) is finite. Here, • ∞ is the L ∞ (R n+2 + )-norm and • * is the Carleson-Dahlberg measure norm defined by • * := C 2 (W ∞ (•)) ∞ with C 2 (g)(x, t) := sup R∋(x,t) 1 |R| ℓ(R) 0 R |g(λ, y, s)| 2 dy ds dλ λ 1/2
, where the supremum is taken over all parabolic cubes R with sidelength ℓ(R) in R n+1 containing (x, t) and W ∞ (•)(λ, y, s) is the essential supremum over the Whitney region given by Λ = (λ/2, 2λ),

Q = B(y, λ) and I = (s -λ 2 , s + λ 2 ). The condition A -A 0 * < ∞ is strictly stronger than A -A 0 ∞ < ∞. Note that A -A 0 s ∼ M -M 0 s
where M 0 corresponds to A 0 just as M corresponds to A. We also remark that A 0 is uniquely defined when using the • * -norm and the ellipticity constants relate to those of A. On the other hand, when using only the L ∞ -norm there are many different possible A 0 . Nevertheless we implicitly assume that the ellipticity constants of A and A 0 are comparable.

The operator related to solving (2.43) with initial data equal to zero writes formally in vectorvalued form as

(S A F ) λ := λ 0 e -(λ-µ)P M 0 E + 0 P (M 0 -M µ )F µ dµ - ∞ λ e (µ-λ)P M 0 E - 0 P (M 0 -M µ )F µ dµ.
Here, E ± 0 = χ ± (P M 0 ) are the projections defined in Section 2.5. We also use the notation F λ = F (λ, •). The integral can be appropriately defined when -1 ≤ s ≤ 0 and enjoys the norm estimate

S A F Es ≤ C A -A 0 s F Es ,
where the E s -norms were defined in Section 2.11 and C depends on ellipticity, s and dimension. Also S A maps into L 2 loc (R + ; R(P )). Then it is indeed true that for any solution of (1.1) with D A u ∈ E s there exists a unique h + ∈ Ḣs,+ P M 0 such that we have an implicit representation (2.44)

(D A u) λ = e -λP M 0 h + + (S A D A u) λ
with equality in E s and h + Ḣs P ∼ D A u Es with implicit constants depending on dimension, s and ellipticity. The boundary behaviour of D A u at λ = 0 is captured by h + and the complementary data

h -:= - ∞ 0 e µP M 0 E - 0 P (M 0 -M µ )(D A u) µ dµ ∈ Ḣs,- P M 0
satisfies for implicit constant depending on dimension, s and ellipticity,

h -Ḣs P A -A 0 s D A u Es . (2.45)
Then, writing h := h + + h -, there are limits

lim λ→0 λ -1 2λ λ D A u -h 2 Ḣs P dµ = 0 = lim λ→∞ λ -1 2λ λ D A u 2 Ḣs P dµ.
In the case s = -1, there is a direct representation of u by means of another operator [5, Thm. 9.2], showing that u -c ∈ C 0 ([0, ∞); L 2 (R n+1 )) for some constant c. Moreover, [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF]Thm. 10.1] can be adapted as well in order to show the control of the non-tangential maximal function by the square function

N * (u -c) 2 D A u E -1
for arbitrary reinforced weak solutions. This uses the dual estimate to (2.24) with M * P * in place of P M when it comes to off-diagonal decay.

When s = 0, the Whitney averages of D A u converge to h almost everywhere on R n+1 . This is best done by showing λ

0 D A u -Sh 2 2 dλ
λ < ∞ with Sh the semigroup extension of h and using Theorem 2.12 for Sh. If -1 < s ≤ 0, then one can adapt the proof of Lemma 13.6 below to show that Whitney averages of u (which can be chosen in L 2 loc (R n+2 + )) converge almost everywhere on R n+1 and the limit turns out to be an element

u 0 ∈ Ḣs/2+1/2 ∂t-∆x ∩ L 2 loc (R n+1
D A u Es ∼ D A u| λ=0 Ḣs P .
In this case, the trace map is an isomorphism onto its range. Let us call H s,+ A the range of this map just for the discussion. With this notation, H s,+ A 0 = H s,+ P M 0 . The conclusion is that if A -A 0 s is small enough, then the equation (2.44) can be used to construct solutions as the operator 1 -S A becomes invertible on E s : given any g ∈ Ḣs P , we can construct a reinforced weak solution to (1.1) with D A u ∈ E s by

D A u = (1 -S A ) -1 (C + 0 g) with C +
0 g the Cauchy extension of g using the operator P M 0 . Set E + A g = h to be the trace of D A u. Taking limits, we find h + = E + 0 g, h -Ḣs P A -A 0 s g Ḣs P and so we obtain that

E + A g = E + 0 g - ∞ 0 e µP M 0 E - 0 P (M 0 -M µ )(1 -S A ) -1 (C + 0 g) µ dµ
is a bounded projection of Ḣs P onto H s,+ A along Ḣs,-P M 0 . Moreover, there is a Lipschitz estimate

E + A -E + 0 A -A 0 s .
Turning eventually to boundary value problems, we can use h + to construct u from a boundary data and obtain the following theorem. Theorem 2.41. Let -1 ≤ s ≤ 0. If A -A 0 s is small enough, then the following assertions hold.

(i) (R) L Es is well-posed if and only if N r : Ḣs,+ P M 0 → ( Ḣs P ) r is an isomorphism. (ii) (N ) L
Es is well-posed if and only if N ⊥ : Ḣs,+ P M 0 → ( Ḣs P ) ⊥ is an isomorphism. In fact, as we have characterized the traces of conormal derivatives of solutions in E s provided that A -A 0 s is small enough, well-posedness of (R) L Es is equivalent to invertibility of N r : H s,+ A → ( Ḣs P ) r . By the Lipschitz estimate on the difference of projectors, we may equivalently replace H s,+ A by H s,+ A 0 = H s,+ P M 0 . The proof for (N ) L Es is the same. This result is not formulated like this in [START_REF] Auscher | Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I[END_REF] or [START_REF] Rosén | Layer potentials beyond singular integral operators[END_REF] but it is an illuminating formulation. Indeed, note that the right hand conditions mean well-posedness for the λ-independent operator L 0 with coefficients A 0 by Theorem 2.15. Hence, this is also a perturbation result in that well-posedness for A 0 implies well-posedness for nearby A in the norm • s . We remark that compatible wellposedness is also stable in this process. (This is a little tedious but not difficult using a Neumann series arguments when computing inverses.) In the same spirit we can use that the right hand sides in Theorem 2.41 are also stable under the change (L 0 , s) to (L * 0 , -1 -s) by Theorem 2.21, to obtain Corollary 2.42. Let -1 ≤ s ≤ 0 and assume that A -A 0 s is small enough. Let L and L * be as above with coefficients A and A * , respectively. Then (BVP ) L Es is (compatibly) well-posed if and only if (BVP ) L * E -1-s is (compatibly) well-posed, where BVP designates either N or R.

Homogeneous fractional Sobolev spaces

We introduce the homogeneous Sobolev space Ḣ1/2 (R) and related spaces in several variables. It will be necessary to have them realised not only within the tempered distributions modulo polynomials but, equivalently and more concretely, within L 2 loc (R) by means of a Riesz potential. In doing so we have to deviate from common literature [START_REF] Adams | Function Spaces and Potential Theory[END_REF][START_REF] Bergh | Interpolation Spaces. An Introduction[END_REF], see also [START_REF] Bourdaud | Localisation et multiplicateurs des espaces de Sobolev homogènes[END_REF], since 1/2 is a critical exponent for homogeneous L 2 -Sobolev space on the real line. For the reader's convenience we shall provide all necessary details.

In the following D 1/2 t denotes the half-order time derivative, defined L 2 (R) → S ′ (R) by the Fourier multiplier τ → |τ | 1/2 , and H t is the Hilbert transform corresponding to τ → i sgn(τ ). The sign convention for the Hilbert transform is in accordance with the textbook [START_REF] Kilbas | Fractional Integrals and Derivatives. Theory and applications[END_REF].

3.1. Spaces on the real line. For v ∈ S(R) there are well-known kernel representations

D 1/2 t v(t) = 1 2 √ 2π R 1 |t -s| 3/2 (v(t) -v(s)) ds, (3.1)
and

H t D 1/2 t v(t) = 1 2 √ 2π R sgn(t -s) |t -s| 3/2 (v(t) -v(s)) ds, (3.2
|D 1/2 t v(t)| + |H t D 1/2 t v(t)| ≤ C min{1, |t| -3/2 } (t ∈ R), (3.3)
where C depends only on the size of v in the topology of S(R). In fact, for boundedness we would split the integral at height |t -s| = 1 and treat both terms separately and in order to reveal the decay we would split at |t -s| = |t|/2. We will freely use these facts in the following without referring to them at each occurrence. Definition 3.1. The space Ḣ1/2 (R) consists of all v ∈ L 2 loc (R) ∩ S ′ (R) such that firstly for every φ ∈ S(R) the product vD

1/2 t φ ∈ L 1 (R) and secondly there is g ∈ L 2 (R) with R v • D 1/2 t φ dt = R g • φ dt (φ ∈ S(R)).
In this case we define D

1/2 t v := g and v Ḣ1/2 (R) := D 1/2 t v L 2 (R) . Remark 3.2. For v ∈ L 2 (R) we have v ∈ Ḣ1/2 (R) if and only if g := |τ | 1/2 v ∈ L 2 (R). Hence, L 2 (R) ∩ Ḣ1/2 (R) = H 1/2 (R) is the usual inhomogeneous fractional Sobolev space.
There is an important growth estimate implicit in Definition 3.1.

Lemma 3.3. Any v ∈ Ḣ1/2 (R) satisfies R |v(t)| dt 1+|t| 3/2 < ∞ and D 1/2 t v = 0 holds if and only if v is constant.
Proof. The estimate follows from v ∈ L 1 loc (R) and since for any φ ∈ C ∞ 0 (R) positive and non-zero at some point, (3.1) yields |D t φ for some φ ∈ S(R). Hence v, ψ = 0, showing that v is supported at the origin. Thus, v is a polynomial and according to our estimate it must be constant.

Further properties of Ḣ1/2 (R) follow from a representation in terms of the Riesz-type potential

I 1/2 t g(t) = 1 √ 2π (-1,1) 1 |t -s| 1/2 g(s) ds + c (-1,1) 1 |t -s| 1/2 - 1 |s| 1/2 g(s) ds ,
where typically g ∈ L 2 (R). A similar modification of the classical Riesz potential was considered in [START_REF] Kurokawa | Riesz potentials, higher Riesz transforms and Beppo Levi spaces[END_REF]. A few words concerning well-definedness: the Hardy-Littlewood-Sobolev inequality [57, Sec. V.1.2] ensures that the integral over (-1, 1) converges absolutely for a.e. t ∈ R and the same argument applies to the integral over c (-1, 1) near the singularity of the kernel. On the remaining part this kernel is controlled by (1 + |s|) -3/2 and the Cauchy-Schwarz inequality applies. This argument reveals that I

1/2 t : L 2 (R) → L 2 loc (R) is bounded. Proposition 3.4. Let g ∈ L 2 (R) and φ ∈ S(R). If T denotes any of the operators 1, H t , or H t D 1/2 t , then I 1/2 t g • D 1/2 t T φ ∈ L 1 (R) and R I 1/2 t g • D 1/2 t T φ dt = R g • T φ dt. Moreover, I 1/2 t H t g agrees up to a constant with the potential - 1 √ 2π (-1,1) sgn(t -s) |t -s| 1/2 g(s) ds + c (-1,1) sgn(t -s) |t -s| 1/2 - sgn(s) |s| 1/2 g(s) ds . Proof. For g ∈ S(R) a calculation, using the Riemann integral ∞ -∞ e is |s| -1/2 ds = √ 2π, reveals R e itτ |τ | 1/2 g(τ ) dτ = √ 2π R 1 |t -s| 1/2 g(s) ds.
Hence, in this case I 1/2 t g agrees with the following function of t up to a constant:

1 2π (-1,1) e itτ -1 |τ | 1/2 g(τ ) dτ + F -1 1c (-1,1) (τ ) g(τ ) |τ | 1/2 (t). (3.4)
This representation of I 1/2 t g is better suited for the extension to general g ∈ L 2 (R). In fact, define v 1 to be the first function in (3.4) and v 2 the second. Due to Plancherel's theorem v 2 2 ≤ g 2 and, using

|e itτ -1| ≤ 2|tτ | 1/4 for t, τ ∈ R, |v 1 (t)| ≤ 1 2π (-1,1) |e itτ -1| |τ | 1/2 | g(τ )| dτ ≤ 4|t| 1/4 g 2 (t ∈ R). (3.5) This proves that the maps g → v j are bounded L 2 (R) → L 2 loc (R), just as is I 1/2
t . Thus, it remains true for all g ∈ L 2 (R) that the function defined in (3.4) coincides with I 1/2 t g up to a constant. The potential representation for I 1/2 t H t g follows by a similar reasoning with H t g in place of g using the Riemann integral ∞ -∞ e is |s| -1/2 sgn(s) ds = i √ 2π. To prove the actual claim, we let g ∈ L 2 (R) and keep on denoting the two summands in (3.4) by v 1 and v 2 . In the cases T = 1 and T = H t we recall from (3.3) the bound

|D 1/2 t T φ(t)| min{1, |t| -3/2 } (t ∈ R),
with implicit constants depending only on the size of φ in S(R). In the case

T = H t D 1/2 t we even have D 1/2 t T φ = ∂ t φ ∈ S(R)
and so this bound holds all the more. After the preceding discussion we may directly argue on

v 1 + v 2 instead of I 1/2 t g. Due to (3.4), R |v 1 (t)||D 1/2 t T φ(t)| dt ≤ 4 g 2 R |t| 1/4 |D 1/2 t T φ(t)| dt < ∞,
thereby justifying the use of Fubini's theorem when computing

R v 1 (t) • D 1/2 t T φ(t) dt = (-1,1) g(τ ) |τ | 1/2 • R D 1/2 t T φ(t)e -itτ dt dτ = 2π (-1,1) g(τ ) • T φ(τ ) dτ,
where in the first step we have used that D 1/2 t T φ has integral zero since its Fourier transform vanishes at the origin. Concerning v 2 , Plancherel's theorem yields

R |v 2 (t)||D 1/2 t T φ(t)| dt ≤ g 2 D 1/2 t T φ 2 < ∞ and R v 2 (t) • D 1/2 t T φ(t) dt = 2π c (-1,1) g(τ ) • T φ(τ ) dτ.
Adding up the corresponding estimates and identities together with a final application of Plancherel's theorem leads us to the claim.

The first consequence of Proposition 3.4 is a Riesz potential representation for Ḣ1/2 (R). It also allows to improve on Lemma 3.3 obtaining a continuous embedding, which is apparently new.

Corollary 3.5. If g ∈ L 2 (R), then I 1/2 t g ∈ Ḣ1/2 (R) and D 1/2 t I 1/2 t g = g. In particular, for any v ∈ Ḣ1/2 (R) there exist c, d ∈ C such that v = c + I 1/2 t D 1/2 t v and R |v(t) -d| dt 1 + |t| 3/2 ≤ 42 v Ḣ1/2 (R) .
Proof. The first statement follows from Proposition 3.4 applied with T = 1. This implies the representation of v by taking g = D

R v • H t D 1/2 t φ dt = - R H t D 1/2 t v • φ dt and R v • ∂ t φ dt = - R H t D 1/2 t v • D 1/2 t φ dt.
The kernel representations (3.1) and (3.2) remain valid for v ∈ Ḣ1/2 (R) in a certain sense.

Corollary 3.7. If v ∈ Ḣ1/2 (R), then for almost every t ∈ c (supp v), D 1/2 t v(t) = - 1 2 √ 2π R v(s) |t -s| 3/2 ds H t D 1/2 t v(t) = - 1 2 √ 2π R sgn(t -s)v(s) |t -s| 3/2 ds. Proof. Let φ ∈ C ∞ 0 (R) be supported in c (supp v).
For the first formula we start out with

R D 1/2 t v(t) • φ(t) dt = R v(t) • D 1/2 t φ(t) dt, represent D 1/2
t φ via (3.1), and apply Fubini's theorem (justified by Lemma 3.3), to obtain

R D 1/2 t v(t) • φ(t) dt = R -1 2 √ 2π R v(s) |t -s| 3/2 ds φ(t) dt.
For the second formula we start out with the first identity in Corollary 3.6 instead.

We stress that our, seemingly new, definition of Ḣ1/2 (R) is in accordance with the relevant literature: It realises the closure of C ∞ 0 (R) for the homogeneous norm D

1/2 t • L 2 (R) .
Corollary 3.8. The space Ḣ1/2 (R)/C is a Hilbert space for the norm • Ḣ1/2 (R) and it contains C ∞ 0 (R) as a dense subspace. Proof. In view of Lemma 3.3, • Ḣ1/2 (R) is a Hilbertian norm on Ḣ1/2 (R)/C. Due to Corollary 3.5 we have an isomorphism

I 1/2 t : L 2 (R) → Ḣ1/2 (R)/C.
Thus, Ḣ1/2 (R)/C is a Hilbert space. L 2 -functions with Fourier transform supported away from 0 are dense in L 2 (R), hence their image under

I 1/2 t is dense in Ḣ1/2 (R)/C. For such g, however, (3.4) defines the L 2 (R)-function F -1 (|τ | -1/2 g(τ )) up to a constant. This proves that the inhomogeneous space H 1/2 (R) = L 2 (R)∩ Ḣ1/2 (R) is dense in Ḣ1/2 (R)/C. The density of C ∞ 0 (R) in H 1/2 (R)
is classical and can be proved by truncation and convolution with smooth kernels.

Spaces of several variables.

The notions introduced so far can be extended straightforwardly to functions of several variables. Definition 3.9. The space Ḣ1/2 (R;

L 2 loc (R n+1 + )) consists of all v ∈ L 2 loc (R n+2 + ) such that v(λ, x, •) ∈ Ḣ1/2 (R) for almost every (λ, x) ∈ R n+1 + and such that K R |D 1/2 t v(λ, x, t)| 2 dt dx dλ < ∞ for every compact set K ⊂ R n+1 + . Similarly, Ḣ1/2 (R; L 2 loc (R n )) is defined as a subspace of L 2 loc (R n+1
). There is an analogue of the embedding stated in Corollary 3.5 for the spaces in several variables. ) by the support assumption on φ. In view of Definition 3.9 the ground space Ėloc for reinforced weak solutions,

Lemma 3.10. Every v ∈ Ḣ1/2 (R; L 2 loc (R n+1 + )) satisfies vD 1/2 t φ, vH t D 1/2 t φ ∈ L 1 (R n+2 + ) if φ ∈ S(R n+2
Ėloc (R n+2 + ) = Ḣ1/2 (R; L 2 loc (R n+1 + )) ∩ L 2 loc (R; W 1,2 loc (R n+1 + )
), is unambiguously defined as a subspace of L 2 loc (R n+2 + ) and within there is the energy space Ė(R n+2

+ ) of those v ∈ Ėloc (R n+2 + ) for which v Ė(R n+2 + ) := D 1/2 t v 2 L 2 (R n+2
+ ) + ∇ λ,x v 2 L 2 (R n+2 + ) 1/2 < ∞.
Note that in Section 2 we simply wrote Ėloc and Ė. We stress the dependence of the domain since for the time being we shall concentrate on the analogous spaces of functions in the variables x and t only. Similar to the situation on the real line, there is a corresponding parabolic Riesz potential given by

I 1/2 t,x g(x, t) = |ξ|∨|τ |<1 e ix•ξ e itτ g(ξ, τ ) |ξ| + |τ | 1/2 dξ dτ + F -1 1 {|ξ|∨|τ |≥1} g(ξ, τ ) |ξ| + |τ | 1/2 .
Here, g ∈ L 2 (R n+1 ), ξ, τ are the Fourier variables corresponding to x, t and a ∨ b = max(a, b).

Lemma 3.11. Let g ∈ L 2 (R n+1 ) and let v = I 1/2 t,x g. Then v ∈ Ė(R n+1 ) with derivatives ∇ x v D 1/2 t v = F -1 1 |ξ| + |τ | 1/2 iξ |τ | 1/2 g and estimates v L 2 + L ∞ g L 2 and v Ė(R n+1 ) ∼ g L 2 . Conversely, each v ∈ Ė(R n+1 ) can be represented as v = c + I 1/2 t,x g for unique g ∈ L 2 (R n+1 ) and c ∈ C.
Proof. Let us denote the two summands in the definition of v = I 1/2 t,x g by v 1 and v 2 . Thanks to Plancherel's theorem we have v 2 2 ≤ g 2 and

|v 1 (t)| ≤ |ξ|∨|τ |<1 | g(ξ, τ )| |ξ| + |τ | 1/2 dξ dτ ≤ |ξ|∨|τ |<1 1 |ξ| 2 + |τ | dξ dτ g 2 (t ∈ R), showing v 1 ∈ L ∞ (R n+1
). This already proves the first of the two estimates for v. For any φ ∈ S(R n+1 ) the functions

v 1 D 1/2 t φ and v 2 D 1/2 t φ are in L 1 (R n+1
) thanks to (3.3). In turn, this allows us to use the theorems of Fubini and Plancherel just as in the proof of Proposition 3.4 to conclude

R n+1 v • ∇ x φ D 1/2 t φ dξ dτ = R n+1 F -1 1 |ξ| + |τ | 1/2 iξ |τ | 1/2 g • φ(ξ, τ ) dξ dτ.
Note that the Fourier multiplication operator appearing on the right-hand side is bounded on L 2 (R n+1 ). Taking φ in tensor form φ(x, t) = φ x (x)φ t (t) reveals v ∈ Ėloc (R n+1 ) and

∇ x v D 1/2 t v = F -1 1 |ξ| + |τ | 1/2 iξ |τ | 1/2 g ∈ L 2 (R n+1 ), (3.6) that is, v ∈ Ė(R n+1
). Finally, the estimate v Ė ≤ g 2 is immediate from Plancherel's theorem and to complete the first part of the lemma we record the reverse estimate

g 2 ≤ |ξ| 1/2 |ξ| + |τ | 1/2 g 2 + |τ | 1/2 |ξ| + |τ | 1/2 g 2 = ∇ x v 2 + D 1/2 t v 2 = v Ė.
We turn to the second part where now

v ∈ Ė(R n+2 + ) is given. Let φ ∈ C ∞ 0 (R n+1 ; C n ). It follows from Remark 3.10 that vD 1/2 t div x φ ∈ L 1 (R n+1
) and this justifies the use of Fubini's theorem when computing

R R n ∇ x v • D 1/2 t φ dx dt = - R R n v • D 1/2 t div x φ dx dt = - R n R v • D 1/2 t div x φ dt dx = - R n R D 1/2 t v • div x φ dt dx. With h := ∇ x v and h θ := D 1/2 t v, Plancherel's theorem lets discover |τ | 1/2 h = i h θ ξ. So, g := F -1 |ξ| + |τ | 1/2 |τ | 1/2 h θ (ξ, τ ) ∈ L 2 (R n+1 )
solves (3.6) and from the first part we deduce

∇ x v = ∇ x I 1/2 t,x g and D 1/2 t v = D 1/2 t I 1/2 t,x g. Taking into account Lemma 3.3, this implies that c = v -I 1/2
t,x g is constant. Finally, g is unique due to the estimate in the first part of the lemma.

As a consequence we obtain that our definition of the energy spaces Ė gives a realisation of the closure of test functions with respect to their homogeneous norm. In particular, they are the same spaces that have been introduced by the third author and collaborators in this context, see [START_REF] Castro | Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients[END_REF][START_REF] Nyström | Square functions estimates and the Kato problem for second order parabolic operators in R n+1[END_REF][START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF].

Corollary 3.12. The space Ė(R n+1 )/C is a Hilbert space for the norm • Ė(R n+1 ) and contains

C ∞ 0 (R n+1
) as a dense subset. The analogous statement holds for Ė(R n+2 + )/C using C ∞ 0 (R n+2 + ) as dense subset.

Proof. The statement for Ė(R n+1 )/C can be obtained by a literal repetition of the proof of Corollary 3.8, replacing I t by I t,x and the multiplier |τ | -1/2 by (|ξ| + |τ |) -1/2 therein. Of course, the same result holds in dimension n + 2. Since even reflection around λ = 0 allows to extend any function in Ė(R n+2 + ) to a function in Ė(R n+2 ), we readily obtain the second claim.

Remark 3.13. (i) Corollary 3.12 can be rephrased as saying that Ė(R n+1 ) is a realisation of the parabolic Sobolev space Ḣ1/2 ∂t-∆x within the tempered distributions modulo constants. The same kind of argument can be used to obtain such realisations for Ḣs ∂t-∆x , 0 < s < 1/2. (ii) Sobolev embeddings in parabolic space show that Ė(R n+1 )/C has a realisation in the Lebesgue space L q (R n+1 ), where q = 2(n+2) n . We shall not use this fact.

Finally, we supply a proof of the trace theorem for Ė(R n+2 + ) that played an important role for the study of energy solutions in Section 2.3. The reader may recall the definition of the parabolic Sobolev space Ḣ1/4 ∂t-∆x from Section 2.7.

Lemma 3.14. The space

Ė(R n+2 + )/C continuously embeds into C([0, ∞); Ḣ1/4 ∂t-∆x ). Conversely, any f ∈ Ḣ1/4 ∂t-∆x has an extension v ∈ Ė(R n+2 + ) such that v| λ=0 = f . Proof. Let v ∈ C ∞ 0 (R n+2 + ).
For any λ 0 > 0 the fundamental theorem of calculus yields

|τ | 1/4 v| λ=λ 0 2 2 + |ξ| 1/2 v| λ=λ 0 2 2 = 2 Re ∞ λ 0 (|τ | 1/2 v, ∂ λ v) + (|ξ| v, ∂ λ v) dλ ≤ ∞ 0 |τ | 1/2 v 2 2 + |ξ| v 2 2 + 2 ∂ λ v 2 2 dλ,
where norms, scalar products and the Fourier transform are in L 2 (R n+1 ). By Plancherel's theorem the left-hand side compares to the norm of v| λ=λ 0 in Ḣ1/4 ∂t-∆x and the right-hand side to the norm of v in Ė(R n+2 ). This proves uniform continuity of λ 0 → v| λ=λ 0 into Ḣ1/4 ∂t-∆x and thus also the limit at 0. Due to Corollary 3.12 we obtain the required embedding by density. Conversely, given f ∈ C ∞ 0 (R n+1 ), we can define an extension to R n+2

+ by v(λ, x, t) = F -1 (e -λ(|ξ| 2 +iτ ) 1/2 f )(x, t). From Plancherel's theorem we can infer v Ė(R n+2 ) f Ḣ1/4 ∂ t -∆x
and so this extension operator extends to Ḣ1/4 ∂t-∆x by density.

The parabolic Dirac operator

In this section we develop the basic operator theoretic properties of the parabolic Dirac operator

P =    0 div x -D 1/2 t -∇ x 0 0 -H t D 1/2 t 0 0   
with maximal domain

D(P ) = f ∈ L 2 : ∇ x f ⊥ ∈ L 2 , H t D 1/2 t f ⊥ ∈ L 2 , div x f -D 1/2 t f θ ∈ L 2 in L 2 = L 2 (R n+1 ; C n+2
). The reader may recall relevant notation from Sections 2.4 and 2.5. Clearly P is closed and its null space is

N(P ) = f ∈ L 2 : f ⊥ = 0, div x f = D 1/2 t f θ in S ′ (R n+1 ) .
Here is an explicit description of R(P ) as a space of distributions.

Lemma 4.1. It holds

R(P ) = f ∈ L 2 : ∇ x f θ = H t D 1/2 t f in S ′ (R n+1 ; C n ) .
Proof. The inclusion '⊂' follows from the definition of P . Conversely assume that f is contained in the right-hand space. Let g ∈ N(P * ), that is, g ⊥ = 0 and div x g = -H t D 1/2 t g θ . On the Fourier side the relation for -and θ-components can be solved by

f = f θ sgn(τ )|τ | 1/2 ξ, g θ = -ξ • g sgn(τ )|τ | 1/2
(a.e. on R n+1 ),

where τ and ξ are the Fourier variables corresponding to t and x. Hence, ( f , g) = 0 and thus (f, g) = 0 by Plancherel's theorem. This proves that f ∈ N(P * ) ⊥ and hence f ∈ R(P ).

Bisectoriality of P M and M P can be obtained by implementing the 'hidden coercivity' of the parabolic operator L discussed in Section 2.3 within the first order framework. Proof. Closedness will follow from bisectoriality since the latter implies that the operator's resolvent set is non-empty. Consider for δ ∈ R the transformation

U δ = 1 1 + δ 2   1 -δH t 0 0 0 1 + δH t 0 0 0 δ -H t   .
Then, as H 2 t = -1,

P U δ = 1 1 + δ 2    0 div x (1 + δH t ) -δD 1/2 t + H t D 1/2 t -∇ x (1 -δH t ) 0 0 -H t D 1/2 t -δD 1/2 t 0 0   .
Using that H * t = -H t and that H t commutes with derivatives in x, we see that P U δ is self-adjoint. Next, we claim that U -1 δ M is accretive for δ > 0 small enough. Indeed, on recalling the definition of M from (2.7), we can write U -1 δ M as

   1 + δH t 0 0 0 1 -δH t 0 0 0 δ + H t       Â⊥⊥ Â⊥ 0 Â ⊥ Â 0 0 0 1    =    (1 + δH t ) Â⊥⊥ (1 + δH t ) Â⊥ 0 (1 -δH t ) Â ⊥ (1 -δH t ) Â 0 0 0 δ + H t   .
Since Re(H t g, g) = 0, we see that the lower block of the matrix on the right is accretive for all δ > 0.

Since the upper block of the product rewrites as  plus a bounded perturbation of size δ, it remains accretive if δ is small enough.

Based on the above we can conclude that P M = (P U δ )(U -1 δ M ) is a product of a self-adjoint and a bounded accretive operator. Such operators have exhaustively been studied in the literature and it is well-known that they are bisectorial and satisfy

R(P M ) = R((P U δ )(U -1 δ M )) = R(P U δ ) = R(P ),
see for instance [START_REF] Auscher | On a quadratic estimate related to the Kato conjecture and boundary value problems[END_REF] or [START_REF] Egert | On Kato's conjecture and mixed boundary conditions[END_REF]Prop. 6.1.17]. Next, M P = M (P M )M -1 is bisectorial by similarity and R(M P ) = M R(P ) holds by accretivity of M .

Remark 4.3.

For parabolic systems M is only supposed to be accretive on the range of P , see Section 14. This property is also satisfied by U -1 δ M if δ is small. In that case, M -1 should be interpreted as the inverse of the restriction M | R(P ) , so that M P and P M are similar on the closures of their ranges and they are null on their null spaces. None of our proofs in the following sections will use the invertibility of M in L ∞ (R n+2 + ; L(C n+2 )).

5.

The correspondence to a first order system: Proof of Theorem 2.2

In this section we prove the correspondence between the parabolic equation (1.1) and the first order system ∂ λ F + P M F = 0. Whenever convenient, the reader may look up the definitions of P and M in (2.7). We always consider P as a closed operator in L 2 with maximal domain D(P ) and we write (• , •) for the inner product on this space. Let us stress that in the following we do not use the bisectoriality of P M and that M may depend on λ. Definition 5.1. A weak solution F to the first order system

∂ λ F + P M F = 0 is a function F ∈ L 2 loc (R + ; R(P )) such that ∞ 0 (F (λ), ∂ λ φ(λ)) dλ = ∞ 0 (M F (λ), P * φ(λ)) dλ (φ ∈ C ∞ 0 (R n+2 + ; C n+2 )). (5.1)
This notion of weak solutions is in accordance with Section 2 since Lemma 4.1 asserts that L 2 loc (R + ; R(P )) is exactly the space H loc introduced beforehand in Section 2.4.

5.1. Conormal differentials are weak solutions. Let u be a reinforced weak solution to (1.1) with the additional property

D A u =    ∂ ν A u ∇ x u H t D 1/2 t u    ∈ L 2 loc (R + ; L 2 ),
where

D 1/2 t
is in the sense of Definition 3.9. We put F := D A u and need to demonstrate that F is a weak solution to the first order system.

To verify the compatibility condition

F ∈ L 2 loc (R + ; R(P )), let φ ∈ C ∞ 0 (R n+1 ; C n
) and note that in view of Lemma 3.10 we have u(λ)

• H t D 1/2 t div x φ ∈ L 1 (R n+1
) for a.e. λ > 0. Hence, we can compute by means of Fubini's theorem

(F (λ) θ , div x φ) = - R n R u(λ) • H t D 1/2 t div x φ dt dx = - R R n u(λ) • div x H t D 1/2 t φ dx dt = (F (λ) , H t D 1/2 t φ).
This extends to all φ ∈ S(R n+1 ) by approximation, showing

∇ x F (λ) θ = H t D 1/2 t F (λ) in the sense of S ′ (R n+1
). Thus, F (λ) ∈ R(P ) thanks to Lemma 4.1.

In order to verify (5.1), let φ ∈ C ∞ 0 (R n+2 + ; C n+2 ). Taking φ ⊥ as a test function in the definition of reinforced weak solutions to the parabolic problem, we find

∞ 0 (F ⊥ , ∂ λ φ ⊥ ) dλ = R n+2 + (A∇ λ,x u) ⊥ • ∂ λ φ ⊥ dx dt dλ = R n+2 + -(A∇ λ,x u) • ∇ x φ ⊥ + H t D 1/2 t u • D 1/2 t φ ⊥ dx dt dλ, which in the language of P and M reads ∞ 0 (F ⊥ , ∂ λ φ ⊥ ) dλ = ∞ 0 ((M F ) , (P * φ) ) + ((M F ) θ , (P * φ) θ ) dλ.
For the -components we deduce, using integration by parts and the definitions of P and M ,

∞ 0 (F , ∂ λ φ ) dλ = R n+2 + ∇ x u • ∂ λ φ dλ = R n+2 + ∂ λ u • div x φ dλ = ∞ 0 ((M F ) ⊥ , (P * φ) ⊥ ) -(∂ λ u, H t D 1/2 t φ θ ) dλ.
As for the θ-component we first note that u

• H t D 1/2 t (∂ λ φ θ ) ∈ L 1 (R n+2
+ ) holds since u is a reinforced weak solution, see Lemma 3.10. Thus, by definition of D 1/2 t and Fubini's theorem

∞ 0 (F θ , ∂ λ φ θ ) dλ = - ∞ 0 R n R u • H t D 1/2 t ∂ λ φ θ dt dx dλ = - R n R ∞ 0 u • ∂ λ H t D 1/2 t φ θ dλ dt dx = ∞ 0 (∂ λ u, H t D 1/2 t φ θ ) dλ,
where we used

H t D 1/2 t φ(•, x, t) ∈ C ∞ 0 (R + ) for (x, t) ∈ R n+1
fixed in the last step. Adding up the previous three identities gives (5.1).

5.2.

Every weak solution is a conormal differential. Conversely, we are now given a weak solution F ∈ L 2 loc (R + ; R(P )) to the first order system and we have to construct a potential u such that u is a reinforced weak solution to the parabolic equation in (1.1) and D A u = F . Such u is necessarily unique up to a constant since if D A u = 0, then ∇ λ,x u = 0 showing that u only depends on t and then H t D 1/2 t u = 0 implies that u is constant, see Lemma 3.3. The construction of a specific u is in two steps.

Step 1: Adjusting the (x, t)-direction. By definition of F we have F (λ) ∈ R(P ) for almost every λ > 0. In this case

∇ x F (λ) θ = H t D 1/2
t F (λ) in the sense of tempered distributions due to Lemma 4.1 or, equivalently, iξ F (λ) θ = i sgn(τ )|τ | 1/2 F (λ) by taking the Fourier transform in (x, t). As usual, (ξ, τ ) is Fourier variable corresponding to (x, t). With these compatibility conditions at hand, we get for almost every λ > 0 that the measurable function

g(λ) := |ξ| + |τ | 1/2 i sgn(τ )|τ | 1/2 F (λ) θ belongs to L 2 (R n+1 ) as | g(λ)| ≤ | F (λ) θ | + | F (λ)
| almost everywhere and the so-defined function g is in L 2 loc (R + ; L 2 (R n+1 )). Now, Lemma 3.11 furnishes a parabolic potential v(λ, x, t) := I

1/2 t,x g(λ, x, t) ∈ L 2 loc (R n+2 + ) that satisfies v(λ) ∈ Ḣ1/2 (R; L 2 loc (R n )) ∩ L 2 loc (R; W 1,2 loc (R n )) for almost every λ > 0 with derivatives ∇ x v(λ) D 1/2 t v(λ) = F -1 1 |ξ| + |τ | 1/2 iξ |τ | 1/2 g(λ) .
Taking into account the relation between F (λ) and F (λ) θ , we obtain

∇ x v H t D 1/2 t v = F (λ) F (λ) θ , (5.2)
so that v already behaves as desired when it comes to derivatives in x-and t-direction. We may also assume that there exists some bounded set Q × I ⊆ R n+1 with positive Lebesgue measure such that

-- Q×I v(λ, x, t) dx dt = 0 (a.e. λ > 0) (5.3)
since otherwise we could subtract this very average from v(λ) for each λ > 0 without affecting any of the other properties of v discussed above.

To compute the transversal derivative

∂ λ v, we let ϕ ∈ C ∞ 0 (R + ), ψ ∈ C ∞ 0 (R n ; C n ), η ∈ C ∞ 0 (R) and test (5.1) with φ(λ, x, t) =   0 ϕ(λ)ψ(x)η(t) 0   .
Spelling out the identity

∞ 0 (F, ∂ λ φ) dλ = ∞ 0 (M F, P * φ) dλ
and taking into account that for the terms involving ϕ only the -component of φ and the ⊥component of P * φ do not vanish, we readily find

- R R n ∞ 0 v • ∂ λ ϕ dλ η div x ψ dx dt = R R n ∞ 0 (M F ) ⊥ • ϕ dλ η div x ψ dx dt.
Since ψ and η were arbitrary, the inner integrals over (0, ∞) differ only by a constant, which we can compute by taking an average over Q × I ⊂ R n+1 and using (5.3). Thus,

∞ 0 v • ∂ λ ϕ + (M F ) ⊥ • ϕ dλ = ∞ 0 -- Q×I (M F ) ⊥ dx dt • ϕ dλ
a.e. as functions defined on R n+1 . This means that in the sense of distributions

∂ λ v = (M F ) ⊥ --- Q×I (M F ) ⊥ dx dt ∈ L 2 loc (R n+2 + ). (5.4)
In particular, we have now found that v ∈ Ḣ1/2 (R;

L 2 loc (R n+1 + ))∩ L 2 loc (R; W 1,2 loc (R n+1 + )
) has the required regularity of a reinforced weak solution.

Step 2: Adjusting u in λ-direction. From (5.2) and (5.4) we obtain

D A v =    A ⊥⊥ A ⊥ 0 0 1 0 0 0 1       ∂ λ v ∇ x v H t D 1/2 t v    =    F ⊥ F F θ    +    A ⊥⊥ A ⊥ 0 0 1 0 0 0 1       -- Q×I (M F ) ⊥ dx dt 0 0    .
Let now w be an anti-derivative of the

L 2 loc -function λ → --Q×I (M F ) ⊥ dx dt, that is, w ∈ W 1,2 loc (R + ) and ∂ λ w = --Q×I (M F ) ⊥ dx dt.
Then the modified potential u := v -w still has the regularity of a reinforced weak solution since w depends only on λ and from above we can read off D A u = F . Hence, u has the required property. Finally, we let ϕ ∈ C ∞ 0 (R n+2 + ) and take φ = [ϕ, 0, 0] as a test function in (5.1). Unravelling the resulting identity gives

∞ 0 R n+1 + A∇ λ,x u • ∇ λ,x ϕ + H t D 1/2 t u • D 1/2 t ϕ dx dt dλ = 0.
This confirms u as a reinforced weak solution to (1.1) and the proof is complete.

Remark 5.2. Independently of the equation (1.1), the argument above shows in fact that any

G ∈ L 2 loc (R + ; L 2 (R n+1 ; C n+2 )) with ∂ λ G -∇ x G ⊥ = 0 and H t D 1/2 t G = ∇ x G θ has a potential v ∈ L 2 loc (R + ; L 2 (R n+1 )), unique up to constants, such that [∂ λ v, ∇ x v, H t D 1/2 t v] = G.

Resolvent estimates

In this section we prove the resolvent estimates stated in Lemma 2.10 and Proposition 2.11.

6.1. Proof of Lemma 2.10. As there is nothing to prove for λ = 0, we can in the following, by scaling (x, t) → (|λ|x, |λ| 1/2 t), assume λ = ±1 as long as implicit constants in all of our estimates only depend on dimension and the ellipticity constants of A. We shall only argue for λ = 1 since the proof for λ = -1 is the same. For 1 < p, q < ∞, we define inhomogeneous parabolic Sobolev spaces H p,q (R n+1 ) by

H p,q (R n+1 ) := L p (R; W 1,q (R n )) ∩ H 1/2,p (R; L q (R n ))
equipped with the mixed norm

u H p,q := |u| + |∇ x u| + |D 1/2 t u| L q (R n ) L p (R) .
We let H * p,q denote the space dual to H p,q with respect to the L 2 (R n+1 ) inner product. For any

f ∈ L 2 (R n+1 ; C n+2 ), there exists a unique f ∈ L 2 (R n+1 ; C n+2 ), such that f =    (A f ) ⊥ f f θ    ,
compare with Proposition 2.1. Spelling out the definitions of P and M reveals that the equation

(1 +iP M ) -1 f = g is equivalent to g ⊥ ∈ H 2,2 with        (A g) ⊥ + i div x (A g) -iD 1/2 t g θ = f ⊥ , g -i∇ x g ⊥ = f , g θ -iH t D 1/2 t g ⊥ = f θ .
Using the second and third equations to eliminate g and g θ in the first one, this is equivalent to the system

       (∂ t + L A ) g ⊥ = f ⊥ -A ⊥ f -i div x (A f ) + iD 1/2 t f θ , g -i∇ x g ⊥ = f , g θ -iH t D 1/2 t g ⊥ = f θ , (6.1)
where

L A := 1 i div x A 1 i∇ x .
The parabolic operator ∂ t + L A admits hidden coercivity in the same spirit as discussed for L in Section 2.3, that is, it can be associated with a bounded and coercive sesquilinear form on H 2,2 . As a consequence of the Lax-Milgram lemma we find that ∂ t + L A is a bounded and invertible operator from H 2,2 to its dual H * 2,2 . Moreover, ∂ t + L A clearly extends to a bounded operator from H p,q to H * p ′ ,q ′ for any 1 < p, q < ∞. All of these bounds depend only upon the ellipticity constants of A. We intend to apply Sneȋberg's lemma [START_REF] Sneȋberg | Spectral properties of linear operators in interpolation families of Banach spaces[END_REF] in order to extend invertibility to a neighbourhood of (2, 2) in the (p, q)-plane. This is possible provided that H p,q and H * p,q form complex interpolation scales. Lemma 6.1. Let ε > 0. For 1 + ε < p 0 , p 1 , q 0 , q 1 < 1 + ε -1 and 0 < θ < 1 the complex interpolation identities

[H p 0 ,q 0 , H p 1 ,q 1 ] θ = H p,q and [H * p 0 ,q 0 , H * p 1 ,q 1 ] θ = H * p,q
up to equivalent norms, where 1 p = 1-θ p 0 + θ p 1 and 1 q = 1-θ q 0 + θ q 1 . Equivalence constants can be chosen only in dependence of ε and n.

Proof. For 1 < p, q < ∞ we abbreviate L p (L q ) = L p (R; L q (R n )). It is well-known that these spaces interpolate according to the rule

[L p 0 (L q 0 ), L p 1 (L q 1 )] θ = L p (L q )
with equal norms, see, for example, [23, Thm. 5.1.1/2]. Let T : S(R n+1 ) → S ′ (R n+1 ) be defined on the Fourier side via multiplication by the symbol

m(τ, ξ) = 1 1 + |τ | 1/2 + |ξ 1 | + • • • + |ξ n | ,
where τ ∈ R and ξ ∈ R n . Then m satisfies Mihlin's condition

τ α 0 ξ α 1 1 • • • ξ αn n ∂ α m ∂ α 0 τ ∂ α 1 ξ 1 • • • ∂ αn ξn ≤ C(α) < ∞ (τ, ξ 1 , . . . , ξ n = 0)
for all multi-indices α ∈ N n+1 0 . The Marcinkiewicz multiplier theorem on L p (L q ), see [50, Cor. 1], yields that T extends to a bounded operator on L p (L q ) for every pair (p, q) ∈ (1, ∞) × (1, ∞). Complex interpolation, starting from the four extremal points with p, q ∈ {1 + ε, 1 + ε -1 }, yields the required uniform bound on the operator norm within their convex hull in the (p, q)-plane. By the same argument applied to the multipliers |τ | 1/2 m(τ, ξ), |ξ|m(τ, ξ) and iξ j /|ξ j |, we see that in fact the extension of T is bounded L p (L q ) → H p,q and invertible. Thus, the required interpolation rules for the H p,q -spaces including uniformity of the equivalence constants follow from those for L p (L q ).

As another consequence every space H p,q contains T (S(R n+1 )) as a dense set since S(R n+1 ) is dense in L p (L q ). Thus, the interpolation rules for the dual spaces H * p,q follow from the duality theorem for the complex interpolation functor [23, Thm. 4.5.1]. Lemma 6.1 allows to apply Sneȋberg's result [START_REF] Sneȋberg | Spectral properties of linear operators in interpolation families of Banach spaces[END_REF], see also [START_REF] Auscher | Non-local self-improving properties: A functional analytic approach[END_REF]App. A] for a quantitative version, to the effect that ∂ t + L A remains invertible for (p, q) in a neighbourhood of [START_REF] Alfonseca | Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L ∞ coefficients[END_REF][START_REF] Alfonseca | Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L ∞ coefficients[END_REF]. Technically speaking, this only applies to complex interpolation scales with one parameter: Seeing ( 1p , 1 q ) on a line passing through ( 12 , 1 2 ), it yields an interval for invertibility around (2, 2) on that line. However, the result is quantitative in that the length of that interval depends only upon upper/lower bounds for ∂ t + L A : H 2,2 → H * 2,2 and the equivalence constants in the interpolation from Lemma 6.1. Therefore we can obtain the same interval on each line, eventually summing up to a two-dimensional neighbourhood in the (p, q)-plane. Let us also remark that all inverses are compatible on the intersection of any two such H * p,q spaces by abstract interpolation theory [START_REF] Kalton | Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations[END_REF]Thm. 8.1]. Returning to the equation (1 +iP M ) -1 f = g, we obtain from (6.1) that if (p, q) is as above, then given f ∈ L 2 (L 2 ) ∩ L p (L q ) we have g L p (L q ) f L p (L q ) with implicit constants depending only on dimension and ellipticity. We conclude by density.

Next, we prove the analogous result for M P . If M -1 is in L ∞ , which is the case for equations, then the result is immediate by similarity. For systems, however, we only have that M : R(P ) → R(M P ) is invertible. In this case, we first use the argument above to find that (1 + iP ) -1 is bounded on L p (L q ) if 1 < p, q < ∞. Therefore the domains D p,q of the maximal realisation of P in L p (L q ), equipped with the graph norm, form a complex interpolation scale. Since M P is bisectorial on L 2 (L 2 ), we have that 1 + iM P is an isomorphism D 2,2 → L 2 (L 2 ). For all other p, q it is at least bounded D p,q → L p (L q ). As above, 1 + iM P remains an isomorphism D p,q → L p (L q ) for (p, q) in a neighbourhood of (2, 2) with compatible inverses. Consequently, the resolvent estimate holds in L p (L q ) for those (p, q). Finally, M * is an operator in the same class as M . Thus, we already have resolvent bounds on L p (L q ) for P M * and M * P and those for M P * and P * M follow by duality. 6.2. Proof of Proposition 2.11. It suffices to prove (2.24) for q = 2 and 0 < ε < ε 0 with ε 0 = δ 0 of Lemma 2.10 and N 0 to be determined. Then the proposition follows by interpolation with the global bound provided by Lemma 2.10 as long as q is close enough to 2 to guarantee the announced decay. We shall only argue for P M as the proof for the other operators is literally the same.

Let h ∈ L 2 (R n+1 ; C n+2 ) = L 2 (L 2 ). Let Q, I, j, k, λ as in the statement and let N ≥ 4. We set J = 4 j I and split

C k (Q × J) = C 1 k ∪ C 2 k with C 1 k := (2 k+1 Q \ 2 k Q) × N k J and C 2 k := 2 k Q × (N k+1 J \ N k J).

We assume that h is supported in C k and hence we can write

h = h 1 + h 2 with h i supported in C i k . To estimate --Q×J |(1 +iλP M ) -1 h 1 | 2
dy ds, we can rely on the same argument as in the elliptic situation since the x support of h 1 is far from Q: More precisely, we have

-- Q×J |(1 +iλP M ) -1 h 1 | 2 dy ds ≤ 1 ℓ(J)|Q| R Q |(1 +iλP M ) -1 h 1 | 2 dy ds
and by the argument in [9, Prop. 5.1], using a commutator with a cut-off function in x-direction only, we obtain for any m ∈ N,

-- Q×J |(1 +iλP M ) -1 h 1 | 2 dy ds 2 -km ℓ(J)|Q| R R n |h 1 | 2 dy ds = 2 -km N k+1 2 kn -- C k (Q×J)
|h| 2 dy ds.

(6.2)

In order to estimate

--Q×J |(1 +iλP M ) -1 h 2 | 2 dy ds, we pick a smooth cut-off function η ∈ C ∞ 0 (N k-1 J), which is valued in [0, 1], equal to 1 on N k-2 J and satisfies (N k ℓ(J)) ∂ t η ∞ 1 uniformly in k, N
and the size of J. With p > 2 in the range of Lemma 2.10 we have

-- Q×J |(1 +iλP M ) -1 h 2 | 2 dy ds ≤ 1 |Q| - J R n |(1 +iλP M ) -1 h 2 | 2 dy ds ≤ 1 |Q| - J R n |(1 +iλP M ) -1 h 2 | 2 dy p/2 ds 2/p ≤ 1 |Q|ℓ(J) 2/p R R n |η(1 +iλP M ) -1 h 2 | 2 dy p/2 ds 2/p . Since η(t)h 2 (x, t) = 0, we can re-express η(1 +iλP M ) -1 h 2 using a commutator η(1 +iλP M ) -1 h 2 = [η, (1 +iλP M ) -1 ]h 2 = (1 +iλP M ) -1 [η, iλP M ](1 +iλP M ) -1 h 2 = (1 +iλP M ) -1 iλ[η, P ]M (1 +iλP M ) -1 h 2 , ( 6.3)
where

[η, P ] =    0 0 -[η, D 1/2 t ] 0 0 0 -[η, H t D 1/2 t ] 0 0    .
The commutators in [η, P ] depend only on the t variable and due to (3.1) and (3.2) they have kernels K(t, s) bounded by

|K(t, s)| ≤ C |η(t) -η(s)| |t -s| 3/2 h -1/2 h -1 φ t -s h (6.4)
with φ(t) = min{|t| -1/2 , |t| -3/2 } and h := N k ℓ(J). Thus, by Young's inequality,

[η, D 1/2 t ] L 2 (R)→L p (R) + [η, H t D 1/2 t ] L 2 (R)→L p (R) h-1/2-((1/2)-(1/p)) .
Using usual vector-valued extension [57, Sec. II.5], it follows that [η, P ] : L 2 (L 2 ) → L p (L 2 ) with norm as above. This is the key point. Now, inserting (6.3) into the ongoing estimate, and applying the L p (L 2 ) boundedness of the resolvent and the just obtained commutator bound, we find

1 |Q|ℓ(J) 2/p R R n |η(1 +iλP M ) -1 h 2 | 2 dy p/2 ds 2/p 1 |Q|ℓ(J) 2/p R R n |iλ[η, P ]M (1 +iλP M ) -1 h 2 | 2 dy p/2 ds 2/p |λ| 2 |Q|ℓ(J) 2/p • 1 (N k ℓ(J)) 1+(1-2/p) R R n |M (1 +iλP M ) -1 h 2 | 2 dy ds.
Since the resolvents of P M are L 2 (L 2 )-bounded, we obtain furthermore

|λ| 2 |Q|ℓ(J) 2 (N k ) 2-2/p R R n |h 2 | 2 dy ds 2 kn (N k ) 1-2/p |λ| 2 ℓ(J) -- C k (Q×J)
|h| 2 dy ds.

We remark that ℓ(J) = 4 j ℓ(I) and λ 2 ∼ ℓ(I) by assumption. Hence, we have proved

-- Q×J |(1 +iλP M ) -1 h 2 | 2 dy ds 2 kn (N k ) 1-2/p -- C k (Q×J)
|h| 2 dy ds.

To conclude the proof, we only have to compare with (6.2) and choose the parameters at our disposal appropriately: First we pick 0 < ε < ε 0 and then p with ε < 1 2 -1 p < ε 0 . For N large enough 2 kn N -2kε (N k ) 1-2/p and given any such choice of N , there is a choice of m large verifying 2 -km N k+1 2 kn N -2kε .

7. Quadratic estimates for P M : Proof of Theorem 2.3

Based on Lemma 4.2 we see that in order to prove Theorem 2.3 it suffices to prove the quadratic estimate (7.1)

∞ 0 λM P (1 +λ 2 M P M P ) -1 h 2 2 dλ λ h 2 2 (h ∈ L 2 )
and the analogous estimate obtained upon replacing M P by M * P * , where L 2 = L 2 (R n+1 ; C n+2 ) as before: In fact, (7.1) for M P implies the analogous estimate for P M since these two operators are similar on the closure of their ranges by Remark 4.3. By the same argument, the estimate for M * P * implies the one for P * M * . Once we have (7.1) for all four operators, an abstract duality argument gives the reverse inequalities for all four operators and hence the claim, see [START_REF] Mcintosh | Operators which have an H ∞ functional calculus[END_REF] or [START_REF] Egert | On Kato's conjecture and mixed boundary conditions[END_REF]Lem. 3.4.10]. Below, we shall only give the proof for M P since the argument for M * P * is identical upon minor changes of the algebra. Note that when M = 1, then these estimates with P or P * can simply be proved using a Fourier transform argument in the (x, t) variables.

In proving (7.1), we shall follow the algorithm developed in [START_REF] Auscher | On a quadratic estimate related to the Kato conjecture and boundary value problems[END_REF]. We set R λ = (1 +iλM P ) -1 for λ ∈ R. Then

Q λ = 1 2i (R -λ -R λ ) = λM P (1 +λ 2 M P M P ) -1
and also 1 2 (R -λ + R λ ) = (1 + λ 2 M P M P ) -1 . Since M P is bisectorial, see Lemma 4.2, the operators R λ , Q λ and (1 +λ 2 M P M P ) -1 are uniformly bounded on L 2 . We define the parabolic approximation of the identity P λ = (1 + λ 2 (∂ t -∆ x )) -1 for λ ∈ R and let it act componentwise on C n+2 valued functions. By Plancherel's theorem the operators P λ are uniformly bounded on L 2 .

Proposition 7.1. For all v ∈ D(P ) the estimate

∞ 0 Θ λ (1 -P λ )P v 2 2 dλ λ P v 2 2 .
Proof. From (2.18) we know that P 2 agrees with ∂ t -∆ x on D(P 2 ) ∩ R(P ). Hence, P λ agrees with (1 + λ 2 P 2 ) -1 on R(P ) and to ease the computations we may think of P λ = (1 + λ 2 P 2 ) -1 just for the proof. Since P λ and P commute and (1 -P λ )v ∈ D(P ), we have

Θ λ (1 -P λ )P v = (Θ λ P )(1 -P λ )v = λ(M P ) 2 (1 +(λM P ) 2 ) -1 (1 -P λ )v. Now, (λM P ) 2 (1 +(λM P ) 2 ) -1 = 1 -(1 +(λM P ) 2 ) -1 is uniformly bounded, hence Θ λ (1 -P λ )P v 2 1 λ (1 -P λ )v 2 .
Using that

1 λ (1 -P λ )v = λP (1 +λ 2 P 2 ) -1 (P v)
we see by the square function estimate for P that

∞ 0 λP (1 +λ 2 P 2 ) -1 (P v) 2 2 dλ λ P v 2 2 .
Next, we perform the principal part approximation. We use the following dyadic decomposition of R n+1 in parabolic dyadic cubes. Let = ∞ j=-∞ 2 j where 2 j := {δ j (k + (0, 1] n+1 ) : k ∈ Z n+1 } with δ j (x, t) = (2 j x, 4 j t). For a parabolic dyadic cube R ∈ 2 j we denote by ℓ(R) = 2 j its sidelength and by |R| = 2 (n+2)j its volume. We set λ = 2 j if 2 j-1 < λ ≤ 2 j . Let the parabolic dyadic averaging operator S λ : L 2 → L 2 be given by

S λ u(x, t) := u R := -- R u(y, s) dy ds = 1 |R| R u(y, s) dy ds
for every (x, t) ∈ R n+1 and λ > 0, where R is the unique parabolic dyadic cube in λ that contains (x, t). We remark that S 2 λ = S λ . Definition 7.2. By the principal part of (Θ λ ) λ>0 we mean the multiplication operators γ λ defined by

γ λ (x, t)ζ := (Θ λ ζ)(x, t)
for every ζ ∈ C n+2 . We view ζ on the right-hand side of the equation above as the constant function valued in C n+2 defined on R n+1 by ζ(x, t) := ζ. We identify γ λ (x, t) with the (possibly unbounded) multiplication operator γ λ : f (x, t) → γ λ (x, t)f (x, t).

Of course ζ /

∈ L 2 but still this definition is justified in virtue of the next lemma.

Lemma 7.3. The operator Θ λ extends to a bounded operator from L ∞ into L 2 loc . In particular we have well defined functions

γ λ ∈ L 2 loc (R n+1 ; L(C n+2 )) with bounds -- R |γ λ (x, t)| 2 dx dt 1 for all R ∈ λ . Here, | • | is the operator norm of L(C n+2 ). Moreover, γ λ S λ is bounded on L 2 with γ λ S λ
1 uniformly for all λ > 0.

Proof. Fix a parabolic cube R ∈ λ and f ∈ L ∞ (R n ; C n+2 ) with f ∞ = 1. Then write f = f 1 + f 2 + . . . where f 1 = f on 2R and 0 elsewhere and if j ≥ 2, f j = f on C j (R) with the notation of Proposition 2.11 and 0 elsewhere. Then apply Θ λ = 1 2i (R -λ -R λ )M , use the off-diagonal estimates of Proposition 2.11 for each term Θ λ f j , with N large enough, and sum to obtain for Θ

λ := ∞ j=1 Θ λ f j the estimate -- R |(Θ λ f )(x, t)| 2 dx dt ≤ C.
If we do this for the constant functions with values describing an orthonormal basis of C n+2 and sum, we obtain an upper bound for the desired average of γ λ . Next, for a function f ∈ L 2 ,

γ λ S λ f 2 2 = R∈ λ R γ λ (x, t) -- R f 2 dx dt R∈ λ |R| -- R f 2 ≤ f 2 2 .
Here, | • | is the usual Hermitian norm on C n+2 .

We have the following principal part approximation of Θ λ by γ λ S λ .

Lemma 7.4. It holds

∞ 0 Θ λ P λ f -γ λ S λ f 2 2 dλ λ f 2 2 (f ∈ L 2 ).
Proof. Let Φ λ be a nice parabolic approximation of the identity on L 2 , that is, the convolution operator by a real valued function

λ -n-2 φ(λ -1 x, λ -2 t) with φ ∈ C ∞ 0 (B(0, 1) × (-1, 1)) satisfying R n+1 φ(x, t) dx dt = 1. Write Θ λ P λ -γ λ S λ = (Θ λ P λ -γ λ S λ P λ ) + (γ λ S λ (P λ -Φ λ )) + (γ λ S λ (Φ λ -S λ )) + (γ λ S 2 λ -γ λ S λ ). (7.3)
Because of S 2 λ = S λ , the last term vanishes. Next, as the γ λ S λ are uniformly bounded as operators on L 2 ,

∞ 0 γ λ S λ (P λ -Φ λ )f 2 2 + γ λ S λ (Φ λ -S λ )f 2 2 dλ λ ∞ 0 (P λ -Φ λ )f 2 2 + (Φ λ -S λ )f 2 2 dλ λ .
A straightforward Fourier transform argument reveals control by f 2 2 for the square function of P λ -Φ λ . The analogous bound for the square function of Φ λ -S λ is done componentwise and for each component it is precisely the statement of [START_REF] Castro | Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients[END_REF]Lem. 2.5]. As for the first term in (7.3), we will show that for α ∈ (0, 1) small enough there is a bound

(7.4) (Θ λ P λ -γ λ S λ P λ )f 2 2 λ 2 ∇ x P λ f 2 2 + λ 4α D α t P λ f 2 2
, where D α t is the fractional derivative of order α on R defined via the Fourier symbol |τ | α . We postpone the proof of this claim until Section 7.3. Hence, we obtain

∞ 0 Θ λ P λ f -γ λ S λ P λ f 2 2 dλ λ ∞ 0 λ∇ x P λ f 2 2 dλ λ + ∞ 0 λ 2α D α t P λ f 2 2 dλ λ 2 2
from a Fourier transform argument on each component of f , given the definition of P λ .

Combining Lemma 7.4 with Proposition 7.1, we obtain the principal part approximation

(7.5) ∞ 0 Θ λ P v -γ λ S λ P v 2 2 dλ λ P v 2 2 (v ∈ D(P )).
Hence, to complete the proof of (7.2) we want to apply Carleson's theorem to conclude that

∞ 0 γ λ S λ P v 2 2 dλ λ γ λ 2 C P v 2 2 (v ∈ D(P )),
where

γ λ 2 C = sup R∈ 1 |R| T (R) |γ λ (x, t)| 2 dx dt dλ λ .
Here T (R) := (0, ℓ(R)] × R is the Carleson box over R. Hence, it remains to prove the Carleson measure estimate γ λ C < ∞, and the claim in (7.4), to finish the proof of the quadratic estimate in (7.1).

The T(b) argument.

We first construct test functions which belong to the range of P . Let η : R → [0, 1] be a smooth non-negative function with support in [-1, 1] which is 1 on

[-1/2, 1/2]. Consider ν(t) = η(t) -η(t -z) for z ∈ R, |z| > 2.
Since ν(0) = 0, we can define h := (-H t D 1/2 t ) -1 ν by Fourier transform. We claim that h ∈ L 2 (R) satisfies |h(t) -h(0)| ≤ C|t| and that we can find some z such that h(0) = 0. Indeed,

h(t) = - 1 2π R e itτ 1 -e izτ i sgn(τ )|τ | 1/2 η(τ ) dτ, so that the cheap estimate |1 -e itτ | ≤ |tτ | yields |h(t) -h(0)| ≤ |t| π R |τ | 1/2 | η(τ )| dτ,
and likewise, using |1 -e izτ | ≤ |zτ | and Plancherel's theorem, it follows that

h 2 2 ≤ |z| 2 2π R |τ || η(τ )| 2 dτ < ∞.
To see the last claim, assume to contrary that h(0) = 0 holds for any |z| > 2. By differentiating the formula for h(0) with respect to z, we find

1 2π R e izτ |τ | 1/2 η(τ ) dτ = 0 (|z| > 2).
This means that the Fourier transform of τ → |τ | 1/2 η(τ ) is compactly supported, hence this function is analytic. In particular, it is analytic at the origin. But τ → η(τ ) has the same property and ). We need also a parameter 0 < δ ≤ 1 which is small and will be chosen later. We let χ be a smooth function on R n , valued in [0, 1], which is 1 on the cube [-1/2, 1/2] n and with support in [-1, 1] n .

η(0) = R η ≥ 1. Hence τ → |τ |
Fix a parabolic dyadic cube R = Q × I, its center being denoted (x Q , t I ). Observe that with our notation ℓ(R) = ℓ(Q) = ℓ(I). We set

χ Q (x) := χ x -x Q ℓ(Q) (x ∈ R n ), h I,δ (t) := δ -1/2 ℓ(I) 1/2 h δ • t -t I ℓ(I) (t ∈ R), f 1 (x) := ζ n+2 - δ 1/2 ζ • (x -x Q ) h(0)ℓ(Q) (x ∈ R n ), f 2 (x) := δ 1/2 ζ 1 (x 1 -x Q,1 ) h(0)ℓ(Q) (x ∈ R n )
and finally we introduce

L ζ R,δ : R n+1 → C n+2 through L ζ R,δ (x, t) :=         χ Q (x)f 1 (x) h I,δ (t) χ Q (x)f 2 (x) h I,δ (t) 0 . . . 0         ((x, t) ∈ R n+1 ).
Clearly L ζ R,δ ∈ L2 , but we do not need its L 2 -norm and so we do not compute it. We claim that L ζ R,δ ∈ D(P ) with (7.6)

P L ζ R,δ 2 2 ≤ Cδ -2 |R| and that (7.7) R |P L ζ R,δ -ζ| 2 dx dt ≤ Cδ|R|
with an absolute constant C. Indeed, we first note that

P L ζ R,δ (x, t) =         ∂ x 1 (χ Q (x)f 2 (x)) h I,δ (t) -∂ x 1 (χ Q (x)f 1 (x)) h I,δ (t) . . . -∂ xn (χ Q (x)f 1 (x)) h I,δ (t) χ Q (x)f 1 (x) (-H t D 1/2 t )h I,δ (t)        
. Furthermore, we observe by a change of variables that

h I,δ L 2 (R) = δ -1 ℓ(I) h L 2 (R) and (-H t D 1/2 t )h I,δ (t) = ν δ • t-t I ℓ(I) , so that (-H t D 1/2 t )h I,δ L 2 (R) = δ -1/2 ℓ(I) 1/2 ν L 2 (R) .
Using that the partial derivatives of the functions of x are bounded (uniformly for δ ≤ 1) by C/ℓ(Q) with support in 2Q, we get

P L ζ R,δ 2
Because of the support properties of χ Q and ν we have for (x, t) ∈ R that

P L ζ R,δ (x, t) =           ∂ x 1 f 2 (x) h I,δ (t) -∂ x 1 f 1 (x) h I,δ (t) . . . -∂ xn f 1 (x) h I,δ (t) f 1 (x)           =           ζ 1 h(0) -1 h δ • t-t I ℓ(I) ζ 2 h(0) -1 h δ • t-t I ℓ(I)
. . .

ζ n+1 h(0) -1 h δ • t-t I ℓ(I) ζ n+2 - δ 1/2 ζ •(x-x Q ) h(0)ℓ(Q)          
, so that at the center of the cube P L ζ R,δ (x Q , t I ) = ζ and by the Lipschitz property of h at 0 and |ζ| = 1, we get

|P L ζ R,δ (x, t) -ζ| ≤ |h(0)| -1 (Cδ + δ 1/2 ) ((x, t) ∈ R).
This completes the proof of (7.6) and (7.7).

We now define the test functions

b ζ R,ε,δ for ε, δ ∈ (0, 1) by b ζ R,ε,δ := P v ζ R,ε,δ v ζ R,ε,δ := (1 +iεℓM P ) -1 L ζ R,δ
, where ℓ = ℓ(R).

Lemma 7.5. There exists

C > 0 such that for each ζ ∈ C n+2 with |ζ| = 1, each parabolic dyadic cube R ⊂ R n+1 and each ε, δ ∈ (0, 1), (7.8) R n+1 |v ζ R,ε,δ -L ζ R,δ | 2 dx dt ≤ C(εℓ) 2 δ -2 |R|, (7.9)
R n+1 |b ζ R,ε,δ -P L ζ R,δ | 2 dx dt ≤ Cδ -2 |R|, (7.10) -- R (b ζ R,ε,δ -ζ) dx dt ≤ C(ε 1/3 δ -1 + δ 1/2 ), (7.11) R |b ζ R,ε,δ -ζ| 2 dx dt ≤ Cδ -2 |R|, (7.12
)

T (R) |γ λ (x, t)S λ b ζ R,ε,δ (x, t)| 2 dx dt dλ λ ≤ Cε -2 δ -2 |R|.
Proof. Using (1 +iλM P ) -1 -1 = -iλM P (1 +iλM P ) -1 , we have

v ζ R,ε,δ -L ζ R,δ = -iεℓ(1 +iεℓM P ) -1 (M P L ζ R,δ
). The inequality (7.6) for P L ζ R,δ and the uniform boundedness of s → (1 +isM P ) -1 M imply (7.8). Applying P , we see that

b ζ R,ε -P L ζ R,δ = -iεℓP (1 +iεℓM P ) -1 (M P L ζ R,δ
). Again the inequality (7.6) and the boundedness of s → sP (1 +isM P ) -1 M imply (7.9). Next, (7.11) follows directly from (7.7) and (7.9), keeping in mind that δ ∈ (0, 1).

As for (7.12), we first note, using (7.5), (7.6) and the fact that b ζ R,ε,δ = P v ζ R,ε,δ , that it suffices to establish the estimate (7.13)

T (R) |Θ λ b ζ R,ε,δ (x, t)| 2 dx dt dλ λ ≤ Cε -2 δ -1 |R|. Now, Θ λ b ζ R,ε,δ = λM P (1 +λ 2 M P M P ) -1 M P (1 +iεℓM P ) -1 L ζ R,δ = λM P (1 +λ 2 M P M P ) -1 (1 +iεℓM P ) -1 (M P L ζ R,δ ) = (λ/εℓ)(1 +λ 2 M P M P ) -1 εℓM P (1 +iεℓM P ) -1 (M P L ζ R,δ ).
Since (1 +λ 2 M P M P ) -1 and εℓM P (1 +iεℓM P ) -1 are bounded uniformly with respect to λ and εℓ, we have

Θ λ b ζ R,ε,δ 2 ≤ C(λ/εℓ) P L ζ R,δ 2 .
Integrating over λ ∈ (0, ℓ], we obtain (7.13).

It remains to establish (7.10). Let ϕ : R n+1 → [0, 1] be a smooth function which is 1 on -2 , where s ∈ (0, 1) still has to be chosen. We can write

(1 -s)Q × (1 -s 2 )I, supported on R, with ∇ x ϕ ∞ ≤ C(sℓ) -1 and ∂ t ϕ ∞ ≤ C(sℓ)
R (b ζ R,ε,δ -ζ) dx dt = R (b ζ R,ε,δ -P L ζ R,δ ) dx dt + R (P L ζ R,δ -ζ) dx dt.
Due to (7.7), the second integral is bounded by δ 1/2 |R|. Next, we split the first integral as

R ϕP (v ζ R,ε,δ -L ζ R,δ ) dx dt + R (1 -ϕ)(b ζ R,ε,δ -P L ζ R,δ ) =: I + II.
Using (7.9), the properties of ϕ and the Cauchy-Schwarz inequality, we obtain

|II| ≤ Cs 1/2 δ -1 |R|.
The Euclidean norm of I ∈ C n+2 can be computed by

|I| 2 = n+2 j=1 R n+1 (v ζ R,ε,δ -L ζ R,δ )P * -→ ϕ j dx dt 2
with -→ ϕ j = (0, . . . , 0, ϕ, 0 . . . , 0) ∈ L 2 and the non-zero entry sitting at j-th position. Now, P * -→ ϕ j can contain two types of terms: the first one are first order derivatives of ϕ in the x variable, and the second one are D

1/2 t ϕ or H t D 1/2
t ϕ acting on the t variable. Both are functions in L 2 with bounds on the same order of (sℓ) -1 |R| 1/2 . For the first one this follows by construction. For the second one, we obtain the order (s -2 ℓ -2 ) 1/2 |R| 1/2 from the L 2 -bounds on ϕ and ∂ t ϕ and the inequality D

1/2 t ϕ 2 ≤ ϕ 1/2 2 ∂ t ϕ 1/2
2 , which follows directly from Plancherel's theorem. Thus, we can apply (7.8) to the effect that

|I| ≤ C(ε/s)δ -1 |R|.
Hence, choosing s = ε 2/3 , we have shown (7.10).

In conclusion, setting b

ζ R,ε := b ζ R,ε,ε 1/6 , we obtain -- R (b ζ R,ε -ζ) dx dt ≤ Cε 1/12 and -- R |b ζ R,ε -ζ| 2 dx dt ≤ Cε -1/3 .
At this point, we can run the stopping time argument presented in [START_REF] Auscher | On a quadratic estimate related to the Kato conjecture and boundary value problems[END_REF]Sec. 3.4] with this family of test functions instead of b w Q,ε there, using Lemma 7.5 exactly as Lemma 3.10 there, to obtain that γ λ (x, t) is a parabolic Carleson function. As the argument is really line by line the same with only few minor cosmetic changes, we omit details and refer the reader to [START_REF] Auscher | On a quadratic estimate related to the Kato conjecture and boundary value problems[END_REF]. This finishes the proof of Theorem 2.3 modulo the proof of (7.4) that we present below. 7.3. Proof of (7.4). To start the proof we first note, for λ > 0 fixed and (x, t) ∈ R n+1 , that

(Θ λ P λ -γ λ S λ P λ )f (x, t) = Θ λ g --- R g (x, t),
where g = P λ f and R is the only parabolic dyadic cube in λ containing (x, t). Let us write R = Q × I as usual. Using the notation of Proposition 2.11 with

C k (R) = C k (Q × I) when k ≥ 1 and C 0 (R) = 2Q × N I, we obtain (Θ λ P λ -γ λ S λ P λ )f 2 2 = R∈ λ |R| -- R Θ λ g --- R g 2 ≤ R∈ λ |R|   k≥0 -- R Θ λ 1 C k (R) g --- R g 2 1/2   2 . Since Θ λ = 1 2i (R -λ -R λ )
M and as the bounded multiplication operator M commutes with 1 C k (R) , we can apply Proposition 2.11 to continue the estimate with

R∈ λ |R|   k≥0 N -kε -- C k (R) g --- R g 2 1/2   2 R∈ λ k≥1 |R|N -2kε -- 2 k Q×N k I g --- R g 2 ,
where in the second step we have used C k (R) is contained in 2 k+1 Q × N k+1 I and has comparable measure, and shifted the index k. Now, we write

g --- R g = g -- Q g + - Q g -- I g . (7.14)
Then, using Poincaré's inequality [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Lem. 7.12/16] in the x variable,

2 k Q g -- Q g 2 2 2k ℓ(Q) 2 2 k Q |∇ x g| 2 .
Hence, integrating over N k I,

R∈ λ k≥1 |R|N -2kε -- 2 k Q×N k I g -- Q g 2 R∈ λ k≥1 |R|N -2kε 2 2k ℓ(Q) 2 -- 2 k Q×N k I |∇ x g| 2 . Since λ ∼ ℓ(Q) and R∈ λ 1 2 k Q×N k I = 2 kn N k , we can continue by λ 2 k≥1 N -2kε 2 2k R n+1 |∇ x g| 2 λ 2 ∇ x g 2 2 ,
where we have assumed N 2ε > 4 by taking N large enough as we may.

For the second term in (7.14), we use the following lemma as a substitute for Poincaré's inequality. Recall that D α t is the fractional derivative of order α on R defined via the Fourier symbol |τ | α . We write H α (R) for space of all h ∈ L 2 (R) with

D α t h ∈ L 2 (R). Lemma 7.6. Assume h ∈ H α (R) with α ∈ (0, 1/2) and let p, q ∈ [1, ∞) be such that (1-α)p < q ≤ p.
Then for any interval J and any N ≥ 2,

- J h -- J h p ds 1/p ℓ(J) α l≥1 N (α-1)l - N l J |D α t h| q ds 1/q .
The analogous inequality with H t D α t h instead of D α t h on the right-hand side also holds and both inequalities remain valid for α = 1/2 and h ∈ Ḣ1/2 (R).

Proof. We treat the estimates with D α t h first. If α < 1/2, then we can represent h as a classical Riesz potential

h(s) = 1 2Γ(α) cos(απ/2) R D α t h(σ) |s -σ| 1-α dσ (a.e. s ∈ R),
see for example [START_REF] Kilbas | Fractional Integrals and Derivatives. Theory and applications[END_REF]Sec. 12.1]. If α = 1/2, then we use the modified representation obtained in Corollary 3.5. Hence, in any case we have

h(s) -h(s ′ ) = C α R 1 |s -σ| 1-α - 1 |s ′ -σ| 1-α D α t h(σ) dσ (a.e. s, s ′ ∈ J)
for a constant C α depending only on α. Averaging first in s ′ and then in s yields

- J h(s) -- J h p ds 1/p ≤ C α - J R Φ(s, σ)|D α t h(σ)| dσ p ds 1/p , where Φ(s, σ) := - J 1 |s -σ| 1-α - 1 |s ′ -σ| 1-α ds ′ is seen to satisfy |Φ(s, σ)| ℓ(J) α 1 ℓ(J) min s -σ ℓ(J) α-1 , s -σ ℓ(J) α-2 (s ∈ J, σ ∈ R),
using either the triangle inequality (if σ ∈ 2J) or the mean-value theorem (if σ ∈ c (2J)). In particular, R Φ(s, σ) dσ ℓ(J) α . So, writing Φ = Φ 1-1/q Φ 1/q , we can apply Hölder's inequality to the inner integral to find

- J R Φ(s, σ)|D α t h(σ)| dσ p ds 1/p ℓ(J) α-α/q - J R Φ(s, σ)|D α t h(σ)| q dσ p/q ds 1/p
and since p ≥ q by assumption, we obtain from Minkowski's inequality

- J h(s) -- J h p ds 1/p ℓ(J) α-α/q R - J Φ(s, σ) p/q ds q/p |D α t h(σ)| q dσ 1/q . (7.15)
Putting C 0 (J) := N J and C l (J) := N l+1 J \ N l J, l ≥ 1, the pointwise bounds for Φ imply

- J Φ(s, σ) p/q ds q/p ℓ(J) α-1 N (α-2)l (σ ∈ C l (J)),
where it is the term for l = 0 that requires the assumption (α -1)p/q > -1. This being said, the claim follows upon splitting R = ∞ l=0 C l (J) on the right-hand side of (7.15). The argument is exactly the same if we want to let H t D 1/2 t h appear on the right-hand side: Indeed, for α < 1/2 we can again rely on a classical representation [47, Sec. 12.1]:

h(s) = 1 2Γ(α) sin(απ/2) R sgn(s -σ)H t D α t h(σ) |s -σ| 1-α dσ (a.e. s ∈ R).
Likewise, for α = 1/2 we have h = c -I

1/2 t H t (H t D 1/2
t h) for some constant c (Corollary 3.5) and the required integral representation for I 1/2 t H t has been obtained in Proposition 3.4.

Returning to the second term in (7.14), a telescoping sum and Lemma 7.6 applied with p = q = 2 and an arbitrary α ∈ (0, 1/2) yield the bound --

2 k Q×N k I - Q g -- I g 2 ≤ - Q - N k I g -- I g 2 - Q (k + 1) 2 k j=0 - N j I g -- N j I g 2 - Q (k + 1) 2 k j=0 ℓ(N j I) 2α l≥1 N (α-1)l - N l N j I |D α t g| 2 .
So, using R∈ λ 1 Q×N l+j I = N l+j , we obtain

R∈ λ k≥1 |R|N -2kε -- 2 k Q×N k I - Q g -- I g 2 k≥1 N -2kε (k + 1) 2 k j=0 N 2jα ℓ(I) 2α l≥1 N (α-1)l R n+1 |D α t g| 2 .
Choosing α < ε allows us to sum and we finally obtain a bound by ℓ(I) 2α D α t g 2 2 . Since λ 2 ∼ ℓ(I), this completes the proof of (7.4) and thus the proof of Theorem 2.3.

Reverse Hölder estimates and non-tangential maximal estimates

The first part of this section contains the proof of the reverse Hölder estimate for the parabolic conormal differential of reinforced weak solutions alluded to in Theorem 2.14. In the second part we shall derive some consequences of this estimate: the L 2 -bounds for the modified non-tangential maximal function N * and a.e. convergence of Whitney averages for reinforced weak solutions as stated in Theorems 2.12 and 2.13. 8.1. Proof of Theorem 2.14. For convenience we will fix some notation which will be used throughout the proof but not always restated in all intermediate results.

We let Λ × Q × I be a parabolic cylinder of sidelength r > 0 defined by Λ = (λ -r, λ + r), Q = B(x, r) and I = (t -r 2 , t + r 2 ] and we assume 8r < λ. Hence, the parabolic enlargement 8Λ × 8Q × 64I is contained in R n+2 + . We fix a smooth cut-off η : R n+2

+ → [0, 1] with support in 2Λ × 2Q × 4I that is 1 on an enlargement 3 2 λ × 3 2 Q × 9 4 I.
For a reason which will become clear later on, we choose η to have the product form

η(µ, y, s) = η Λ (µ)η Q (y)η I (s),
where η I is symmetric about the midpoint of I. To keep control over implicit constants we may also assume that η is obtained from the one fixed cut-off η 0 for the parabolic cylinder with sidelength r = 1 centered at the origin (λ, x, t) = (0, 0, 0) by the change of variables

η(µ, y, s) = η 0 µ -λ r , y -x r , s -t r 2 .
We denote the translates of an interval J by J k = kℓ(J) + J, k ∈ Z. For the sake of clarity we give a name to the translation sums

(v) := k∈Z 1 1 + |k| 3/2 --- 4Λ×4Q×I k |v|,
where v is a measurable function on R n+2 + . For most of the proof we work with 4Λ × 4Q on the right-hand side and assume only 4r < λ. It is only in the final step where we shall enlarge to 8Λ × 8Q as in the statement of the theorem. All bounds below will depend only on n and the ellipticity constants of A, but for simplicity we keep on using the symbol .

To begin with, let us recall the classical local estimates. Proofs can be found, for example, in Sections 3.2 and 4.1 of [START_REF] Auscher | On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients[END_REF]. (The proof for Caccioppoli's inequality there assumes f = 0 but the argument is the same when f = 0).

Lemma 8.1 (Caccioppoli). Let

Ω ⊂ R n+2 + be open and c > 1. Let Λ × Q × I ⊂ Ω be a parabolic cylinder of sidelength r > 0 such that cΛ × cQ × c 2 I ⊂ Ω. Given f ∈ L 2 loc (Ω), every weak solution u to the inhomogeneous problem ∂ t u -div λ,x A∇ λ,x u = div λ,x f in Ω satisfies --- Λ×Q×I |∇ λ,x u| 2 1 r 2 --- cΛ×cQ×c 2 I |u| 2 + --- cQ×cΛ×c 2 I |f | 2 ,
with an implicit constant depending only on c, n and the ellipticity constants of A.

Lemma 8.2 (Reverse Hölder). Let Ω ⊂ R n+2

+ be open and c > 1. Suppose that u is a weak solution to ∂ t u -div λ,x A∇ λ,x u = 0 in Ω. Then for every parabolic cylinder

Λ × Q × I ⊂ Ω such that cΛ × cQ × c 2 I ⊂ Ω, --- Λ×Q×I |u| 2 1/2 --- cΛ×cQ×c 2 I |u|,
with an implicit constant depending only on c, n and the ellipticity constants of A.

The following variant of Lemma 7.6 will be our instrument for localising in time.

Lemma 8.3. Let p, q ∈ [1, ∞) satisfy p/2 < q ≤ p. Then for each interval J ⊂ R and every h ∈ Ḣ1/2 (R),

- J h -- J h p ds 1/p ℓ(J) k∈Z 1 1 + |k| 3/2 - J k |D 1/2 t h| q ds 1/q . The analogous inequality with H t D 1/2 t h instead of D 1/2
t h on the right-hand side also holds. Proof. The proof follows that of Lemma 7.6 verbatim, the only difference being that in (7.15) we use the uniform estimate

- J Φ(s, σ) p/q ds q/p 1 ℓ(J) 1 1 + |k| 3/2 (σ ∈ J k )
and split R = k∈Z J on the right-hand side.

With these estimates at hand, we can already prove a reverse Hölder inequality of required type for the spatial gradient of reinforced weak solutions.

u -c = u --- 2Λ×2Q u + -- 2Λ×2Q u -c.
Poincaré's inequality in the spatial variables [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Lem. 7.12/16] allows for an estimate 1 r ---

4Λ×4Q×16I u --- 2Λ×2Q u --- 4Λ×4Q×16I |∇ λ,x u| |k|≤8 1 1 + |k| 3/2 --- 4Λ×4Q×I k |∇ λ,x u|.
As for the averages in time, we abbreviate

f (s) := -- 2Λ×2Q u(µ, y, s) dµ dy.
Since the reinforced weak solution u belongs to the class Ḣ1/2 (R; L 2 loc (R n+1 + )), we obtain f ∈ Ḣ1/2 (R) and due to Fubini's theorem,

H t D 1/2 t f (s) = -- 2Λ×2Q H t D 1/2
t u(µ, y, s) dµ dy, (8.1) see also Lemma 3.10. By a telescoping sum 1 r ---

4Λ×4Q×16I -- 2Λ×2Q u -c = 1 ℓ(I) - 16I f -- I f 1 ℓ(16I) - 16I f -- 16I f , so that Lemma 8.3 applies to the effect that 1 r --- 4Λ×4Q×16I -- 2Λ×2Q u -c k∈Z 1 1 + |k| 3/2 - 16I+16kℓ(I) |H t D 1/2 t f | k∈Z 1 1 + |k| 3/2 - I k |H t D 1/2 t f |,
where in the last step we have split 16I + 16kℓ(I) into translates of I. We conclude by plugging in the identity (8.1). Note that we could also decide to let D 1/2 t f appear in (8.1) and our final estimate.

Due to the non-locality of the half-order derivative in time, reverse Hölder estimates for H t D 1/2 t u and D 1/2 t u are much harder to get. A part of the the following argument was inspired by [START_REF] Hofmann | L 2 solvability and representation by caloric layer potentials in time-varying domains[END_REF].

Since H t D 1/2 t
annihilates constants, we can write the average to be estimated as ---

Λ×Q×I |H t D 1/2 t u| 2 = --- Λ×Q×I |H t D 1/2 t (u -c)| 2 ,
where c := ---2Λ×2Q×I u and the same can be done with D

1/2 t in place of H t D 1/2 t .
We treat a local version first. Lemma 8.5. Let u be a reinforced weak solution to (1.1) and let c = ---2Λ×2Q×I u. Then ---

Λ×Q×I |H t D 1/2 t (η(u -c))| 2 1/2 (|∇ λ,x u| + |H t D 1/2 t u|), where H t D 1/2 t can be replaced with D 1/2 t
on either side of the inequality.

Proof. We begin with a parabolic rescaling of R n+1 + , setting u(µ, y, s) = u(rµ, ry, r 2 s) and similarly η, A, as well as Λ = (r -1 λ -1, r -1 λ + 1) and similarly Q, I. Then, 1 r n+1

Λ×Q×I |H t D 1/2 t (η(u -c))| 2 dµ dy ds = Λ× Q× I |H t D 1/2 t ( η( u -c))| 2 dµ dy ds ≤ R H t D 1/2 t ( η( u -c)) 2 L 2 (R n+1 ) ds.
Let F t denote the Fourier transform in t-direction only. For the moment take for granted that all of the subsequent integrals are finite. We shall check this in the further course of the proof anyway. The Hilbert space valued version of Plancherel's theorem [57, Sec. II.5] yields

R H t D 1/2 t ( η( u -c)) 2 L 2 (R n+1 ) ds = 2π R i sgn(σ)|σ| 1/2 F t ( η( u -c)) 2 L 2 (R n+1 ) dσ = 2π R F t ( η( u -c)), |σ|F t ( η( u -c)) 2 dσ
with the duality brackets denoting the W 1,2 (R n+1 )-W -1,2 (R n+1 ) duality. Since these are (isomorphic to) Hilbert spaces as well, we can apply Cauchy-Schwarz' inequality and Plancherel's theorem backwards in order to obtain altogether 1 r n+1

Λ×Q×I |H t D 1/2 t (η(u -c))| 2 dµ dy ds ≤ R η( u -c) 2 W 1,2 (R n+1 ) ds 1/2 R ∂ t ( η( u -c)) 2 W -1,2 (R n+1 ) ds 1/2 . (8.2)
Above we instantly used |i sgn(σ)| ≤ 1 and so the same analysis would apply if we started out with D

1/2 t (η(u -c)). As η is supported in 2Λ × 2Q × 4I, the first integral on the right is bounded by R η( u -c) 2 W 1,2 (R n+1 ) ds 2Λ× 2Q× 4I | u -c| 2 + |∇ λ,x u| 2 dµ dy ds = 1 r n+3 2Λ×2Q×4I |u -c| 2 + r 2 |∇ λ,x u| 2 dµ dy ds. Next, let φ ∈ C ∞ 0 (R n+1 ) and χ ∈ C ∞ 0 (R)
, where φ acts in the λ-and x variables and χ acts in t-direction. Using that u -c is a weak solution to a parabolic problem (1.1) with coefficients A,

R R n+1 η( u -c) • φ dµ dy ∂ t χ ds = 2Λ× 2Q× 4I ( u -c) • ∂ t ( ηφχ) -( u -c) • φχ∂ t η dµ dy ds = 2Λ× 2Q× 4I A∇ λ,x ( u -c) • ∇ λ,x ( ηφχ) -( u -c) • φχ∂ t η dµ dy ds = 4I 2Λ× 2Q A∇ λ,x u • ∇ λ,x ( ηφ) -( u -c) • φ∂ t η dµ dy χ ds.
This implies the bound for the second integral

R ∂ t ( η( u -c)) 2 W -1,2 (R n+1 ) ds 2Λ× 2Q× 4I | u -c| 2 + |∇ λ,x u| 2 dµ dy ds = 1 r n+3 2Λ×2Q×4I |u -c| 2 + r 2 |∇ λ,x u| 2 dµ dy ds.
Returning to (8.2) and clearing the powers of r, we find --- We turn to the corresponding non-local terms, which can be written as ---

Λ×Q×I |H t D 1/2 t (η(u -c))| 2 dµ dy ds --- 2Λ×2Q×4I 1 r 2 |u -c| 2 + |∇ λ,
Λ×Q×I |H t D 1/2 t (1 -η)(u -c)| 2 = --- Λ×Q×I |H t D 1/2 t (1 -η I )(u -c)| 2
and similarly with D

1/2 t in place of H t D 1/2
t , since η Λ and η Q are independent of t and satisfy η Λ η Q = 1 on Λ × Q. Similar to the proof of Lemma 8.4 we separate

(1 -η I )(u -c) = (1 -η I ) u --- 2Λ×2Q u + (1 -η I ) -- 2Λ×2Q u ---- 2Λ×2Q×I u . (8.3)
Both addends on the right-hand side are in Ḣ1/2 (R; L 2 loc (R n+1 + )). To see this, we first note that u is contained therein as a reinforced weak solution. As in the proof of Lemma 8.5 we also have

φu ∈ L 2 (R; W 1,2 (R n+1 )) ∩ H 1 (R; W -1,2 (R n+1 )) (φ ∈ C ∞ 0 (R n+2 + ))
and thus φu ∈ Ḣ1/2 (R; L 2 loc (R n+1 + )). In particular, we obtain η I u ∈ Ḣ1/2 (R; L 2 loc (R n+1 + )) and for the other terms we note that Ḣ1/2 -regularity in time inherits to averages in space in virtue of Fubini's theorem, see also Lemma 3.10.

Next, we shall see that the estimate of the first term on the right-hand side of (8.3) can once more be reduced to Lemma 8.4. Lemma 8.6. Let u be a reinforced weak solution to (1.1). Then ---

Λ×Q×I H t D 1/2 t (1 -η I ) u --- 2Λ×2Q u 2 1/2 (|∇ λ,x u| + |H t D 1/2 t u|), where H t D 1/2 t can be replaced with D 1/2 t
on either side of the inequality.

Proof. We introduce v = (1 -η I )(u ---2Λ×2Q u). By construction v(µ, y, •) is supported in c ( 94 I) and v(µ, y, •) ∈ Ḣ1/2 (R) for almost every µ, y. This justifies to use the representation

H t D 1/2 t v(µ, y, s) = 1 2 √ 2π R sgn(s -σ)
|s -σ| 3/2 v(µ, y, σ) dσ from Corollary 3.7 almost everywhere on Λ × Q × I and we can decompose R = j∈Z I j on the right-hand side to obtain the pointwise estimate

|H t D 1/2 t v(µ, y, s)| 1 r j∈Z\{0} 1 1 + |j| 3/2 - I j |v(µ, y, σ)| dσ.
Due to Minkowski's inequality we can bound the average over Λ × Q × I by ---

Λ×Q×I |H t D 1/2 t v| 2 1/2 1 r -- Λ×Q j∈Z 1 1 + |j| 3/2 - I j |v| 2 1/2 ≤ 1 r j∈Z 1 1 + |j| 3/2 --- Λ×Q×I j |v| 2 1/2 .
The very same argument applies to D 1/2 t v and yields the same bound. Now, by definition of v and Poincaré's inequality in the spatial variables ---

Λ×Q×I j |v| 2 --- 2Λ×2Q×4I j u --- 2Λ×2Q u 2 r 2 --- 2Λ×2Q×4I j |∇ λ,x u| 2 ,
so that altogether, ---

Λ×Q×I |H t D 1/2 t v| 2 1/2 j∈Z 1 1 + |j| 3/2 --- 2Λ×2Q×4I j |∇ λ,x u| 2 1/2 .
The upshot of all this is that each integral above falls under the scope of Lemma 8.4, giving after an index-shift the bound

j∈Z 1 1 + |j| 3/2 k∈Z 1 1 + |k -j| 3/2 --- 4Λ×4Q×I k |∇ λ,x u| + |H t D 1/2 t u| .
It remains to remark that the occurring discrete convolution satisfies

j∈Z 1 1 + |j| 3/2 • 1 1 + |k -j| 3/2 1 1 + |k| 3/2 (k ∈ Z) (8.4)
as can be checked by distinguishing the cases j ≤ |k|/2 and j > |k|/2.

At this stage of the proof we have reduced the matter to two final reverse Hölder estimates, which are the ones for

H t D 1/2 t (1 -η I ) -- 2Λ×2Q u ---- 2Λ×2Q×I u and its counterpart with D 1/2 t in place of H t D 1/2
t . Note that these functions only depend on the t variable. As for the odd half-order derivative H t D 1/2 t , the estimate will be a consequence of the following lemma. It is at this point where the symmetry assumption on the cut-off η I comes into play and in fact the argument would not go through for the even half-order derivative D 1/2 t . Lemma 8.7. Let J be a bounded interval and η : R → [0, 1] be a smooth cut-off function with support in 4J that is identically 1 on 9 4 J. Suppose furthermore that η is symmetric about the midpoint of J.

If f ∈ Ḣ1/2 (R) is such that (1 -η)f ∈ Ḣ1/2 (R), then almost everywhere on J, H t D 1/2 t (1 -η) f -- J f k∈Z 1 1 + |k| 3/2 - J k |D 1/2 t f | + |H t D 1/2 t f | ds.
Proof. After a shift we may assume for simplicity that J is centered at the origin. Let w :=

H t D 1/2 t ((1 -η)(f --J f ))
. Thanks to the support properties of η, Corollary 3.7 yields

w(τ ) = -1 2 √ 2π R sgn(τ -s) |τ -s| 3/2 (1 -η(s)) f (s) -- J f ds (a.e. τ ∈ J) (8.5)
and this defines w as a continuous function on J. For the rest of the proof we will consider w only on J and stick with this particular representative.

Step 1: The bound for w(τ ) -w(0). Since s / ∈ 9 4 J whenever the integrand in (8.5) is non-zero, we obtain from the mean-value theorem

|w(τ ) -w(0)| R |τ | |s| 5/2 |1 -η(s)| f (s) -- J f ds. Decomposing R = k≥0 (J k ∪ J -k ), we find |w(τ ) -w(0)| 1 ℓ(J) ∞ k=1 1 1 + |k| 5/2 - J k f -- J f + - J -k f -- J f . (8.6)
Now, for every k ≥ 0 the telescoping argument

- J k f -- J f ≤ - J k f -- J k f + 6 k j=1 - J j ∪J j-1 f -- J j ∪J j-1 f
brings us in a position where Lemma 8.3 applies in the form

1 ℓ(J) - J k f -- J f m∈Z 1 1 + |m| 3/2 - J k+m |H t D 1/2 t f | + k j=1 - J j+m ∪J j-1+m |H t D 1/2 t f | ≤ 2 m∈Z k j=0 1 1 + |m -j| 3/2 - Jm |H t D 1/2 t f |,
where the second step follows from a simple regrouping of terms. For the corresponding average over J -k on the left-hand side we obtain an analogous bound with an inner sum ranging over -k ≤ j ≤ 0, so that altogether we sum over |j| ≤ k. These estimates inserted on the right-hand side of (8.6) lead us to

|w(τ ) -w(0)| m∈Z j∈Z k≥|j| 1 1 + |k| 5/2 1 1 + |m -j| 3/2 - Jm |H t D 1/2 t f |.
By a Riemann sum argument on the sum over k and (8.4), j∈Z k≥|j|

1 1 + |k| 5/2 1 1 + |m -j| 3/2 j∈Z 1 1 + |j| 3/2 1 1 + |m -j| 3/2 1 1 + |m| 3/2 , giving the required bound |w(τ ) -w(0)| m∈Z 1 1 + |m| 3/2 - Jm |H t D 1/2 t f |.
Step 2: The bound for w(0). The 'kernel' in the formula (8.5) for w(0) is odd since η is an even function. Hence, we can omit the average -J f and for a later purpose we write

w(0) = 1 2 √ 2π lim S→∞ (-S,S)
sgn(s) |s| 3/2 (1 -η(s))f (s) ds. Due to Corollary 3.5 we can represent f as a potential

f (s) = c + 1 √ 2π (-1,1) g(ρ) |s -ρ| 1/2 dρ + c (-1,1) g(ρ) |s -ρ| 1/2 - g(ρ) |ρ| 1/2 dρ with g = D 1/2
t f and c a constant. Since these integrals converge absolutely for almost every s ∈ R, see Section 3, we can plug in this representation into the one for w(0) and apply Fubini's theorem.

During this procedure the terms c and g(ρ)|ρ| -1/2 depending not on s vanish again by oddness. So, after a substitution s = ρσ,

w(0) = lim S→∞ 1 4π R |σ|≤ S |ρ| sgn(σ) |σ| 3/2 • 1 -η J (σρ) |1 -σ| 1/2 dσ g(ρ) ρ dρ. (8.7)
Let us inspect the inner integral for fixed S ≥ ℓ(J): In the case |ρ| < ℓ(J) we can simply use the support properties of η to bound this in absolute value by

9ℓ(J ) 8|ρ| ≤|σ| 1 |σ| 3/2 • 1 |1 -σ| 1/2 dσ |ρ| ℓ(J) ,
where we have used the estimate |σ| ≤ 9|1 -σ| on the domain of integration in the second step.

If ℓ(J) < |ρ| < 8ℓ(J), then the estimate above remains true for the simple reason that both sides are comparable to absolute constants. If finally |ρ| ≥ 8ℓ(J), then we additionally make use of the symmetry of η to rewrite the integral under consideration as

|σ|≤ S |ρ| sgn(σ) |σ| 3/2 1 |1 -σ| 1/2 -1 dσ - |σ|≤ 4ℓ(J ) |ρ| sgn(σ) |σ| 3/2 1 |1 -σ| 1/2 -1 η(σρ) dσ.
The first term is bounded uniformly in S and ρ since ||1 -σ| -1/2 -1| |σ| for |σ| small, and in the limit for S → ∞ it converges to the value C ∈ R of the corresponding integral over the real line. Similarly, the second term can be controlled by ℓ(J) 1/2 /|ρ| 1/2 since we have |σ| ≤ 1/2 in the domain of integration. This justifies to use Lebesgue's theorem when passing to the limit in S in (8.7). As a result,

|w(0)| - 8J |g(ρ)| dρ + C c (8J) g(ρ) ρ dρ + ℓ(J) c (8J) |g(ρ)| |ρ| 3/2 dρ. (8.8)
The truncated Hilbert transform in the middle term can be handled similar to the classical proof of Cotlar's lemma but for the reader's convenience we repeat the short argument. So, we let g 1 be equal to g on 8J and zero otherwise. Then for any 0 < ε < 2ℓ(J) and s ∈ J,

c (8J) g(ρ) ρ dρ = c (8J) 1 ρ - 1 s -ρ g(ρ) dρ + |s-ρ|>ε g(ρ) s -ρ dρ - |s-ρ|>ε g 1 (ρ) s -ρ dρ.
The mean-value theorem applies to the first term so that in the limit ε → 0,

c (8J) g(ρ) ρ dρ c (8J) ℓ(J)|g(ρ)| |ρ| 2 dρ + |H t g(s)| + |H t g 1 (s)| (a.e. s ∈ J).
In view of Tchebychev's inequality and the weak-type bound for the Hilbert transform of

g 1 ∈ L 1 (R)
we can find some s ∈ J such that simultaneously

|H t g(s)| - 8J |H t g| and |H t g 1 (s)| - 8J |g|.
Returning to (8.8) and plugging in g = D

1/2 t f , |w(0)| ℓ(J) c (8J) |D 1/2 t f | dρ |ρ| 3/2 + - 8J |H t D 1/2 t f | + |D 1/2 t f | dρ.
Now, the usual split up into translates of J gives

|w(0)| m∈Z 1 1 + |m| 3/2 - Jm |D 1/2 t f | + |H t D 1/2 t f |,
which, together with the outcome of Step 1, establishes the required bound even uniformly on J. The claim follows by averaging over τ ∈ J.

Applying this lemma to f = --2Λ×2Q u and η = η I on J = I completes the reverse Hölder estimate for H t D 1/2 t u with an enlargement to 4Λ × 4Q in space on the right-hand side: Corollary 8.8. Let u be a reinforced weak solution to (1.1). Then ---

Λ×Q×I H t D 1/2 t (1 -η I ) -- 2Λ×2Q u ---- 2Λ×2Q×I u 2 1/2 (|D 1/2 t u| + |H t D 1/2 t u|).
The missing piece in the proof of Theorem 2.14 is the corresponding estimate with D 1/2 t on the lefthand side. We call the reader's attention to the fact that this will note be achieved by an argument for functions on R but by resorting to the full reverse Hölder estimate for ---2Λ×2Q×4I |H t D 1/2 t u| 2 that we have just completed. Since we are using this estimate already for the enlargement 2Λ × 2Q in space, we finally obtain an enlargement 8Λ × 8Q on the right-hand side as stated in Theorem 2.14. Lemma 8.9. Let u be a reinforced weak solution to (1.1). Then ---

Λ×Q×I D 1/2 t (1 -η I ) -- 2Λ×2Q u ---- 2Λ×2Q×I u 2 1/2 k∈Z 1 1 + |k| 3/2 --- 8Λ×8Q×I k |∇ λ,x u| + |D 1/2 t u| + |H t D 1/2 t u| dλ dx dt.
Proof. Let f := --2Λ×2Q u and h := f --I f . Rewriting the left-hand side of the estimate in question by means these two functions reveals that it suffices to prove

- I |D 1/2 t (η I h)| 2 1/2 + - I |D 1/2 t f | 2 1/2 k∈Z 1 1 + |k| 3/2 --- 8Λ×8Q×I k |∇ λ,x u| + |D 1/2 t u| + |H t D 1/2 t u| dλ dx dt, (8.9)
where we also used D

1/2 t h = D 1/2
t f as the half-order derivative annihilates constants. We treat both terms separately.

Step 1: The first term in (8.9). Since H 2 t = -1 as bounded operators on L 2 (R),

- I |D 1/2 t (η I h)| 2 1/2 1 ℓ(I) R |H t D 1/2 t (η I h)| 2 1/2 .
Let η 4I have the same properties as η I but for the interval 4I instead of I. In particular, η 4I is supported in 16I and identically 1 on 9I, which contains the support of η I . Using the relation

1 -η 4I = (1 -η I )(1 -η 4I
) in the second line of the following computation, we can use commutators to expand

H t D 1/2 t (η I h) = H t D 1/2 t h -H t D 1/2 t ((1 -η I )(1 -η 4I + η 4I )h) = H t D 1/2 t h -H t D 1/2 t ((1 -η 4I )h) -[H t D 1/2 t , 1 -η I ](η 4I h) -(1 -η I )H t D 1/2 t (η 4I h) = η I H t D 1/2 t f -η I H t D 1/2 t ((1 -η 4I )h) + [H t D 1/2 t , η I ](η 4I h).
Pick some p ∈ (1, 2). From the kernel bounds (6.4) and Young's inequality we obtain that the commutator

[H t D 1/2 t , η I ] is bounded L p (R) → L 2 (
R) with norm controlled by ℓ(I) -1/p . Thus, the various support assumptions and the definition of h yield

- I |D 1/2 t (η I h)| 2 1/2 - 4I |H t D 1/2 t f | 2 1/2 + - 4I H t D 1/2 t (1 -η 4I ) f -- I f 2 1/2 + 1 ℓ(I) - 16I f -- I f p 1/p .
Plugging in the definition of f , the first term is bounded by the L 2 average of H t D

1/2 t u on 2Λ×2Q×4I, which can be controlled as desired since we have already completed the reverse Hölder estimate stated in Theorem 2.14 for H t D 1/2 t u on the left-hand side with an enlargement factor of 4 in the spatial directions. The second term can be controlled by Lemma 8.7 applied with η = η 4I on J = 4I: Indeed, we take the average of f over 1 4 J instead of J here but the only change in the argument would be to use translates of 1 4 J in (8.6), too. Finally, for the third term we use a telescopic sum and then Lemma 8.3 with q = 1 to obtain

1 ℓ(I) - 16I f -- I f p 1/p 1 ℓ(I) - 16I f -- 16I f p 1/p k∈Z 1 1 + |k| 3/2 - (16I) k |D 1/2 t f |,
and so all we have to do is to decompose the translates of 16I into those of I. We note that this estimate required p < 2. This completes the treatment of the first term in (8.9).

Step 2: The second term in (8.9). Using again H 2 t = -1 and the L 2 boundedness of the Hilbert transform, we find

- I |D 1/2 t f | 2 1/2 - 4I |H t D 1/2 t f | 2 1/2 + - I |H t (1c (4I) H t D 1/2 t f )| 2 1/2 .
From Step 1 we know how to handle the first integral on the right. As for the second one, we put g := H t D 1/2 t f and obtain from the singular integral representation of the Hilbert transform

H t (1c (4I) H t D 1/2 t f )(t) = c (4I) g(ρ) ρ -t dρ (a.e. t ∈ I).
Now, we can follow the Cotlar-type argument succeeding (8.8) to deduce the uniform bound

|H t (1c (4I) H t D 1/2 t f )(t)| m∈Z 1 1 + |m| 3/2 - Im |D 1/2 t f | + |H t D 1/2 t f | (a.e. t ∈ I).
We conclude by averaging over t ∈ I and plugging in the definition of f . Remark 8.10. As the reader may have noticed, we could have also written Lemma 8.9 as a statement on real functions, more in the spirit of Lemma 8.7. In fact, the proof above has revealed the following estimate, using the notation and assumptions of Lemma 8.7:

- J D 1/2 t (1 -η) f -- J f 2 1/2 - 4J |H t D 1/2 t f | 2 1/2 + k∈Z 1 1 + |k| 3/2 - J k (|D 1/2 t f | + |H t D 1/2 t f |).
8.2. Proof of Theorem 2.12. Again Λ × Q × I denotes a parabolic cylinder in R n+2 + but to provide for the non-tangential character of the estimates in question we shall now assume its sidelength to be comparable to its distance to the boundary, say Λ = (7λ, 9λ), Q = B(x, λ), and I = (t -λ 2 , t + λ 2 ), for convenience.

Recall that for any measurable function F on the upper half-space R n+2 + the non-tangential maximal function is given by

N * F (x, t) = sup λ>0 --- Λ×Q×I |F (µ, y, s)| 2 dµ dy ds 1/2
, where the supremum is taken over the family of parabolic cubes described above.

It is well-known that the non-tangential maximal function can be controlled by the corresponding square function. For convenience and later reference we include the short argument proving Proof. Spelling out the norm N * F 2 2 gives up to a multiplicative constant depending only on n,

R n+1 sup λ>0 R n+2 + 1 (8µ/9,8µ/7) (λ)1 B(y,λ) (x)1 (s-λ 2 ,s+λ 2 ) (t) λ n+3 |F (µ, y, s)| 2 dµ dy ds dx dt,
where 1 E denotes the characteristic function of a set E. Hence, the lower estimate required in the lemma simply follows on pulling the supremum over λ > 0 outside the double integral and applying Tonelli's theorem. Likewise, for the upper estimate we use

1 λ n+3 • 1 (8/9µ,8/7µ) (λ)1 B(y,λ) (x)1 (s-λ 2 ,s+λ 2 ) (t) ≤ 2 n+3 µ n+3 • 1 B(y,2µ) (x)1 (s-4µ 2 ,s+4µ 2 ) (t),
pull the supremum over λ > 0 into the triple integral, and apply Tonelli's theorem. Now, we apply the first part of the lemma to 1 (0,9ε/8) (µ)F (µ, y, s) in place of F , where ε > 0. It follows

R n+1 sup 0<λ<ε --- Λ×Q×I |F (µ, y, s)| 2 dµ dy ds dx dt ε 0 F µ 2 2 dµ µ ,
proving the a.e. convergence of averages claimed in the second part of the lemma.

Let us recall that the backward parabolic conormal differential of reinforced weak solutions is defined by

D A u(λ, x, t) =    ∂ ν A u(λ, x, t) ∇ x u(λ, x, t) D 1/2 t u(λ, x, t)    .
Using this notation, we can formulate a short hand variant of Theorem 2.14 that is better suited to maximal functions. Lemma 8.12. For any reinforced weak solution u to (1.1) and any parabolic cube

Λ × Q × I with 8Λ ⊂ (0, ∞), --- Λ×Q×I |D A u| 2 dµ dy ds 1/2 m≥0 2 -m --- 8Λ×8Q×4 m I |D A u| + | D A u| dµ dy ds.
Proof. Observe that the right hand sides of the reverse Hölder inequalities in Theorem 2.14 involve integrals of |D A u| + | D A u|. To convert the translates on intervals into the formulation above, use that for any measurable function f on the real line

k∈Z 1 1 + |k| 3/2 - I k |f | ≤ - I |f | + m≥0 4 -3m/2 4 m ≤|k|<4 m+1 - I k |f | ≤ m≥0 2 -m - 4 m I |f |.
Note that if we introduce the operator matrices

H θ t =    1 0 0 0 1 0 0 0 H t    and P =    0 div x -H t D 1/2 t -∇ x 0 0 -D 1/2 t 0 0    , then D A u = H θ t D A u, H θ t commutes
with M and intertwines P and P with P H θ t = H θ t P , so that the operator P M has the same operator theoretic properties as P M . Also we have the equation ∂ λ D A u + P M D A u = 0 and the link with the parabolic equation guarantees the off-diagonal estimates for P M and M P as well. Alternatively, this can be seen by a verbatim repetition of the proof of Proposition 2.11. Thus, in the following we may argue with D A u as we do with D A u and we shall only provide the details involving D A u.

After these preparations we are ready to prove Theorem 2.12.

Step 1: The non-tangential estimate. Let h ∈ R(P M ) and let F = Sh be its semigroup extension F (µ, y, s) = e -µ[P M ] h(y, s).

The lower bound for N * f follows from Lemma 8.11 and strong continuity of the semigroup:

N * F 2 2 lim λ→0 - 9 8 λ 7 8 λ e -µ[P M ] h 2 2 dµ = h 2 2 .
We turn to the interesting upper estimate. In view of (2.12) we have spectral decompositions h = h + + h -, where h ± = χ ± (P M )h ∈ H ± (P M ), and thus F = F + + F -, where

F ± = Sh ± . Since F + = C + 0 h + solves ∂ λ F + + P M F + = 0 in the weak sense in R + , Theorem 2.
2 yields a representation F + = D A u for some reinforced weak solution u to (1.1). Similarly, for λ < 0, the reflection

G - λ = (F -) -λ = (C - 0 h -) λ solves ∂ λ G -+ P M G -= 0 on
R -so that the similar analysis can be performed. So from now on, we may as well assume h = h + to simplify matters and that F = C + 0 h = D A u for some reinforced weak solution u to (1.1). We fix (x, t) ∈ R n+1 and take Λ × Q × I as above. Setting F = D A u, Lemma 8.12 asserts

--- Λ×Q×I |F | 2 dµ dy ds 1/2 m≥0 2 -m --- 8Λ×8Q×4 m I |F | + | F | dµ dy ds.
To control the right hand side, as mentioned above, it is enough to obtain bounds for the terms involving F using the L 2 estimate on h; those for F would lead to the same estimate with P replaced by P and h replaced by

H θ t h, which is in L 2 as well. We introduce R µ := (1 + iµP M ) -1 and split F µ = (F µ -R µ h) + R µ h.
After an application of the Cauchy-Schwarz inequality, the same argument as in the proof of Lemma 8.11 yields

R n+1 sup λ>0 m≥0 2 -m --- 8Λ×8Q×4 m I |F µ -R µ h| dµ dy ds 2 dx dt m≥0 2 -m R n+1 sup λ>0 --- 8Λ×8Q×4 m I |F µ -R µ h| 2 dµ dy ds 2 dx dt m≥0 2 -m ∞ 0 F µ -R µ h 2 2 dµ µ = ∞ 0 F µ -R µ h 2 2 dµ µ .
Now, we can use the functional calculus for P M to write

F µ -R µ h = ψ(µP M )h with ψ(z) = e -[z] -(1 + iz) -1 .
As ψ is holomorphic on some double sector containing the spectrum of P M and has polynomial decay at 0 and ∞, the quadratic estimates for P M obtain in Theorem 2.3 yield

∞ 0 F µ -R µ h 2 2 dµ µ = ∞ 0 ψ(µP M )h 2 2 dµ µ ∼ h 2 2 .
The remaining term is

m≥0 2 -m --- 8Λ×8Q×4 m I
|R µ h| dµ dy ds. (8.10) Let q ∈ (1, 2) be such that we have Proposition 2.11 on L q off-diagonal decay for R µ at our disposal. With N 0 ∈ N as provided by this proposition we define for J ⊂ R and B ⊂ R n the annuli

C 0 (B × J) = 2B × N 0 J, C k (B × J) = (2 k+1 B × N k+1 0 J) \ (2 k B × N k 0 J)
and we obtain an estimate --

8Q×4 m I |R µ h| dy ds ≤ -- 8Q×4 m I |R µ h| q dy ds 1/q k≥0 N -εk 0 -- C k (8Q×4 m I) |h| q dy ds 1/q k≥0 N -εk 0 - 2 k+4 Q - N k+1 0 4 m I |h| q dy ds 1/q
for some ε > 0 and implicit constants independent of m ≥ 0 and µ ∈ 8Λ. By means of the q-adapted maximal operators acting separately on each variable by

M q t f (x, t) = sup J∋t - J |f (x, s)| q ds 1/q and M q x f (x, t) = sup B∋x - B |f (y, t)| q dy 1/q
, where J ⊂ R and B ⊂ R n denote intervals and balls, respectively, the ongoing estimate can be completed as --

8Q×4 m I |R µ h| dy ds M q x (M q t h)(x, t) (m ≥ 0, µ ∈ 8Λ).
Since q < 2, the q-adapted maximal operators are bounded on L 2 . Whence, the term defined in (8.10) has non-tangential maximal function bounds

R n+1 sup λ>0 m≥0 2 -m --- 8Λ×8Q×4 m I |R µ h| dµ dy ds 2 dx dt M q x M q t h 2 2 h 2 2 .
Altogether, we obtain

N * (F ) 2 2 R n+1 sup λ>0 m≥0
2 -m ---

8Λ×8Q×4 m I |F (µ, y, s)| dµ dy ds 2 dx dt h 2 2 (8.11)
as required.

Step 2: Almost everywhere convergence of averages. In order to complete the proof of the theorem we have yet to show that for any h ∈ R(P M ) there is convergence

lim λ→0 --- Λ×Q×I |Sh(µ, y, s) -h(x, t)| 2 dµ dy ds = 0 (a.e. (x, t) ∈ R n+1 ). (8.12)
As we have the L 2 estimate on the maximal function, it suffices to prove this for h in some dense subset of R(P M ).

Lemma 8.13. There exists δ 0 > 0 depending only on dimension and the ellipticity constants of

A such that if | 1 p -1 2 | < δ 0 , then {h ∈ R(P M ) ∩ D(P M ) : P M h ∈ L p (R n+1 ; C n+2 )} is dense in R(P M ).
Proof. We take δ 0 as in Lemma 2.10, thereby guaranteeing that all resolvents

R µ = (1 + iµP M ) -1 , µ ∈ R, map L p ∩ L 2 into itself where L p = L p (R n+1 ; C n+2 ). For m ∈ N define T m ∈ L(L 2 ), T m h = imR im P M R i/m h.
Since P M is densely defined and bisectorial, the T m are uniformly bounded with respect to m and on R(P M ) they converge strongly to the identity, see [START_REF] Egert | On Kato's conjecture and mixed boundary conditions[END_REF]Prop. 3.2.2] for an explicit statement. Now, let h ∈ R(P M ). We can choose functions h m ∈ L p ∩ L 2 such that h m → h in L 2 . Then also T m h m → h in L 2 . However, by definition we have T m h m ∈ R(P M ) ∩ D(P M ) and expanding

P M T m h m = imh m + R i/m h m + m 2 R im h m -imR im R i/m h m , we read off P M T m h m ∈ L p ∩ L 2 as desired.
We complete the proof of the theorem by demonstrating (8.12) if h ∈ R(P M ) ∩ D(P M ) satisfies P M h ∈ L p (R n+1 ; C n+2 ) for some p > 2. As in Step 1 we let R µ = (1 + iµP M ) -1 and ψ(z) = e -[z] -(1 + iz) -1 . We start out by estimating Thanks to Lemma 8.11 and the quadratic estimates for P M , the first term vanishes for a.e. (x, t) ∈ R n+1 in the limit λ → 0. The third term vanishes for every (parabolic) Lebesgue point of h ∈ L 2 (R n+1 ), that is, for a.e. (x, t) ∈ R n+1 . Since h ∈ D(P M ), the middle term can be rewritten as

--- Λ×Q×I |µR µ P M h(y, s)| 2 dµ dy ds λ 2 --- Λ×Q×I |R µ P M h(y, s)| 2 dµ dy ds λ 2 M 2 x M 2 t (P M h)(x, t) 2 ,
where the second step follows from the L 2 off-diagonal estimates for R µ similar to the estimate of (8.10). Hence, these terms vanish in the limit λ → 0 if only the maximal function M 2 x M 2 t (P M h)(x, t) is finite. This happens to be almost everywhere thanks to our additional assumption on h: In fact, M 2 t and M 2 x are bounded on L p (R n+1 ) since p > 2 and thus

P M h ∈ L p (R n+1 ; C n+2 ) implies M 2 x M 2 t (P M h) ∈ L p (R n+1
). Remark 8.14. Although it is not needed for our applications, we observe that the almost everywhere convergence (8.12) holds also for h ∈ N(P M ). In this case, as (Sh) λ = h for all λ > 0, we can control the averages by the parabolic Hardy-Littlewood maximal operator M 2 (x,t) h(x, t), use its weak-type (2,2) and Lebesgue differentiation. 8.3. Proof of Theorem 2.13. The estimates for the semigroup extension with respect to M P are easily obtained from those for P M .

Since M : R(P M ) → R(M P ) is an isomorphism, it is a matter of fact for the functional calculus of these type of operators that there is an intertwining relation

e -µ[M P ] h = M e -µ[P M ] M -1 h (h ∈ R(M P )). (8.13)
In fact, we can directly verify (λ -M P ) -1 M = M (λ -P M ) -1 for λ in a suitable double sector in the complex plane and extend this relation to the Cauchy integral defining the semigroups above.

Since M is a bounded multiplication operator, it acts boundedly on L 2 (Λ × Q × I; C n+2 ) for every Whitney region Λ × Q × I. Let now h ∈ R(M P ) = M R(P ). Since M -1 h ∈ R(P M ) falls under the scope of Theorem 2.12 and as M -1 : R(M P ) → L 2 (R n+1 ; C n+2 ) is bounded (though not necessarily a multiplication operator), the upper N * -bound and the almost everywhere convergence of Whitney averages for the M P -semigroup follow from (8.13) and the corresponding result for the P M -semigroup. Finally the lower bound N * (e -•[M P ] h) 2 h 2 follows from Lemma 8.11 and the strong continuity of the semigroup for M P just as in the case of P M .

Duality results for λ-independent operators

In this section we prove Theorem 2.21 and the abstract Green's formula announced in Proposition 2.22. The main step is the following. We use the notation introduced in Section 2.12. The relevant Sobolev spaces have been introduced in Section 2.7. Recall that we gather -and θ-components of vectors in a single component denoted r. ), (( Ḣs P ) ⊥ , ( Ḣ-s-1 P * ) ⊥ ) and (( Ḣs P ) r , ( Ḣ-s-1 P * ) r ) are pairs of orthogonal spaces. Remark 9.2. In order to make the statement of Lemma 9.1 meaningful, we identify spaces of type X ⊥ and X r as subspaces of X via X ⊥ ∼ = X ⊥ ⊕ {0} and X r ∼ = {0} ⊕ X r .

Proof. With N as defined in (2.28) and P * the adjoint of P in the canonical L 2 (R n+1 ; C n+2 ) inner product, N and P * anti-commute: P * N = -N P * . So, with M = N M * N = N M * N -1 , we obtain that the adjoint of P M in the complex form

(f, N g) = R n+1 f • N g dx dt in L 2 (R n+1 ; C n+2
) is -M P * . Thus, the dual of the Sobolev space Ḣs P M in the extension of this duality is Ḣ-s M P * . Next, the formal relation P * ( M P * ) = (P * M )P * allows one to show that P * extends to an isomorphism between Ḣ-s M P * and Ḣ-s-1

P * M
. Therefore ( Ḣs P M , Ḣ-s-1

P * M
) is pair of dual spaces for the duality form β(f, g) = (f, N P * -1 g).

As this very argument also applies to M = 1, we see that ( Ḣs P , Ḣ-s-1 P * ) are dual spaces for the same duality form. As P and P * swap the ⊥-and r-components we have the announced duality for the ⊥and r-spaces as well as the orthogonality. And from the anti-commutation of N and P * , we derive the announced duality and orthogonality for the spectral spaces.

The proof of Theorem 2.21 now follows by applying abstract results. We shall make the same identifications as in Remark 9.2.

Proof of Theorem 2.21. We have a pair of projectors (χ + (P M ), χ -(P M )) associated with the splitting Ḣs P = Ḣs,+ P M ⊕ Ḣs,-P M and due to the identifications made above, the component maps (N ⊥ , N r ) become the projectors associated with Ḣs P = ( Ḣs P ) ⊥ ⊕ ( Ḣs P ) r . In the duality β their dual pairs are (χ -(P * M ), χ + (P * M )) and (N r , N ⊥ ) corresponding to the dual splittings Ḣ-s-1 . Applying an abstract result on pairs of projections [18, Lem. 13.2], we have that invertibility of the latter is exactly equivalent to invertibility of N ⊥ : Ḣ-s-1,+

P * = Ḣ-s-1,- P * M ⊕ Ḣ-s-
P * M → ( Ḣ-s-1 P * ) ⊥ .
By the analogue of Theorem 2.15 for the backward equation, this in turn is equivalent to well-posedness of (N ) L * E -1-s . The regularity problem can be handled in the same manner. Finally, preservation of compatible well-posedness can be obtained unravelling this argument.

A proof of the abstract Green's formula can be given by an approximation procedure as in [START_REF] Auscher | Representation and uniqueness for boundary value elliptic problems via first order systems[END_REF]Thm. 1.7]. There, it was mentioned that one can also use the simultaneous duality above and this is the approach that we shall present here. . We have (h, N P * -1 g) = 0 by the orthogonality relation and we claim that

(h, N P * -1 g) = ∂ ν A u| λ=0 , w| λ=0 -u| λ=0 , ∂ ν A * w| λ=0 .
Indeed, ⊥ and r-component of h and N P * -1 g can be computed in appropriate dual spaces described in the argument below and we write (h, N P * -1 g) = h ⊥ , (N P * -1 g) ⊥ + h r , (N P * -1 g) r . First,

h ⊥ = ∂ ν A u| λ=0 ∈ ( Ḣs P ) ⊥ = Ḣs/2 ∂t-
D A S λ-µ f µ dµ = + e -εP M χ + (P M )F λ-ε + e εP M χ -(P M )F λ+ε -e -RP M χ + (P M )F λ-R -e RP M χ -(P M )F λ+R .
Applying -P -1 to both sides, we have an equality in L 2 (R; Ḣ1 P ). Finally, taking the ⊥-component,

we obtain in L 2 (R; Ḣ1/2 ∂t-∆x ) = E ′ , ε<|λ-µ|<R S λ-µ f µ dµ = -(P -1 e -εP M χ + (P M )F λ-ε ) ⊥ -(P -1 e εP M χ -(P M )F λ+ε ) ⊥ + (P -1 e -RP M χ + (P M )F λ-R ) ⊥ + (P -1 e RP M χ -(P M )F λ+R ) ⊥ .
This defines uniformly bounded operators from Ḣ-1/2 ∂t-∆ λ,x to E ′ . Next, we take strong limits in E ′ . The terms corresponding to R → ∞ tend to 0 using the bounded holomorphic functional calculus of P M and that F λ±R tend to 0 in L 2 (R; Ḣ1 P ). The terms corresponding to ε → 0, converge to

-(P -1 χ + (P M )F ) ⊥ -(P -1 χ -(P M )F ) ⊥ = -(P -1 F ) ⊥ = u.
This proves the limit in the statement.

Proof of Theorem 2.30. To L there correspond P and M = 1 0 0 A with A = A 0 0 1 as in Section 5. Let f ∈ Ḣ-1/2 ∂t-∆x . As f 0 ∈ Ḣ-1 P , we find, using the functional calculus of P M on Ḣ-1

P , R ε S λ f dλ = (P M P ) -1 (e -RP M -e -εP M )χ + (P M ) f 0 ⊥ .
Similarly, we obtain

-ε -R S λ f dλ = -(P M P ) -1 (e εP M -e RP M )χ -(P M ) f 0 ⊥ .
Here, we have used that by construction P M P is an isomorphism from Ḣ1 P onto Ḣ-1 P . Hence, taking strong limits for the semigroup on each spectral space of Ḣ-1 P , and adding terms we obtain lim ε→0, R→∞ ε≤|λ|≤R

S λ f dλ = -(P M P ) -1 (χ + (P M ) + χ -(P M )) f 0 ⊥ = -(P M P ) -1 f 0 ⊥ .
From the form of L we see that P M P is block diagonal and the scalar block is precisely ∂ tdiv x A(x, t)∇ x = L. This gives us the conclusion.

12. Well-posedness results for λ-independent coefficients

In this section we shall prove our well-posedness results formulated in Theorem 2.32 and obtain the resolution of the Kato problem for parabolic operators. We also supply proofs of the related Propositions 2.20 and 2.34. We shall need again the Sobolev spaces introduced in Section 2.7.

12.1. The proof of Theorem 2.6. For A (x, t) any bounded and uniformly elliptic (n × n)-matrix we let

M :=    1 0 0 0 A 0 0 0 1    , P M =    0 div x A -D 1/2 t -∇ x 0 0 -H t D 1/2 t 0 0   
As P M has a bounded holomorphic functional calculus on R(P ) by Theorem 2.3, the operator sgn(P M ) is a bounded involution on this space. Therefore P M and [P M ] = sgn(P M )P M share the same domain with P M h 2 ∼ [P M ]h 2 . But [P M ] = (P M ) 2 by definition and

(P M ) 2 =    L 0 0 0 -∇ x div x A ∇ x D 1/2 t 0 -H t D 1/2 t div x A ∂ t    .
Specialising to the first component, we see that the domain of

√ L is H 1/2 (R; L 2 (R n ))∩L 2 (R; W 1,2 (R n )) together with the homogeneous estimate √ L 2 ∼ ∇ x f 2 + H t D 1/2 t f 2 = ∇ x f 2 + D 1/2
t f 2 This solves the parabolic Kato problem. 12.2. The block case (i). More generally, we assume that the coefficients A(x, t) are in block form

A := A ⊥⊥ 0 0 A .
Since results for this case will be re-used in the context of more general coefficients, we shall write A(x, t) = A b (x, t) and M = M b to avoid confusion. As usual we concatenate -and θ-components of vectors to a single component denoted r. Similarly, we represent operators acting on them as (2 × 2)-matrices. For example, M = M b can be written in block form From Section 2.7 we know that the bounded functional calculus for P M b extends to the Sobolev spaces Ḣs P M b = Ḣs P , -1 ≤ s ≤ 0, and by density of R(P ) ∩ Ḣs P in Ḣs P , the formula (12.1) extends, too. In particular, 1 ± s ⊥⊥ (P M b ) and 1 ± s rr (P M b ) all act as the identity on the respective components of Ḣs P . Thus, Proposition 2.25 yields compatible well-posedness of (R) L Es and (N ) L Es .

M b = A -1 ⊥⊥ 0 0 M rr ,
12.3. The triangular cases (ii) and (iv). We adopt notation from the previously discussed block case. It suffices to prove well-posedness at regularity s = 0: Indeed, in view of compatible wellposedness for block matrices, we can perturb along A := {τ 1 +(1 -τ )A : 0 ≤ τ ≤ 1} to the heat equation and obtain compatible well-posedness at regularity s = 0 from Proposition 2.20. Then the intermediate cases follow from interpolation with (compatible) well-posedness in the energy class by means of Theorem 2.18.

Appealing to Theorem 2.23, we shall check invertibility of s ⊥r (P M ) and s r⊥ (P M ) for A lowerand upper-triangular, respectively. The absence of one of the entries of A will allow us to transfer invertibility from regularity s = -1/2 to s = 0. In the context of elliptic systems this strategy has been put into place in [START_REF] Auscher | On L 2 solvability of BVPs for elliptic systems[END_REF]. The argument presented below constitutes a simplification even in the elliptic case. In the following we utilize the block coefficients A b := A ⊥⊥ 0 0 A and the corresponding multiplication operator M b . In the case of triangular matrices A we will see that the Sobolev spaces adapted to P M b carry substantial information also on P M . Note that

P M b =    0 div x A -D 1/2 t -∇ x A -1 ⊥⊥ 0 0 -H t D 1/2 t A -1 ⊥⊥ 0 0   
acts independently on ⊥-and r-components and hence so does The fact that we use these operators associated with the coefficients A b different from A reflects that well-posedness, in contrast to a priori representations, cannot be comprised only by the functional calculus arising from A. Below, we shall frequently and without further mentioning use Lemma 2.7 to the effect that Ḣs P M b = Ḣs P = Ḣs P M holds for -1 ≤ s ≤ 0.

Proof of (ii). As we have seen, it suffices to prove that s ⊥r (P M ) : ( Ḣ0 P ) r → ( Ḣ0 P ) ⊥ is invertible if A is lower-triangular. Taking into account (12.2), this is actually equivalent to Of course we want to interpolate between (12.4) and (12.6) but interpolation of invertibility requires that the two obtained inverses agree on their common domain of definition. For the moment, we can at least state for all f ∈ ( Ḣ1/2

P M b ) r ∩ ( Ḣ-1/2 P M b ) r the comparability T r (P M )f ( Ḣ-1/2 P M b )r + T r (P M )f ( Ḣ-3/2 P M b )r ∼ f ( Ḣ1/2 P M b )r + f ( Ḣ-1/2 P M b )r . Hence, T r (P M ) : ( Ḣ1/2 P M b ) r ∩ ( Ḣ-1/2 P M b ) r → ( Ḣ-1/2 P M b ) r ∩ ( Ḣ-3/2 P M b ) r (12.7)
is bounded from above and below. Here, the intersection spaces carry their natural sum norms. Since the bounded maps in (12.4) and (12.6) depend Lipschitz continuously on M , the same holds true for T r (P M ) when considered a bounded map between these two intersection spaces. We claim that in the block case M = M b it is also onto. Indeed, since the diagonal entries of sgn(P M b ) vanish by (12.1), we can improve the calculation in (12.5) to the effect that

0 T r (P M b )f = P M b sgn(P M b ) 0 f . Now, for each g ∈ ( Ḣ-1/2 P M b ) r ∩ ( Ḣ-3/2 P M b ) r we can define f ∈ ( Ḣ1/2 P M b ) r ∩ ( Ḣ-1/2 P M b ) r via 0 f := (P M b ) -1 sgn(P M b ) 0 g and obtain T r (P M b )f = g. By the method of continuity, perturbing M to M b along M (τ ) = τ M b + (1 -τ )M
, we obtain that the map in (12.7) is an isomorphism. In particular, this implies that the inverses obtained in (12.4) and (12.6) are compatible. Eventually, real interpolation midway between (12.4) and (12.6) for T r (P M ) and its unambiguously defined inverse yields the required isomorphy T r (P M ) : ( Ḣ0

P M b ) r ∼ = -→ ( Ḣ-1 P M b ) r .
The interpolation argument uses [START_REF] Auscher | Holomorphic functional calculi of operators, quadratic estimates and interpolation[END_REF]Thm. 5.3] for the real interpolation of the full Ḣs P M b -spaces and [START_REF] Bergh | Interpolation Spaces. An Introduction[END_REF]Thm. 6.4.2] for the transfer to their complemented subspaces ( Ḣs P M b ) r . The proof for upper-triangular matrices follows along the same lines and we shall only give the main steps.

Proof of (iv). First, the question of invertibility of s r⊥ (P M ) : ( Ḣ0 P ) ⊥ → ( Ḣ0 P ) r is equivalent to proving that 12.4. The triangular cases (iii) and (v). This is a consequence of Theorem 2.21: (iii) is equivalent to (ii) and (v) is equivalent to (iv).

T ⊥ (P M ) : f → N ⊥ P M b 0 s r⊥ (P M )f
12.5. The Hermitian case (vi). From the proof presented below it will become clear that we do neither use real coefficients nor equations but self-adjointness. It will also become clear at what instance we have to assume that coefficients do not depend on the time variable. The argument follows that of [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF] using some integral identities to obtain Rellich estimates.

Let h ∈ H + (P M ). Then F = F (λ, •) = e -λP M h satisfies ∂ λ F = -P M F in the strong sense with strong limits lim λ→∞ ∂ k λ F = 0, k ≥ 0, and lim λ→0 F = h in L 2 = L 2 (R n+1 ; C n+2 ). Let N be the reflection matrix introduced in (2.28). A computation shows that

P * N + N P =    0 0 (1 +H t )D 1/2 t 0 0 0 (1 -H t )D 1/2 t 0 0   .
Next, by definition of M in (2.7), the Hermitian condition

A * = A is equivalent to M * N = N M .
Using the Hermitian inner product (• , •) on L 2 , we obtain

∂ λ (N F, M F ) = (N P M F, M F ) + (N F, M P M F ) = (M F, P * N M F ) + (N M F, P M F ) = (M F, (P * N + N P )M F ) = 2 Re((1 +H t )F θ , D 1/2 t (M F ) ⊥ )
. By integration in λ and the limits above,

(N h, M h) = -2 ∞ 0 Re((1 +H t )F θ , D 1/2 t (M F ) ⊥ ) dλ. Using D 1/2 t (M F ) ⊥ = -H t ∂ λ F θ ,
and integration by parts with vanishing boundary terms,

∞ 0 D 1/4 t (M F ) ⊥ 2 2 dλ = -2 Re ∞ 0 λ(D 1/4 t ∂ λ (M F ) ⊥ , D 1/4 t (M F ) ⊥ ) dλ = 2 Re ∞ 0 λ(∂ λ (M F ) ⊥ , H t ∂ λ F θ ) dλ ≤ 2 ∞ 0 λH t ∂ λ F θ 2 2 dλ λ 1/2 ∞ 0 (λM ∂ λ F ) ⊥ 2 2 dλ λ 1/2 ≤ C h 2 2
, where the last step follows from the boundedness of H t and M on L 2 and the square function estimates for P M , see Theorem 2.3.

Thus, by the Cauchy-Schwarz inequality and the boundedness of H t , (12.8)

|(N h, M h)| h 2 ∞ 0 D 1/4 t F θ 2 2 dλ 1/2 .
The remaining integral on the right is the difficult term. In order to justify some calculations, we use the following lemma which we shall prove in Section 12.8 below.

u = ∂ t u ∈ L 2 loc (R + ; L 2 (R n+1 )) and that div λ,x A∇ λ,x u = ∂ λ (A∇ λ,x u) ⊥ + div x (A∇ λ,x u) .
Since F is the semigroup extension of h, we know that (A∇ λ,x u) ⊥ = F ⊥ is smooth in λ as a function valued in L 2 . Therefore we have div x (A∇ λ,x u) ∈ L 2 loc (R + ; L 2 (R n+1 )) and thanks to the lemma also

∇ x F θ = ∇ x H t D 1/2
t u is contained in this space. This allows us to obtain for almost every λ > 0 the following identities:

D 1/4 t F θ 2 2 = (D 1/2 t F θ , F θ ) = (∂ t u, H t D 1/2 t u) = (∂ λ (A∇ λ,x u) ⊥ + (div x (A∇ λ,x u) , H t D 1/2 t u) = (∂ λ (A∇ λ,x u) ⊥ , H t D 1/2 t u) -((A∇ λ,x u) , ∇ x H t D 1/2 t u) = d dλ ((A∇ λ,x u) ⊥ , H t D 1/2 t u)) -((A∇ λ,x u) ⊥ , ∂ λ H t D 1/2 t u)) -((A∇ λ,x u) , ∇ x H t D 1/2 t u) = - d dλ (F ⊥ , F θ ) -(A∇ λ,x u, H t D 1/2 t ∇ λ,x u) = - d dλ (F ⊥ , F θ ) - 1 2 (∇ λ,x u, [A, H t D 1/2 t ]∇ λ,x u),
where we used the self-adjointness of A and the skew-adjointness of H t D (12.10) in the sense that the left hand integral exists provided the right hand integral exists. By Murray's theorem [START_REF] Murray | Commutators with fractional differentiation and BMO Sobolev spaces[END_REF], the commutator between a bounded function a on the real line and D

∞ 0 D 1/4 t F θ 2 2 dλ = -(h ⊥ , h θ ) - ∞ 0 (∇ λ,x u, [A, H t D 1/2 t ]∇ λ,x u) dλ,
∞ 0 D 1/4 t F θ 2 2 dλ |(h ⊥ , h θ )| + D 1/2 t A L ∞ (BMO) h 2 Ḣ-1/2 P .
Here, we wrote L ∞ (BMO) := L ∞ (R n+1 ; BMO(R)). However, there is no chance in our situation that the integral involving the commutator converges just assuming h ∈ H + (P M ). It is at this point that we have to assume that A does not depend on the t variable. In this case [A, H t D 1/2 t ] = 0 and we obtain from (12.8) and (12.10) the estimate

|(N h, M h)| h 2 h ⊥ 1/2 2 h θ 1/2 2 . By definition of N in (2.28) we have |(h, M h) -2(h ⊥ , (M h) ⊥ )| = |(N h, M h)| = |(h, M h) -2(h , (M h) ) -2(h θ , h θ )|.
Taking into account the accretivity condition h 2 2 Re(h, M h), we deduce the Rellich estimate

h ⊥ 2 2 ∼ h 2 2 ∼ h 2 2 + h θ 2 2 = h r 2 2
. This proves that N ⊥ : H + (P M ) → (H P ) ⊥ and N r : H + (P M ) → (H P ) r have lower bounds. Up to now we have made the qualitative assumption that A is Lipschitz continuous in (x, t). However, since the Lipschitz norm of A does not enter the estimate above, we may remove this extra assumption by an approximation argument that we shall present in the proof of Proposition 2.34 below. For the time being we admit that this is possible.

Thus, we can apply Proposition 2.20 with A = {τ 1 +(1 -τ )A : 0 ≤ τ ≤ 1} to obtain compatible well-posedness of (R) L E 0 and (N ) L E 0 from the analogous result for the heat equation in Section 12.2. By reversing time, the same result holds for the backward equation with coefficients A(x) and duality (Theorem 2.21) along with A = A * yields compatible well-posedness of (R) L E -1 and (N ) L E -1 for the forward equation. Finally, by Theorem 2.18 and (compatible) well-posedness in the energy classes for s = -1/2, we obtain the intermediate cases.

12.6. The case of constant coefficients (vii). In this case, the maps N ⊥ and N r are Fourier multiplier operators in the context of the Mihlin multiplier theorem, and their invertibility is done explicitly in [START_REF] Nyström | L 2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients[END_REF], when s = 0. The other cases follow by reversing time, duality and interpolation just as in the Hermitian case above. 12.7. Proof of Proposition 2.34. We recall from (2.23) that for u an energy solution to (1.1) with bounded and measurable coefficients and h the boundary data of D A u, there is comparability , we can follow the proof of (iv) and obtain from (12.8) and (12.11) that

h r Ḣ-1/4 ∂ t -∆x ∼ h Ḣ-1/2 P ∼ h ⊥ Ḣ-1/4 ∂ t -∆x . ( 12
|(N h, M h)| h 2 |(h ⊥ , h θ )| 1/2 + h Ḣ-1/2 P .
Taking into account (2.23), the same reasoning as before reveals

h r 2 2 + h r 2 Ḣ-1/4 ∂ t -∆x ∼ h 2 2 + h 2 Ḣ-1/4 ∂ t -∆x ∼ h ⊥ 2 2 + h ⊥ 2 Ḣ-1/4 ∂ t -∆x , (12.13)
where implicit constants depend only on ellipticity of A, dimension and D 1/2 t A L ∞ (BMO) . So far, this estimate holds under the a priori assumption that A is Lipschitz continuous in (x, t). Let us now remove this additional regularity and assume D 1/2 t A ∈ L ∞ (BMO) only. By convolution with smooth positive kernels we can find a sequence of Lipschitz matrices A j with uniform ellipticity bounds that converge towards A almost everywhere. Since D

1/2 t A j L ∞ (BMO) ≤ D 1/2 t A L ∞ (
BMO) holds due to the translation invariance of the L ∞ (BMO)-norm, the bounds (12.13) are uniform in j for h j ∈ H + (P M j ) ∩ Ḣ-1/2,+ P M j , where M j is the matrix corresponding to A j . We want to obtain the same equivalence for all h ∈ H + (P M ) ∩ Ḣ-1/2,+ P M

. To this end we shall use in a crucial way well-posedness of the BVPs in the energy class, see Section 2.3. Consider for example the Neumann problem. Let f ∈ L 2 (R n+1 ) ∩ Ḣ-1/4 ∂t-∆x and let u and u j be the unique energy solutions to (1.1) with coefficients A and A j and conormal derivative equal to f , respectively. Theorem 2.8 creates two vectors h, h j ∈ Ḣ-1/2,+ P with h ⊥ = (h j ) ⊥ = f . Due to (12.12) the (h j ) r are uniformly bounded in Ḣ-1/4 ∂t-∆x . The hidden coercivity estimate (2.22) implies that the u j are uniformly bounded in the energy space Ė(R n+2 + ), which means that the D A j u j are uniformly bounded in L 2 (R n+2 + ; C n+2 ). Without loss of generality we may assume weak convergence. Passing to the limit in the variational formulation

R n+2 + A j ∇ λ,x u j • ∇ λ,x v + H t D 1/2 t u j • D 1/2 t v dλ dx dt = -f, v| λ=0 (v ∈ Ė),
we discover that the weak limit solves the equation with coefficients A and by well-posedness in the energy class we deduce D A j u j → D A u weakly in L 2 (R n+2 + ; C n+2 ). Now, Lemma 3.14 yields u j | λ=0 → u| λ=0 weakly in Ḣ1/4 ∂t-∆x and hence (h j ) r → h r weakly in Ḣ-1/4 ∂t-∆x . Thus, we can pass to the limit inferior in (12.13) for h j and get the one-sided inequality

h r 2 2 + h r 2 Ḣ-1/4 ∂ t -∆x h ⊥ 2 2 + h ⊥ 2 Ḣ-1/4 ∂ t -∆x .
For the reverse inequality we argue similarly using the energy solutions with fixed regularity data.

As usual, (12.13) and the method of continuity (see the proof of Proposition 2.20 below for the specific application) allows us to conclude that the regularity problem with data u| λ=0 ∈ Ḣ1/2

N • : H + (P M ) ∩ Ḣ-1/2,+ P M → (R(P ) ∩ Ḣ-1/2 P )
• is an isomorphism, where • designates either r or ⊥. Now, (12.13) provides lower bounds for this map and if A is in block form, then Proposition 2.16 and Theorem 2.32(i) yield invertibility. This yields (i) and the first claim in (iii).

We perform a duality argument to conclude the remaining assertions. First, the same analysis applies to the regularity problem for the backward in time equation with coefficients A * (x, t) since the hypothesis D (which is the dual of Ḣ0,+

1/2 t A ∈ L ∞ (BMO)
P * M ∩ Ḣ-1/2,+ P * M in that duality) with h r = [∇ x f, H t D 1/2 t f ].
Then F = e -λP M h is the parabolic conormal differential of a reinforced weak solution u with u| λ=0 = f . By construction u is the sum of two solutions u 1 + u 2 with D A u 1 ∈ E -1 and D A u 2 ∈ E -1/2 and owing to Theorem 2.8 it is unique under all weak solutions with this property. This proves (ii). The corresponding statement on the Neumann problem is obtained in the same way. 12.8. Proof of Lemma 12.1. This is a consequence of Caccioppoli-type arguments for weak solutions. Assume first that u is a weak solution of ∂ t u -div λ,x A(x, t)∇ λ,x u = 0. We let Λ × Q × I be a parabolic cylinder in R n+2 + of size r (not necessary of Whitney type) and Λ × Q × I denote an enlargement contained in R n+2 + . Since A is Lipschitz continuous with respect to X = (λ, x), Caccioppoli's inequality (Lemma 8.1) and the classical method of difference quotients imply that for i = 0, . . . , n the function v = ∂ X i u is a weak solution to ∂ t v -div λ,x A∇ λ,x v = div λ,x (∂ X i A)∇ λ,x u. This being said, we can sum up Caccioppoli's inequality for all these solutions and obtain

Λ×Q×I |∇ 2 λ,x u| 2 1 r 2 Λ× Q× I |∇ λ,x u| 2 + ∇ λ,x A 2 ∞ Λ× Q× I |∇ λ,x u| 2 .
Since now u has second order derivatives in X within L 2 loc (R n+2 + ), we can interpret the equation

∂ t u = div λ,x A∇ λ,x u in L 2 loc (R n+2
v = ∂ t u is a weak solution to ∂ t v -div λ,x A∇ λ,x v = div λ,x (∂ t A)∇ λ,x u. Caccioppoli's inequality for this solution v reads Λ×Q×I |∇ λ,x ∂ t u| 2 1 r 2 Λ× Q× I |∂ t u| 2 + ∂ t A 2 ∞ Λ× Q× I |∇ λ,x u| 2 .
Adjusting the enlargements, we have obtained

Λ×Q×I |∂ t u| 2 + |∇ 2 λ,x u| 2 + |∇ λ,x ∂ t u| 2 Λ× Q× I |∇ λ,x u| 2 ,
with an implicit constant depending on dimension, ellipticity, r and

∇ λ,x A ∞ + ∂ t A ∞ .
If in addition u is a reinforced weak solution with D A u ∈ H loc , then we can sum these estimates in x and t. More precisely, fix Λ = (r, 2r), Λ = (r/2, 4r) and choose translates Q k × I k of Q × I forming a partition of R n+1 . Then the cubes Qk × Ĩk have finite overlap and summing up, we obtain (12.9). By interpolation in time between, using for example Plancherel's theorem, the estimates for ∇ λ,x u and ∂ t ∇ λ,x u, we obtain the one for D 1/2 t ∇ λ,x u and also for H t D 1/2 t ∇ λ,x u as the Hilbert transform is isometric.

where • , • denotes the duality pairing on Ė(R n+2 ). Hidden coercivity reveals that L * : Ė(R n+2 ) → Ė(R n+2 ) * is an isomorphism and we say that H is the energy solution to L * H = φ on R n+2 . A similar discussion applies to energy solutions of the inhomogeneous equation for L. We remark that we work on the whole space R n+2 to avoid the discussion of boundary values.

13.1. Proof of Theorem 2.37. For the sake of clean arrangement we shall postpone proofs of technical lemmas used throughout the main argument until Section 13.2.

Let us assume that u is a weak solution to Lu = 0 in R n+2

+ that satisfies N * (u) ∈ L 2 (R n+1 ) and for almost every (x, t) ∈ R n+1 , lim λ→0 --- W ((x,t),λ) |u| = 0, (13.1)
where for the argument we prefer to use W ((x, t), λ) := (λ/2, 4λ) × B(x, λ) × (t -4λ 2 , t + 4λ 2 ) as the Whitney regions defining N * . We need to show u = 0 almost everywhere.

Let φ ∈ C ∞ 0 (R n+2 + ). For a reason that will become clear in the course of the proof, we test u not against φ but against ∂ 2 λ φ and we shall show (u, ∂ 2 λ φ) = 0, (13.2) where here and throughout (• , •) denotes the inner product on L 2 (R n+2 + ). This suffices to conclude: Indeed, ∂ 2 λ u = 0 implies u(λ, x, t) = f (x, t) + λg(x, t) for some f, g ∈ L 2 loc (R n+1 ) and from (13.1) and Lebesgue's differentiation theorem we deduce f = 0 almost everywhere. As now u(λ, x, t) = λg(x, t), we obtain for any λ > 0 by means of Tonelli's theorem

R n+1 |g(x, t)| 2 dx dt = 1 λ 2 R n+1 --- W ((x,t),λ) |u| 2 dx dt ≤ R n+1 1 λ 2 N * (u)(x, t) 2 dx dt. (13.3) Since N * (u) ∈ L 2 (R n+1
) by assumption, we discover g = 0 almost everywhere on letting λ → ∞.

We turn to the proof of (13.2). Let ε ∈ (0, 1) and R ∈ (1, ∞) be two degrees of freedom that will be chosen sufficiently small and large, respectively. Let η = η(λ) be smooth with η(λ) = 0 for λ ≤ 1 and η(λ) = 1 for λ ≥ 2 and put θ(λ) = η(λ/ε) -η(λ/R). For ε and R sufficiently small and large, respectively, we have (u,

∂ 2 λ φ) = (θu, ∂ 2 λ φ) since φ has compact support in R n+2 + . Since ∂ 2 λ φ ∈ Ė(R n+2
) * , we can introduce the dual equation by writing ∂ 2 λ φ = L * v for some v ∈ Ė(R n+2 ). We have Lemma 13.1. The functions θu and θv are test functions for the reinforced weak formulation of parabolic equations, that is, they are in the closure of

C ∞ 0 (R n+2 + ) for the norm ∇ λ,x • 2 + D 1/2 t • 2 .
Since θ does not depend on t, we obtain in particular u ∈ Ḣ1/2 (R; L 2 loc (R n+1 + )), that is, u is a reinforced weak solution, see also Corollary 3.6. In this way we can justify the calculation

(u, ∂ 2 λ φ) = (θu, L * v) = (∇ λ,x (θu), A * ∇ λ,x v) + (H t D 1/2 t (θu), D 1/2 t v) = (u∇ λ,x θ, A * ∇ λ,x v) -(∇ λ,x u, A * ∇ λ,x θv) + (∇ λ,x u, A * ∇ λ,x (θv)) + (H t D 1/2 t u, D 1/2 t (θv)) = (u∇ λ,x θ, A * ∇ λ,x v) -(∇ λ,x u, A * ∇ λ,x θv),
where in the third step we have used that θ = θ(λ) is real valued and the final step follows from the reinforced formulation of the equation Lu = 0. At this stage of the proof we want to bring the existence hypothesis into play. So, we need further properties of v stated in the following

Lemma 13.2. Let φ ∈ C ∞ 0 (R n+2 ). The energy solution to L * v = ∂ 2 λ φ satisfies v ∈ C ∞ (R; L 2 ∩ Ḣ1/2 ∂t-∆x ). Moreover, N * ,1 ( v-v 0 λ ) ∈ L 2 (R n+1
), where v 0 := v| λ=0 .

Since in particular v 0 ∈ Ḣ1/2 ∂t-∆x , the assumption provides a reinforced weak solution to L * w = 0 that satisfies N * ( D A * w) ∈ L 2 (R n+1 ) and w| λ=0 = v 0 in the sense of Ḣ1/2 ∂t-∆x . The backward conormal differential D A * has been introduced in Section 2.12. Due to v 0 ∈ L 2 (R n+1 ) we can realise w by an additive constant such that w| λ=0 = v 0 holds also in L 2 (R n+1 ). Upon replacing v with w, the same line of reasoning as above then reveals 0 = (θu, L * w) = (u∇ λ,x θ, A * ∇ λ,x w) -(∇ λ,x u, A * ∇ λ,x θw), invoking the following lemma to justify calculations.

Lemma 13.3. The function θw is in the closure of C ∞ 0 (R n+2 + ) for the norm ∇ λ,x • 2 + D 1/2 t • 2 .
Putting things together, we conclude

(u, ∂ 2 λ φ) = (u∇ λ,x θ, A * ∇ λ,x (v -w)) -(∇ λ,x u, A * ∇ λ,x θ(v -w)).
Here, both u and v -w vanish at λ = 0 and in fact this convergence at the boundary was the reason to introduce w. On recalling our choice of θ, we obtain the straightforward bound

|(u, ∂ 2 λ φ)| I ε + I R + II ε + II R , ( 13.4)
where

I σ := - 2σ σ R n+1 |u||∇ λ,x (v -w)|, II σ := - 2σ σ R n+1
|∇ λ,x u||v -w|.

We prepare the limiting arguments ε → 0 and R → ∞. Define parabolic cylinders 1 2 W ((x, t), λ) := (λ, 2λ) × B(x, λ) × (t -λ 2 , t + λ 2 ), of which W ((x, t), λ) are enlargements. By an averaging trick in (x, t) and Tonelli's theorem we conclude

I σ = - 2σ σ R n+1 -- B(x,σ)×(t-σ 2 ,t+σ 2 ) |u||∇ λ,x (v -w)| dy ds dx dt dµ ≤ R n+1 --- 1 2 W ((x,t),σ) |u| 2 dµ dy ds 1/2 --- 1 2 W ((x,t),σ) |∇ λ,x (v -w)| 2 dµ dy ds 1/2 dx dt R n+1 --- W ((x,t),σ)
|u| dµ dy ds ---

W ((x,t),σ) v -w µ dµ dy ds dx dt ≤ N * ,1 (u) L 2 (R n+1 ) N * ,1 v -w λ L 2 (R n+1 ) , ( 13.5)
where in the second to last step we have used the reverse Hölder inequality (Lemma 8.2) on u and Caccioppoli's inequality (Lemma 8.1) followed by the reverse Hölder inequality on v -w. The latter can be justified for σ sufficiently small or large, in which case L * (v -w) = ∂ 2 λ φ = 0 holds on a neighbourhood of W ((x, t), σ). By interchanging the roles of u and v -w we also get

II σ R n+1 --- W ((x,t),σ) u µ dµ dy ds --- W ((x,t),σ)
|v -w| dµ dy ds dx dt

N * ,1 (u) L 2 (R n+1 ) N * ,1 v -w λ L 2 (R n+1 ) . ( 13.6)
In both estimates N * ,1 (u) ∈ L 2 (R n+1 ) holds by assumption and writing

v -w = (v -v 0 ) -(w -v 0 ), we obtain N * ,1 ( v-w λ ) ∈ L 2 (R n+1
) from Lemma 13.2 and the subsequent result. This is a parabolic version of [START_REF] Kenig | The Neumann problem for elliptic equations with nonsmooth coefficients[END_REF]Thm. 3.1] but in contrast to its elliptic counterpart, it is limited to solutions to the homogeneous equation as will become clear from the proof given in Section 13.2. The upshot is that we can compute limits in (13.5) and (13.6) by means of Lebesgue's theorem. From (13.1) we conclude I ε → 0 and II ε → 0 as ε → 0. In order to obtain I R → 0 and II R → 0 as R → ∞, we use N * ,1 (u) ∈ L 2 (R n+1 ) and the generic lemma on non-tangential maximal functions:

Lemma 13.5. If u ∈ L 1 loc (R n+2 + ) satisfies N * ,1 (u) ∈ L 2 (R n+1 ), then for every (x, t) ∈ R n+1 , lim λ→∞ --- Λ×Q×I |u| = 0.
Going back to (13.4), we see that (13.2) holds and the proof of Theorem 2.37 is complete.

13.2. Non-tangential estimates for energy solutions. Here, we supply the auxiliary results used in the proof of Theorem 2.37. As a general observation, but also to set a strategy of proof that will be re-used several times, we begin with an extension of Lemma 3.14 to the effect that the trace of the homogeneous energy space can also be understood in the sense of non-tangential convergence. This is a parabolic analogue of [4, Thm. 6.42].

Lemma 13.6. (13.8)

For each v ∈ Ė(R n+2 + ) the trace v 0 := v| λ=0 ∈ Ḣ1/4 ∂t-∆x can be realised in L 2 loc (R n+1 ) such that N * ,1 ( v-v 0 √ λ ) ∈ L 2 (R n+1
Since F ∈ L 2 (R n+2 + ) by assumption on v, we obtain V ∈ L 2 (R n+1 ) from the L 2 boundedness of the maximal operators. In particular, we may concentrate on fixed (x, t) ∈ R n+1 for which V (x, t) is finite.

Let h(λ) := ---Λ×Q×I v, noting that for us Λ × Q × I depends only on λ as we have fixed (x, t). From (13.8) we derive that for every δ ∈ (0, 1) there is a constant C δ such that for every λ > 0 we have an estimate |h(λ) -h(δλ)| ≤ C δ √ λ. This implies that lim λ→0 h(λ) =: w 0 (x, t) exists. Therefore we can use a telescopic sum of the estimates (13.8) for Whitney regions of sidelength 2 -k λ, k ∈ N, to obtain the stronger estimate ---Λ×Q×I |v -w 0 (x, t)| dµ dy ds √ λV (x, t). (13.9) Moreover, given (x, t), (y, s) ∈ R n+1 , we can use (13.8) on a Whitney region of length λ ∼ |x -y| + |t -s| 1/2 to get |w 0 (x, t) -w 0 (y, s)| (|x -y| + |t -s| 1/2 ) 1/2 (V (x, t) + V (y, s)).

In particular, this implies w 0 ∈ L 2 loc (R n+1 ) ∩ S ′ (R n+1 ) and together with (13.9) we finally obtain ---Λ×Q×I |v -w 0 | dµ dy ds √ λ(V (x, t) + M x M t (V )(x, t)).

Since V ∈ L 2 (R n+1 ), this implies N * ,1 ( v-w 0 √ λ ) ∈ L 2 (R n+1 ). In order to conclude, we have yet to check that w 0 = v 0 holds on R n+1 in the sense of distributions modulo constants. To this end let φ ∈ C ∞ 0 (R n+1 ). Using Fubini's theorem, we perform an averaging trick in (x, t) and write where W ((x, t), λ) := (λ, 2λ)×B(x, λ)×(t-λ 2 , t+λ 2 ). The integral on the right is bounded uniformly in λ since N * ,1 ( v-w 0 √ λ ) ∈ L 2 (R n+1 ) and φ has compact support. Thus, -2λ λ v dµ → w 0 in the sense of distributions as λ → 0. On the other hand, due to Lemma 3.14 we have v(λ, •) → v 0 as λ → 0 in Ḣ1/4

∂t-∆x and hence in the sense of distributions modulo constants.

We adapt this strategy to give the proofs of Lemmas 13.4 and 13.2.

Proof of Lemma 13.4. Since notation is set up for the forward equation throughout the paper, we prefer to argue for a reinforced weak solution to Lu = 0 that satisfies N * (D A u) ∈ L 2 (R n+1 ). This reduction is justified by the usual transposition of results between forward and backward equations, see also Section 2.12. The proof follows that of Lemma 13.6 verbatim, once we have shown that there exists U ∈ L 2 (R n+1 ) such that

--- Λ×Q×I u ---- Λ×Q×I u λU (x, t).
Here, Λ × Q × I is a fixed Whitney region of sidelength λ centered at (x, t). Let F = D A u. From (13.7) we know that we can take U (x, t) := sup

Λ ′ ×Q ′ ×I ′ ∞ m=0
2 -m ---

Λ ′ ×Q ′ ×4 m I ′ |F |,
with the supremum over all Whitney regions Λ ′ × Q ′ × I ′ centered at (x, t). However, this is not the non-tangential maximal function that we control by assumption and it is for this reason that we have to assume that u is a reinforced weak solution to the homogeneous equation. Indeed, F is a solution to the first order system (2.10), see Theorem 2.2, and as N * (F ) ∈ L 2 (R n+1 ) implies the Dini condition sup λ>0 -2λ λ F L 2 (R n+1 ) dµ < ∞, Theorem 2.5 provides a representation F (λ, •) = e -λ[P M ] h for some h ∈ H + (P M ). Now, U 2 h 2 < ∞ follows from (8.11).

Proof of Lemma 13.2. For any ψ ∈ C ∞ 0 (R n+2 ) we have ∂ λ ψ ∈ Ė(R n+2 ) * . Thus, can define L * -1 (∂ λ ψ) by hidden coercivity and there is an a priori estimate

L * -1 (∂ λ ψ) Ė ∂ λ ψ Ė * ≤ ψ 2 .
In particular, we have ∂ λ L * -1 (∂ λ ψ) ∈ L 2 (R n+2 ). As L * has λ-independent coefficients, we can use the method of difference quotients in Ė(R n+2 ) and Caccioppoli's estimate (Lemma 8.1) to deduce

∂ λ L * -1 (∂ λ ψ) = L * -1 (∂ 2 λ ψ) ∈ Ė(R n+2 ).
Applying these observations with ψ = φ, ∂ λ φ reveals v, ∂ λ v ∈ Ė(R n+2 ) ∩ L 2 (R n+2 ), that is, the four functions v, ∇ x v, H t D 1/2 t v, ∂ λ v as well as their λ-derivatives are in L 2 (R n+2 ). Integration in λ shows that they can be defined continuously with values in L 2 (R n+1 ). Thus, v ∈ C(R; L 2 ∩ Ḣ1/2 ∂t-∆x ) and ∂ λ v ∈ C(R; L 2 ). Now, we can iterate this argument taking ψ = ∂ k λ φ, k ≥ 2, to obtain C ∞ (R; L 2 ∩ Ḣ1/2 ∂t-∆x ). Let us turn to the non-tangential estimate. Due to Lemma 13.6 we only need to check it for Whitney regions Λ × Q × I centered at (x, t) ∈ R n+1 with sidelength λ ≤ 1. The same lemma asserts that v attains its trace v 0 in the sense of almost everywhere convergence of Whitney averages. to obtain, taking into account 2 -k+1 λ ≤ 2, --

2 -k Q×4 m-k I - 2 -k+1 λ 2 -k λ |∇ x v| + |H t D 1/2 t v| -- 2 -k Q×4 m-k I 2 0 |∇ x ∂ λ v| + |H t D 1/2 t ∂ λ v| + -- 2 -k Q×4 m-k I |∇ x v 0 | + |H t D 1/2 t v 0 |.
For the integrals over ∂ λ v in (13.10) a similar use of the fundamental theorem of calculus leads us to --

2 -k Q×4 m-k I - 2 -k+1 λ 2 -k λ |∂ λ v| -- 2 -k Q×4 m-k I 2 0 |∂ 2 λ v| + -- 2 -k Q×4 m-k I |∂ λ v| λ=0 |.
Due to v ∈ C ∞ (R; L 2 ∩ Ḣ1/2 ∂t-∆x ) we have

F 0 := |∇ x v 0 | + |H t D 1/2 t v 0 | + |∂ λ v| λ=0 | ∈ L 2 (R n+1
) and

F := 2 0 |∇ x ∂ λ v| + |H t D 1/2 t ∂ λ v| + |∂ 2 λ v| dµ ∈ L 2 (R n+1 ).
The previous two estimates then show that the right-hand side of (13.10) can be controlled by λ M x M t (F + F 0 )(x, t) and L 2 boundedness of the maximal operators yields the claim.

Next, we supply a general fact on weak solutions implying in particular that every weak solution to (D) L 2 is reinforced. This was announced already in Section 2.17 and used in Lemma 13.1. Lemma 13.7. Let φ ∈ C ∞ 0 (R n+2 + ) and let u be a weak solution to Lu = φ on R n+2 + that satisfies N * ,1 (u/ω) ∈ L 2 (R n+1 ) for some locally bounded, measurable ω : (0, ∞) → (0, ∞) depending only on λ. Then for any θ ∈ C ∞ 0 (0, ∞), depending only on λ, it holds θu ∈ L 2 (R n+2 + ) ∩ Ė(R n+2 + ). Proof. The statement we have to prove is purely qualitative and so we shall not care about dependence of implicit constants. First, we pick a > 0 and c 0 < 1 such that supp(θ) ⊂ (c -1 0 a, c 0 a). It will turn out convenient to use W ((x, t), λ) := ((2c 0 ) -1 λ, 2c 0 λ) × B(x, 2λ) × (t -4λ 2 , t + 4λ 2 ) as Whitney cylinder around (x, t) to define N * ,1 . We also introduce 1 2 W ((x, t), λ) := ((c 0 ) -1 λ, c 0 λ) × B(x, λ) × (t -λ 2 , t + λ 2 ). By an averaging trick in (x, t), we find Since φ is compactly supported, there is a parabolic cube Q × I ⊆ R n+1 such that Lu = 0 in a neighbourhood of W ((x, t), a) for every (x, t) ∈ c (Q × I). This makes the reverse Hölder estimate for weak solutions (Lemma 8.2) applicable and we deduce --- Here the second integral is finite by assumption on u and so is the first one since u as a weak solution is locally square-integrable. So far, this proves θu ∈ L 2 (R n+2 + ). Next, we claim ∇ λ,x (θu) ∈ L 2 (R n+2 + ; C n+1 ). Indeed, ∇ λ,x (θu) = u∇ λ,x θ + θ∇ λ,x u, where the first term can be treated just as θu above and for the second one the only change in the argument is to apply Caccioppoli's estimate (Lemma 8.1) to eliminate the gradient of u before using a reverse Hölder estimate as in (13.12). Hence, θu ∈ L 2 (R; W 1,2 (R n+1 + )). At this point, we have all the ingredients to follow the interpolation argument given in the proof of Lemma 8.5

N * ,1 v max{1, λ} ≤ N * ,1 v -v 0 λ + M x M t (v 0 ).
By Lemma 13.2 and L 2 boundedness of the maximal operators the right-hand side is in L 2 (R n+1 ). Thus, Lemma 13.7 yields θv ∈ L 2 (R n+2 + ) ∩ Ė(R n+2 + ). In this inhomogeneous energy space we can approximate θv by the usual cut-off and mollification procedure. Since θv is supported away from the boundary λ = 0, these approximants are in C ∞ 0 (R n+2 + ) as required. Referring to Lemma 13.4 instead, the same argument applies to θw and for θu we may simply use the assumption N * (u) ∈ L 2 (R n+1 ).

We complete the treatment of postponed auxiliary results with the Proof of Lemma 13.5. As usual, let Λ × Q × I be a Whitney region centered around (x, t) with sidelength λ. By a change of Whitney parameters we may assume N * ,1 is defined with respect to the larger regions Λ × 2Q × 4I. With this convention, we have for every (z, r) ∈ Q × I, ∂t-∆x , which by construction satisfy u 1,0 + u 2,0 = f in the sense of distributions modulo constants. In particular, we may modify u by an additive constant to obtain also u 2,0 = f -u 1,0 ∈ L 2 (R n+1 ).

Next, Theorem 2.9 provides a representation u 1 (λ, •) = c -(e -λ[M P ] h) ⊥ for some unique c ∈ C and h ∈ H + (M P ). Since u 1,0 ∈ L 2 (R n+1 ), we must have c = 0, in which case Theorem 2.13 yields N * (u 1 ) ∈ L 2 (R n+1 ) and a.e. convergence of Whitney averages of u 1 towards u 1,0 as λ → 0. As for u 2 , we recall that D A u 2 ∈ E -1/2 precisely means u 2 ∈ Ė(R n+2 + ). Thus, Lemma 13.6 yields N * ,1 ( In proving u = 0, we can follow the proof of Theorem 2.37. Indeed, given φ ∈ C ∞ 0 (R n+2 + ), we can define v := L * -1 (∂ 2 λ φ) as before. Since the regularity assumption on A is preserved under a reversal of time and taking adjoints, we also have at hand Proposition 2.34 for the adjoint equation. In particular, since Lemmas 3.14 and 13.2 pay for v| λ=0 ∈ Ḣ1/2 ∂t-∆x ∩ Ḣ1/4 ∂t-∆x , we can find a reinforced weak solution to L * w = 0 with trace w| λ=0 = v| λ=0 and the same estimate D A * u ∈ E 0 as before. As an additional feature, D A * u ∈ E -1/2 , that is, w ∈ Ė(R n+2 + ). Now that the same setup is settled, there are only two instances in the proof of Theorem 2.37 at which the precise form of the non-tangential control for u matters, namely (13.3) and the limiting argument relying on (13.5) and (13.6). For the former, having L 2 -control only on N * (u/ √ λ) is sufficient. For the latter, we need a different argument only when σ = R is large, in which case we simply distribute the factor 1/µ to obtain Here, the averages on u/ √ λ are under control by assumption and for those of (v -w)/ √ λ we can use the additional information on w: As v -w ∈ Ė(R n+2 + ) has trace 0, the required L 2 -control is due to Lemma 13.6. This being said, we can complete the proof as in the case of Theorem 2.37.

u 2 -u 2,0 √ λ ) ∈ L 2 (R
I σ + II σ R n+1 ---

Miscellaneous generalisations and open problems

14.1. Systems. All of our results apply to parabolic systems without any changes but in the ellipticity condition. More precisely, we can deal with systems of m equations given by

∂ t u α - n i,j=0 m β=1 ∂ i A α,β
i,j (λ, x, t)∂ j u β = 0, α = 1, . . . , m, in R n+2 + , where ∂ 0 = ∂ λ and ∂ i = ∂ x i if i = 1, . . . , n, and where we impose the following assumptions on the matrix A(λ, x, t) = (A α,β i,j (λ, x, t)) α,β=1,...,m i,j=0,...,n . First, A(λ, x, t) = (A α,β i,j (λ, x, t)) α,β=1,...,m i,j=0,...,n ∈ L ∞ (R n+2 ; L(C m(1+n) ))

is bounded and measurable. Second, A satisfies, for some κ > 0 and uniformly for all λ, the ellipticity condition Re for all f = (f α j ) α=1,...,m j=0,...,n ∈ L 2 (R n+1 ; C m(1+n) ) with curl x (f α j ) j=1,...,n = 0 for all α. 14.2. Adding lower order terms to the elliptic part. There is no real difficulty to adapt our techniques and proofs to the case of (strictly elliptic) equations with lower order terms:

Lu = ∂ t u -div X (A(X, t)∇ X u + bu) + c∇ X u + du = 0
where b, c are n + 1 vectors and d is a scalar, all coefficients are bounded and measurable, and we assume pointwise ellipticity (in the case of one equation). In this case the associated parabolic conormal differential becomes and the corresponding parabolic Dirac operator is given by

D A u(λ, x, t) =      ∂ ν A u(λ, x, t) + b ⊥ u(λ,
P =       0 div x 1 -D 1/2 t -∇ x 0 0 0 1 0 0 0 -H t D 1/2 t 0 0 0       , M =       B ⊥⊥ B ⊥ b ⊥ 0 B ⊥ B b 0 c ⊥ c d 0 0 0 0 1      
, where

   B ⊥⊥ B ⊥ b ⊥ B ⊥ B b c ⊥ c d    =    1 0 0 A ⊥ A b c ⊥ c d       A ⊥⊥ A ⊥ b ⊥ 0 1 0 0 0 1    -1
.

The upshot is the proof of the square function implied in Theorem 2.3. One can either adapt the proof given for the pure second order case or prove a lower order perturbation result. We do not get into details. 14.3. The setup of cylindrical domains. We may replace (X, t) ∈ R n+2 + by (X, t) ∈ B × R, where B is the unit ball of R n+1 . In that case, the normal independence of the coefficients means that they do not depend on the radial variable as in [START_REF] Kenig | The Neumann problem for elliptic equations with nonsmooth coefficients[END_REF]. Adapting [START_REF] Auscher | Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems II[END_REF] to the parabolic setting, one can develop the first order approach. Again the proof of the square function in Theorem 2.3 is the main part and should be handled by mixing arguments from here with those of [START_REF] Auscher | Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems II[END_REF].

We remark that if t belongs to the unbounded line, then this makes it impossible to treat the Rellich estimates in the self-adjoint case with t-dependent coefficients via Fredholm methods even though the angular variable lives on the sphere. However, one should be able to treat boundary value problems with initial condition on bounded intervals [0, T ] imposing that the commutators between coefficients and half-order time derivatives are compact, uniformly in the angular variable. We leave this idea and further details to interested readers. 14.4. Degenerate parabolic equations and systems. In the setting of the parabolic upper halfspace, it may also be interesting to consider the case of degenerate equations and systems. To expand a bit on this in the case of equations, one may consider equations λ 1 (X, t)∂ t u -div X A(X, t)∇ X u = 0, (14.1) with κλ 2 (X, t)|ξ| 2 ≤ Re(A(X, t)ξ • ξ), |A(X, t)ξ • ζ| ≤ Cλ 2 (X, t)|ξ||ζ|, where the weights λ 1 and λ 2 are non-negative measurable functions subject to conditions.

In the case when λ 1 (X, t) = w(x) = λ 2 (X, t) for some w ∈ A 2 (R n ), one may try to develop a weighted version of our paper by following [START_REF] Auscher | Boundary value problems for degenerate elliptic equations and systems[END_REF]. Again, the matter is to prove the weighted analogue to Theorem 2.3 for λ-independent coefficients. We claim that this is possible. It is an interesting open problem to construct test functions in the case when the weight also depends on the t variable and is, for instance, an A 2 weight for the parabolic doubling space R n+1 .

Although not directly related, it is here worth noting that in the case of real parabolic equations, interior regularity of weak solutions to (14.1), when λ 1 (X, t) = w(X, t) = λ 2 (X, t), or λ 1 (X, t) = 1 and λ 2 (X, t) = w(X, t), was studied in [START_REF] Chiarenza | Degenerate parabolic equations and Harnack inequality[END_REF][START_REF] Chiarenza | Pointwise estimates for degenerate parabolic equations[END_REF] under various integrability conditions on the weight. We remark that the conclusions there in terms of Muckenhoupt conditions are not exactly as for the elliptic case, showing that the A 2 condition is not the optimal one to obtain regularity. 14.5. Boundary value problems with other spaces of data. Another possible development is the formalisation of the first order setup to approach boundary value problems with data in other spaces than the L 2 based Sobolev spaces.

Starting with L p spaces, it amounts to first study a priori estimates enjoyed by the Cauchy extension and next to identify trace spaces of solutions having some square function control and prove that their conormal differentials are given by a Cauchy extension.

As a consequence of the results in Section 2.9, in particular, Lemma 2.10 and Proposition 2.11, one can extend the bounded holomorphic functional calculus of P M to L p for p near 2 and this implies (after some work) the alluded square function estimates. However, if one wants to develop results covering larger ranges of p, one should look at Hardy space theory and uniqueness results as studied in [START_REF] Auscher | Representation and uniqueness for boundary value elliptic problems via first order systems[END_REF][START_REF] Auscher | Functional Calculus for first order systems of Dirac type and boundary value problems[END_REF] for the elliptic situation. This theory has been extended in [START_REF] Amenta | Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach[END_REF] to cover fractional Besov-Hardy-Sobolev spaces. While the strategy is in place, the details could reveal more complicated again due to the presence of half-order derivatives in time.

14.6. Higher order problems and more. One can also study other types of problems either for the purpose of boundary value problems with λ independent coefficients or to prove the Kato square root estimate.

We have in mind parabolic systems with higher order elliptic part in divergence form. The first order method for the elliptic case was discussed in [START_REF] Auscher | On a quadratic estimate related to the Kato conjecture and boundary value problems[END_REF].

One could also think of replacing ∂ t by other operators. If we replace it by ∂ 2 t , we come back to an elliptic problem. But we could also think of ∂ 3 t or fractional derivatives. In principle, the algebraic formalism can be set but the conditions to proceed with the analysis are unclear.

Finally, one could try to treat elliptic parts of some fractional order but in this case the scary thing is the lack of spatial off-diagonal estimates.

Proposition 2 . 25 .

 225 Let s ∈ [-1, 0]. (i) If 1 ± s ⊥⊥ (P M ) is invertible between the respective components of Ḣs P and the inverse agrees with the one for s = -1/2, then (R) L Es is compatibly well-posed on R n+2 ∓ . (ii) If 1 ± s rr (P M ) is invertible between the respective components of Ḣs P and the inverse agrees with the one for s = -1/2, then (N ) L Es is compatibly well-posed on R n+2 ∓ .

P

  and all bounded holomorphic functions b : S µ → C.

2 . 31 .

 231 23 and Corollary 2.24 in the language of layer potentials become Theorem Let s ∈ [-1, 0]. The following hold true.

Theorem 2 . 33 .

 233 Let s ∈ [-1, 0] and u be a solution to Lu = 0 with D A u ∈ E s (on either the upper or the lower half-space). All eight representations in (2.38) -(2.

1 / 2 t

 12 φ(t)| ∼ |t| -3/2 for |t| large. Clearly the constant functions are in Ḣ1/2 (R) and satisfy D 1/2 t v = 0. Conversely, let v ∈ Ḣ1/2 (R) be of that kind. We can write any ψ ∈ S(R) with Fourier transform supported away from 0 as ψ = D 1/2

1 / 2 t 2 tCorollary 3 . 6 .

 12236 v and then using Lemma 3.3. From the proof of Proposition 3.4 we know that (3.4) with g = D 1/v reproduces v up to a constant d ∈ C and therefore the final estimate follows from the bounds for v 1 and v 2 obtained therein, see (3.5) and the lines above. Next, Proposition 3.4 with T = H t and T = H t D 1/2 t yields the fractional integration by parts formulae that justify to speak of reinforced weak solutions in the first place. Let v ∈ Ḣ1/2 (R) and φ ∈ S(R). Then vH t D 1/2 t φ ∈ L 1 (R) and there are fractional integration by parts formulae

Lemma 4 . 2 .

 42 The operators P M and M P are closed and bisectorial on L 2 with R(P M ) = R(P ) and R(M P ) = M R(P ).

7. 1 .

 1 Reduction to a Carleson measure. As Q λ vanishes on N(M P ) = N(P ), it is enough to prove the quadratic estimate (7.1) for h ∈ R(M P ). By density, h ∈ R(M P ) suffices. So, setting Θ λ = Q λ M , it amounts to showing (

Lemma 8 . 4 .

 84 Let u be a reinforced weak solution to (1.1) and let c = ---2Λ×2Q×I u. Then

x u| 2 |u -c| dµ dy ds 2 +

 22 dµ dy ds. Lemma 8.2 applied to the weak solution u -c provides control by 1 r 2 ---4Λ×4Q×16I ---2Λ×2Q×4I |∇ λ,x u| 2 dµ dy ds and we conclude by Lemma 8.4.

Lemma 8 . 11 .|F | 2

 8112 For any measurable function F on the upper half-space R n+2 + holds true where F µ = F (µ, •). If the rightmost integral is finite, then for a.e. (x, t) ∈ R n+1 , dµ dy ds = 0.

  , y, s) -h(x, t)| 2 dµ dy ds ---Λ×Q×I |ψ(µP M )h(y, s)| 2 dµ dy ds + ---Λ×Q×I |R µ h(y, s) -h(y, s)| 2 dµ dy ds + --Q×I |h(y, s) -h(x, t)| 2 dy ds.

Lemma 9 . 1 (

 91 Simultaneous duality

Proof of Proposition 2 . 22 .

 222 Set h = D A u| λ=0 ∈ Ḣs,+ P M and g = D A * w| λ=0 ∈ Ḣ-s-1,+ P * M

.

  For sgn(P M b ) we have already introduced such a representation as a (2 × 2)-matrix in (2.30): sgn(P M b ) = s ⊥⊥ (P M b ) s ⊥r (P M b ) s r⊥ (P M b ) s rr (P M b ) , which by the holomorphic functional calculus is bounded on R(P ) and satisfies sgn(P M b ) 2 = 1. On the other hand, M b commutes with the matrix N introduced in (2.28) since both are in block form. Hence, P M b N = -N P M b , which carries over to the functional calculus by sgn(P M b )N = -N sgn(P M b ). These two properties of sgn(P M b ) can only hold if its diagonal blocks vanish and its off-diagonal blocks are inverses of each other, that is, sgn(P M b ) = 0 s ⊥r (P M b ) s ⊥r (P M b ) -1 0 . (12.1)

  (P M b ) 2 . So, by the very definition of homogeneous Sobolev spaces we can decompose Ḣs P M b = ( Ḣs P M b ) ⊥ ⊕ ( Ḣs P M b ) r for any s ∈ R, see Section 2.7, and besides the usual isomorphy P M b : Ḣs P M b → Ḣs-1 P M b we have the restricted isomorphisms P M b : ( Ḣs P M b ) ⊥ ⊕ {0} ∼ = -→ {0} ⊕ ( Ḣs-1 P M b ) r (12.2) and P M b : {0} ⊕ ( Ḣs P M b ) r ∼ = -→ ( Ḣs-1 P M b ) ⊥ ⊕ {0}. (12.3)

T 2 P

 2 r (P M ) : f → N r P M b s ⊥r (P M )f 0 providing an isomorphism ( Ḣ0 P M b ) r → ( Ḣ-1 P M b ) r and it is this property that we shall check. From Corollary 2.24 and (12.2) we can infer T r (P M ) ) depends Lipschitz continuously on M , see the discussion before Theorem 2.19, also T r (P M ) depends Lipschitz continuously on M as a bounded operator between the Sobolev spaces above. Now, we use the lower-triangular structure of M for the one and only time to see that for f ∈ ( Ḣ1/2 P M b ) r ∩ ( Ḣ-1/2 P M b ) r we can also write T r (P M )f = N r P M b s ⊥r (P M )f 0 = N r P M sgn(P M ) 0 f = N r sgn(P M )P M 0 f = s r⊥ (P M )N ⊥ P M b 0 f . (12.5) Hence, (12.3) and Corollary 2.24 yield that T r (P M ) extends by density to an isomorphism T r (P M ) same argument as before we have Lipschitz continuous dependence on M .

  acts as an isomorphism ( Ḣ0P M b ) ⊥ → ( Ḣ-1 P M b ) ⊥ . Due to Corollary 2.24 and (12.3) it certainly is invertible from ( Ḣ-1/2 P M b ) ⊥ into ( Ḣ-3/2 P M b ) ⊥ .The fact that M is upper-triangular lets us discover the identityT ⊥ (P M )f = N ⊥ P M b 0 s r⊥ (P M )f = N ⊥ P M sgn(P M ) f 0 = N ⊥ sgn(P M )P M f 0 = s r⊥ (P M )N r P M b f 0 for f ∈ ( Ḣ1/2 P M b ) ⊥ ∩ ( Ḣ-1/2 P M b ) ⊥ . Therefore T ⊥ (P M ) extends to an isomorphism from ( Ḣ1/2 P M b ) ⊥ to ( Ḣ-1/2 P M b )⊥ and we conclude by interpolation as in the proof of (ii).

Lemma 13 . 4 .

 134 Let w be a reinforced weak solution to L * w = 0 on R n+2 + satisfying N * ( D A * w) ∈ L 2 (R n+1 ). Then there exists a trace w 0 ∈ L 2 loc (R n+1 ) such that for almost every (x, t) ∈ R n+1 , lim λ→0 ---Λ×Q×I |w -w 0 (x, t)| dµ dy ds = 0. Furthermore, N * ,1 ( w-w 0 λ ) ∈ L 2 (R n+1 ) and ∇ x w 0 = ( D A * w) | λ=0 . An analogous statement holds for reinforced weak solutions to Lu = 0 upon replacing D A * by D A .

2 t 0 (

 20 ) and for almost every(x, t) ∈ R n+1 , lim λ→0 ---Λ×Q×I |v -v 0 (x, t)| dµ dy ds = 0. Proof. Abbreviating F := |∇ λ,x v| + |H t D1/v|, we can infer from Poincaré's inequality as in the proof of Lemma 8.4, step follows by a simple rearrangement of terms as in the proof of Lemma 8.12. Let M x and M t denote again the maximal operators in the x and t variable, respectively. By Hölder's inequality, M x M t F µ )(x, t) 2 dµ 1/2=:√ λV (x, t).

  , y, s) dµ -w 0 (y, s) φ(y, s) dy ds = √ λ

2 t 0 ∂

 20 Therefore we can apply (13.7) iteratively on the Whitney regions of sidelength 2 -k λ, k ∈ N, centered at (x, t) to find by a telescopic sum---Λ×Q×I |v -v 0 | λ sup k∈N m≥0 2 -m --2 -k Q×4 m-k I -2 -k+1 λ 2 -k λ |∂ λ v| + |∇ x v| + |H t D 1/2 t v| . (13.10) In order to treat the integrals over |∇ x v| + |H t D 1/v|, we write v(µ, y, s) = v 0 (y, s) + µ λ v(σ, y, s) dσ ((µ, y, s) ∈ R n+2 + )

1 2 W 2 N 2 W

 222 ((x,t),a)|u|2 ---W ((x,t),a) |u| * ,1 (u/ω)(x, t) 2 ((x, t) ∈ c (Q × I)). (13.12)Note that we have used the local boundedness of ω in the second step. Thus, ((x,t),a)|u| 2 dx dt + c (Q×I)N * ,1 (u/ω) 2 (x, t) dx dt.

3 .

 3 , s)| dy ds ≤ 2 n+2 N * ,1 (u)(z, r). , s)| dy ds ≤ 2 n+2 --Q×I N * ,1 (u)(z, r) dz dr λ -1-n/2 N * ,1 (u) L 2 (R n+1 )and the conclusion follows.13.Proof of Theorem 2.40.Let f ∈ L 2 (R n+1 ) ⊂ L 2 (R n+1 ) + Ḣ1/4∂t-∆x and let u be the corresponding solution to Lu = 0 provided by part (ii) of Proposition 2.34, written as the sum of two solutions u 1 + u 2 to the same equation with D A u 1 ∈ E -1 and D A u 2 ∈ E -1/2 . Thanks to Theorem 2.8 there are tracesu 1,0 := u 1 | λ=0 ∈ L 2 (R n+1 ), u 2,0 := u 2 | λ=0 ∈ Ḣ1/4

W

  ((x,t),σ) u √ µ dµ dy ds ---W ((x,t),σ)v -w √ µ dµ dy ds dx dt.

  λ, x, t)f (x, t) • f (x, t)) dx dt ≥ κ t)| 2 dx dt,

  As the boundary problems for s = -1/2 are well-posed by the method of energy solutions in Section 2.3, compatible well-posedness reduces to another simple statement.

	L Es is well-posed if and only if N r : Ḣs,+ P M → ( Ḣs P ) r is an isomorphism. (ii) (N ) L Es is well-posed if and only if N ⊥ : Ḣs,+ P M → ( Ḣs P ) ⊥ is an isomorphism. In each case, ontoness is equivalent to existence and injectivity to uniqueness, respectively.
	Proposition 2.16. The following assertions hold for -1 ≤ s ≤ 0. (i) (R) L Es is compatibly well-posed if and only if N r : Ḣs,+ P M → ( Ḣs P ) r is an isomorphism and its inverse agrees with the one at s = -1/2 on ( Ḣs P ) r ∩ ( Ḣ-1/2 P ) r . (ii) (N ) L Es is compatibly well-posed if and only if N ⊥ : Ḣs,+ P M → ( Ḣs P ) ⊥ is an isomorphism and its inverse agrees with the one at s = -1/2 on ( Ḣs P ) ⊥ ∩ ( Ḣ-1/2

P

) ⊥ .

Proposition 2.20. Let

  s ∈ [-1, 0]. Let A be a class of matrices A(x, t) with uniform ellipticity bounds that is arcwise connected. If this class contains one matrix A 0 for which the problem (BVP ) L 0

	Es is (compatibly) well-posed and if all the corresponding maps N • have lower bounds on the trace space Ḣs,+ P M for each A ∈ A, then (BVP ) L Es is (compatibly) well-posed for all A ∈ A.
	2.12.

The backward equation and duality results. The

	equation dual to (1.1) is the backward
	equation

we also have reverse Hölder inequalities for reinforced weak solutions of the backward equation. At this occasion let us recall that the reverse Hölder inequalities of Theorem 2.14 already contain both H t D

			. It
	is only when coming back to (2.26) that we have to use a different correspondence.
	As the reversal of time maps solutions of the backward equation L * v = 0 to a solution of a forward equation with coefficients A * (x, -t), 1/2 t v and D 1/2 t v on the left hand side. All other results
	for the backward equation, including non-tangential estimates as in Theorem 2.12, are obtained by
	literally repeating the arguments provided for the forward equation and we shall not give further
	details.
	With the construction outlined above we can formulate duality results for the boundary value
	problems.
	Theorem 2.21. Let L and L * be as above with λ-independent coefficients and let -1 ≤ s ≤ 0. Then (BVP ) L Es is (compatibly) well-posed if and only if (BVP ) L * E -1-s is (compatibly) well-posed.
	In a similar spirit we obtain a generalisation of Green's formula, which is of independent interest
	and worth stating as it avoids integration by parts. Recall that the dual of	Ḣs/2 ∂t-∆x can be identified
	with the inner product on L 2 (R n+1 ). Ḣ-s/2 -∂t-∆x , which coincides with	Ḣ-s/2 ∂t-∆x up to equivalent norms, and the duality pairing extends

Proposition 2.22. Let

  -1 ≤ s ≤ 0. For any weak solution u to (1.1) with D A u ∈ E s and any weak solution w to (2.26) with D A * w ∈ E -s-1 it holds (2.29) ∂ ν A u| λ=0 , w| λ=0 = u| λ=0 , ∂ ν A * w| λ=0 .

	Here, the first pairing is the Ḣs/2+1/2 ∂t-∆x , Ḣ-s/2-1/2 -∂t-∆x sesquilinear duality. Ḣs/2 ∂t-∆x , Ḣ-s/2 -∂t-∆x sesquilinear duality while the second one is the

  S = 1 and z + z = 1, we see that on ℓ 2 (N) × ℓ 2 (N) we have a pair of complementary projections . Its inverse is the Dirichlet to Neumann operator Γ L,+ DN g := ∂ ν A v| λ=0 , where v is the energy solution to the Dirichlet problem for

	χ + :=	zSS * S *	S zS * S	,	χ -:=	S * S -zSS * -S -S * zS * S
	for which both 1 ± s rr are invertible, while s r⊥ = 2S * has one-dimensional nullspace and s ⊥r = 2S is not onto.
	Upon interchanging ⊥-and r-coordinates, the same two examples give the same conclusions with the roles of ⊥ and r reversed. In Section 2.3 we have obtained well-posedness of both boundary value problems for the energy
	class on the upper half-space but the discussion there applies verbatim to the lower one. These are
	precisely the problems (R) L E -1/2 and (N ) L E -1/2 , see Section 2.8. Hence, we can record
	Corollary 2.24. Invertibility of the six operators in Theorem 2.23 above always holds at regularity
	s = -1/2.
	The inverses for s = -1/2 have an illuminating interpretation in terms of boundary operators. Indeed, let us define the Neumann to Dirichlet operator on the upper half-space by
			Γ L,+ N D f :=	∇ x t H t D 1/2	u| λ=0 ,
	+ where u is the energy solution to the Neumann problem for Lu = 0 on R n+2 to well-posedness, this defines an isomorphism ( Ḣ-1/2 P ) ⊥ → ( Ḣ-1/2	with data f . Owing
	+ Lv = 0 on R n+2 solution to Lu = 0 on the upper half-space has a trace in with regularity data g. Hence, the parabolic conormal differential of an energy Ḣ-1/2 P = Ḣ-1/2 P M given by D A u| λ=0 = [f, Γ L+ N D f ]. However, due to Theorem 2.8 these traces also describe the spectral space Ḣ-1/2,+ . P M Consequently,
	(2.32)	Ḣ-1/2,+ P M	= h ∈	Ḣ-1/2 P	: h r = Γ L,+ N D h ⊥ ,
	which can be rephrased as saying that for h ∈ are equivalent. Unravelling the latter equivalence gives Ḣ-1/2 P the conditions sgn(P M )h = h and h r = Γ L,+ N D h ⊥

2 , . . .) be the right shift with S * : (a n ) → (a 2 , a 3 , a 4 , . . .) its adjoint and let z ∈ C be a root of z 2 -z + 1. Taking into account the relations S * P ) r

  while the inequality f 2 N * (u) 2 is an easy consequence of the non-tangential convergence at the boundary. Compatible well-posedness was already stated in Theorem 2.21 and is preserved as well. Comparing with Theorem 2.32, we also obtain

Corollary 2.39. (D) L

2 is compatibly well-posed if A is either of block form, lower triangular, Hermitian and time independent, or constant.

  ). Thus, the boundary equation can be understood in L 2 loc (R n+1 ) with ∇ x u 0 = h and H t D So far, we have seen that all conormal differentials D A u ∈ E s of reinforced weak solutions to (1.1) have a trace h = D A u| λ=0 and that this trace maps E s boundedly into Ḣs

	1/2

t u 0 = h θ . P . If now A -A 0 s is small enough, then h Ḣs P ∼ h + Ḣs P as a consequence of (2.45). Hence,

  ) valid for a.e. t ∈ R, see[47, §12.1]. These formulae can be used to prove that D

	1/2 t v and H t D t v are 1/2
	integrable functions satisfying

  + ) is compactly supported in λ and x. A similar statement holds for Ḣ1/2 (R; L 2 loc (R n )). Proof. Corollary 3.5 and the growth estimate (3.3) yield the global integrability conditions with v -d in place of v, where d ∈ L 1 loc (R n+2 + ) does not depend on t. However, dD

	L 1 (R n+2 +	1/2 t φ and dH t D t φ are in 1/2

  1/2 should be analytic at the origin, which is absurd. We now fix z with |z| > 2 such that all the properties above hold. Let ζ ∈ C n+2 with |ζ| = 1. Let ζ i denote the components of ζ and ζ = (ζ 2 , . . . , ζ n+1

  The first estimate is a direct consequence of Lemmas 8.1 and 8.2 applied to the weak solution u -c. For the second estimate we separate averages in space and time by writing

	1 r	---	1/2 t u|),
	where H t D t u could be replaced with D 1/2	1/2

4Λ×4Q×16I |u -c| (|∇ λ,x u| + |H t D t u on the right-hand side. Proof.

  ). Let -1 ≤ s ≤ 0. There is a pairing simultaneously realising the pairs of spaces ( Ḣs

	spaces. Moreover, in this pairing, the pairs ( Ḣs,± P M , Ḣ-s-1,± P

P , Ḣ-s-1 P * ), ( Ḣs,± P M , Ḣ-s-1,∓ P * M ), (( Ḣs P ) ⊥ , ( Ḣ-s-1 P * ) r ) and (( Ḣs P ) r , ( Ḣ-s-1 P * ) ⊥ ) as dual * M

  1,+

	P * M Es . By Theorem 2.15 this is well-posed if Let us consider for example the Neumann problem (N ) L and Ḣ-s-1 P * = ( Ḣ-s-1 P * ) r ⊕ ( Ḣ-s-1 P * ) ⊥ , respectively. and only if N ⊥ : Ḣs,+ P M → ( Ḣs P ) ⊥ is an isomorphism. In the duality above, the adjoint of N ⊥ : Ḣs,+ P M → ( Ḣs

P ) ⊥ is χ -(P * M ) : ( Ḣ-s-1 P * ) r → Ḣ-s-1,-P * M

  ∆x and using Lemma 2.26 and the isomorphism property of π P . Note that D A u ∈ L 2 (R; R(P )). Following the strategy of proof of Theorem 2.2, we see that u = L -1 f if and only if ∂ λ F + P M F = f 0 in the weak sense, where F = D A u. Recall that for vectors F ∈ C n+2 we gather here F and F θ in

	is well-defined and a two-sided inverse for N ⊥ : Ḣs,+ P M ∩ assumption it extends to a bounded operator ( Ḣs P ) ⊥ → Ḣs,+ Ḣ-1/2,+ P M P M . This yields the sought-after inverse → ( Ḣs P ) ⊥ ∩ ( Ḣ-1/2 P ) ⊥ . Again by for N ⊥ : Ḣs,+ P M → ( Ḣs P ) ⊥ and compatibility on ( Ḣ-1/2 P ) ⊥ follows by construction.
	11. Inverting parabolic operators by layer potentials
	Here, we provide the remaining proofs for results on layer potentials discussed in Section 2.14,
	where the relevant notation has been introduced.
	Proof of Theorem 2.29. Let f ∈ ∂t-∆ λ,x one component denoted F r . Replacing Ḣ-1/2 ∂t-∆ λ,x and u ∈ Ḣ1/2 f 0 by the differential equation and integrating by parts, it
	follows that we have, in L 2 (R; R(P )) and for almost every λ > 0,
	ε<|λ-µ|<R
	P * ) ⊥ =	Ḣ-s/2 -∂t-∆x .

* , (N P * -1 g) ⊥ = w| λ=0 ∈ (π P * Ḣ-s M P * ) ⊥ = ( Ḣ-s

  Assuming the hypothesis of Lemma 12.1 and using the correspondence of F with a reinforced weak solution u, we observe that F θ = H t D It will be simpler to come back to the usage of u in the following calculation. Note that div λ,x A∇ λ,x

	Lemma 12.1. Assume that the coefficients of L are Lipschitz continuous with respect to (x, t) and that u is a reinforced weak solution of (1.1) with D A u ∈ H loc . Then ∇ 2 λ,x u, ∂ t u and ∇ x ∂ t u are in L 2 loc (R + ; L 2 (R n+1 )) and for any r > 0,
	(12.9)	r	2r	∇ 2 λ,x u 2 2 + ∂ t u 2 2 + ∇ x ∂ t u 2 2 dλ ≤ C	4r r/2	∇ λ,x u 2 2 dλ
				1/2 t u and D	1/2

with C depending on r, the ellipticity constants of A and ∇A ∞ . In particular, a similar estimate holds for

∇ x H t D 1/2 t u. t F θ = ∂ t u.

  ] denotes the commutator between the multiplication by A and H t D

	[A, H t D	1/2	1/2 t	in the last line and 1/2 t . Integrating the
	equality above and recalling the limits for F , we obtain

t

  .12)Here, h r = [∇ x u, H t D ∂t-∆x ) n+1 and we forget the power of n + 1 by abuse of notation. Moreover, implicit constants depend only on ellipticity of A and dimension since these estimates were obtained from the Lax-Milgram lemma. Given h ∈ H

	1/2 t u]| λ=0 belongs to (	Ḣ-1/4

+ (P M ) ∩ Ḣ-1/2,+ P M

  A u ∈ E 0 ∩ E -1/2 .Indeed, by Theorems 2.5 and 2.8 this amounts to showing that

	and the Neumann problem with data ∂ ν A u| λ=0 ∈ Ḣ0 ∂t-∆x ∩ solutions with D	Ḣ1/4 ∂t-∆x ∂t-∆x are well-posed in the class of ∂t-∆x ∩ Ḣ-1/4

  is preserved in changing t to -t and in taking adjoints. By the argument of Theorem 2.21, each f ∈ L 2 (R n+1 )+

	Ḣ1/4 ∂t-∆x corresponds to a unique h ∈ Ḣ-1,+ P M +	Ḣ-1/2,+ P M

  + ). It follows that Λ×Q×I |∂ t u| 2 can be bounded in the same way as Λ×Q×I |∇ 2 λ,x u| 2 above. Eventually, we use that A is also Lipschitz continuous with respect to t to conclude likewise that

  to first show ∂ t (θu) ∈ L 2 (R; W -1,2 (R n+1 + )) and then deduce the missing piece of informationH t D 1/2 t (θu) ∈ L 2 (R n+2 + ). The fact that here u solves the inhomogeneous equation Lu = φ does not pose any difficulties in this argument since we have θφ ∈ L 2 (R n+2 + ). As a direct application we obtain the Proofs of Lemma 13.1 and Lemma 13.3. Let us start with θv and let v 0 := v| λ=0 . On noting that for almost every (λ, x, t) ∈ R n+2

+ , v(λ, x, t) max{1, λ} ≤ v(λ, x, t) -v 0 (x, t) λ + |v 0 (x, t)|,

we deduce the pointwise bound

  n+1 ) and a.e. convergence of Whitney averages of u 2 to u 2,0 as λ → 0. Adding up, we obtain Whitney average convergence of u towards the boundary data f . Moreover, on splitting for almost every (λ,x, t) ∈ R n+2 * ,1 (u 1 ) + N * ,1 u 2 -u 2,0 √ λ + M x M t (u 2,0 )and we have seen that all three maximal functions are in L 2 (R n+1 ). The reverse Hölder inequality in Lemma 8.2 finally yields N * ( λ} ) ∈ L 2 (R n+1 ) as required. It remains to prove uniqueness. Let u be a weak solution to Lu = 0 with N * ( λ} ) ∈ L 2 (R n+1 ) such that for almost every (x, t) ∈ R n+1 ,

		+ ,
	u(λ, x, t) max{1, √ λ}	≤ |u 1 (λ, x, t)| +	u 2 (λ, x, t) -u 2,0 (x, t) √ λ	+ |u 2,0 (x, t)|,
	we get a pointwise bound
	N * ,1	u max{1,	√ max{1, λ} ≤ N u √
					u max{1,	√
			lim λ→0	---

Λ×Q×I |u| = 0.

≤ C(ℓ(Q) n-2 δ -2 ℓ(I) 2 + ℓ(Q) n δ -1 ℓ(I)) δ -2 |R|.

P.A. and M.E. were supported by the ANR project "Harmonic Analysis at its Boundaries", ANR-12-BS01-0013. P.A. was also supported by the NSF grant no. DMS-1440140, while being in residence at the MSRI in Berkeley, California, during the spring 2017 semester. M.E. was also supported by a public grant as part of the FMJH and thanks MSRI for hospitality. K.N. was supported by a grant from the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. K.N. thanks Université Paris-Sud for support and hospitality.

To compute the other part, we may symmetrise the argument by changing s to -s -1 and observing that (h, N P * -1 g) = -(N P -1 h, g) and that N P -1 swaps r-and ⊥-components. Hence h r , (N P * -1 g) r ) = -(N P -1 h) ⊥ , g ⊥ formally and it remains to identify what this means. We have (N P -1 h) ⊥ = u| λ=0 ∈ ( Ḣs+1 P ) ⊥ = Ḣs/2+1/2 ∂t-∆x and g ⊥ = ∂ ν A * w| λ=0 ∈ ( Ḣ-s-1 P * ) ⊥ = Ḣ-s/2-1/2 -∂t-∆x . The result follows.

10. The operator sgn(P M )

We provide the rather algebraic proofs of the results on sgn(P M ) stated in Section 2. [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF]. The appearing Sobolev spaces have been introduced in Section We keep on representing vectors by ⊥-and r-components and operators on spaces X = Ḣs P by the associated (2 × 2)-matrices and let i ⊥ : X ⊥ → X ⊥ ⊕ {0} and i r : X r → {0} ⊕ X r be the canonical embeddings. Then the maps N ⊥ , N r are just the coordinate maps with this representation. From (2.31) we obtain

With this notation set up, we are ready to give the Proof of Theorem 2.23.

(i) Based on Theorem 2.15 and its counterpart for the lower half-space, (R) L

Es is well-posed on both half-spaces if, and only if, the two operators N r : Ḣs,± P M → ( Ḣs P ) r are invertible. By an abstract principle for pairs of complementary projections [START_REF] Auscher | Representation and uniqueness for boundary value elliptic problems via first order systems[END_REF]Lem. 13.6], this in turn is equivalent to invertibility of the composite map N r χ + (P M ) i ⊥ : ( Ḣs P ) ⊥ → ( Ḣs P ) r , which in view of (10.1) exactly means that s r⊥ (P M ) : ( Ḣs P ) ⊥ → ( Ḣs P ) r is invertible. In order to avoid confusion, we note that in [START_REF] Auscher | Representation and uniqueness for boundary value elliptic problems via first order systems[END_REF] the results are equivalently formulated using the projectors

, respectively, via a convergent Neumann series. Using the relations • ; and it is invertible for τ = 0 again by assumption and since χ + 0 is a projection. Thus, the method of continuity yields invertibility for all |τ | ≤ δ. Iterating this process finitely many times, we obtain invertibility for every τ ∈ [0, 1].

Finally, we turn to compatible well-posedness. From Theorem 2.15 and well-posedness of the BVPs in the energy class, we can infer for every τ that N

) • is invertible. Together with the assumption, this entails lower bounds for N

) • , the intersections being Hilbert spaces for the sum of the two norms. Now, the crucial observation is that invertibility between these intersection spaces is equivalent to compatibility of the two inverses on ( Ḣs P ) • and ( Ḣ-1/2 P ) • since their intersection is dense in both of them. This being said, we can simply repeat the argument above, replacing systematically Ḣs P by Ḣs P ∩ Ḣ-1/2 P , to deduce that the inverses of N • are compatible for all τ provided they are for τ = 0.

Uniqueness for Dirichlet problems

Here, we give the proofs of the uniqueness results on Dirichlet problems with non-tangential maximal function control stated in Theorems 2.37 and 2.40. For simplicity, we often fix Whitney parameters in the definition of N * in such a way that Λ = (λ, 2λ), Q = B(x, λ), I = (t -λ 2 , t + λ 2 ) is a Whitney cylinder of sidelength λ > 0, where the center (x, t) ∈ R n+1 will be clear from the context. We shall freely use that different choices of Whitney parameters yield non-tangential maximal functions with comparable L 2 -norms, see Section 2.10. We also introduce a variant of the non-tangential maximal function

where the subscript is reminiscent of the fact that we are using L 1 -averages instead of L 2 -averages. Of course, N * ,1 (u) ≤ N * (u) due to Hölder's inequality and if Lu = 0 in the weak sense, then N * (u) and N * ,1 (u) are comparable in L 2 -norm thanks to the reverse Hölder estimate in Lemma 8.2.

In both theorems we have to deduce that a certain weak solution to Lu = 0 vanishes on R n+2 + . In order to bring into play the assumptions on the dual equations, we shall use test functions of the form φ = L * H. Similar to Section 2.3 we use the homogeneous energy space Ė(R n+2 ) (considered modulo constants) to define