Laurent Baratchart
email: laurent.baratchart@sophia.inria.fr

Juliette Leblond
email: juliette.leblond@sophia.inria.fr

Stéphane Rigat
email: rigat@cmi.univ-mrs.fr

Emmanuel Russ
email: emmanuel.russ@univ-cezanne.fr

Hardy spaces of the conjugate Beltrami equation

Keywords: Hardy spaces, conjugate Beltrami equation, trace, non-tangential maximal function

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents 1 < p < ∞. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation div(σ∇u) = 0 with L p -boundary data.

Background and motivation

Classical Hardy spaces of the disk or the half-plane lie at the crossroads between complex and Fourier analysis, and many developments in spectral theory and harmonic analysis originate in them [START_REF] Nikolskii | Treatise on the shift operator[END_REF][START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF][START_REF] Stein | Introduction to Fourier analysis on euclidean spaces[END_REF]. From a spectral-theoretic point of view, the shift operator and its various compressions play a fundamental role and stress deep connections between function theory on the one hand, control, approximation, and prediction theory on the other hand [START_REF] Nikolskii | Operators, functions, and systems: an easy reading[END_REF][START_REF] Parfenov | Estimates of the singular numbers of a Carleson operator[END_REF][START_REF] Peller | Hankel Operators and their Applications[END_REF]. In particular, Hankel and Toeplitz operators on Hardy classes team up with standard functional analytic tools to solve extremal problems where a function, given on part or all of the boundary, is to be approximated by traces of analytic or meromorphic functions [START_REF] Adamjan | Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem[END_REF][START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF][START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF][START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF][START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF][START_REF] Baratchart | An L p analog to the AAK theory[END_REF][START_REF] Chalendar | Approximation in reflexive Banach spaces and applications to the invariant subspace problem[END_REF][START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimumenergy transfer functions of linear systems[END_REF][START_REF] Prokhorov | On L p -generalization of a theorem of Adamjan, Arov, and Krein[END_REF]. Such techniques are of particular relevance to identification and design of linear control systems [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data, 10th conference in Analysis and optimization of systems[END_REF][START_REF] Baratchart | Robust identification in the disc algebra from band-limited data[END_REF][START_REF] Doyle | Feedback Control Theory[END_REF][START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF][START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimumenergy transfer functions of linear systems[END_REF][START_REF] Peller | Hankel Operators and their Applications[END_REF][START_REF] Seyfert | Extraction of Coupling Parameters For Microwave Filters: Determination of a Stable Rational Model from Scattering Data[END_REF]. In recent years, on regarding the Laplace equation as a compatibility condition for the Cauchy-Riemann system, analogous extremal problems were set up to handle inverse Dirichlet-Neumann issues for 2-D harmonic functions [START_REF] Baratchart | Recovery of pointwise sources or small inclusions in 2D domains and rational approximation[END_REF][START_REF] Baratchart | How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?[END_REF][START_REF] Baratchart | 2-D inverse problems for the Laplacian: a meromorphic approximation approach[END_REF][START_REF] Jacob | A constrained approximation problem arising in parameter identification[END_REF][START_REF] Jaoua | Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains[END_REF]. Laying grounds for a similar approach to inverse problems involving more general diffusion elliptic equations in the plane has been the initial motivation for the authors to undertake the present study. The equations we have in mind are of the form div(σ∇u) = 0, σ real-valued, 0 < c < σ < C,

which may be viewed, upon setting ν = (1σ)/(1 + σ), as a compatibility condition for the conjugate Beltrami equation

∂f = ν ∂f , f = u + iv, ν ∈ R, |ν| < κ < 1. (2)
This connection between (1) and ( 2) was instrumental in [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF] for the solution of Calderón's conjecture in dimension 2. Equation ( 2) breaks up into a system of two real equations that reduces to the Cauchy-Riemann system when σ = 1. It differs considerably from the better known Beltrami equation: ∂f = ν ∂f whose solutions are the so-called quasiregular mappings, which are complex linear and have been extensively studied by many authors, see [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF][START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF][START_REF] Astala | Elliptic Partial Differential Equations and Quasiconformal mappings in the plane[END_REF][START_REF] Bojarski | Homeomorphic solutions of Beltrami systems[END_REF][START_REF] Clop | Beltrami equations with coefficients in the Sobolev space W 1[END_REF][START_REF] David | Solutions de l'équation de Beltrami avec µ ∞ = 1[END_REF][START_REF] Gutlianskii | On the degenerate Beltrami equations[END_REF][START_REF] Iwaniec | What's new for the Beltrami equation? in National Research Symposium on Geometric Analysis and Applications[END_REF][START_REF] Iwaniec | Geometric Function Theory and Non-linear Analysis[END_REF][START_REF] Iwaniec | The Beltrami Equation[END_REF][START_REF] Lehto | Quasiconformal mappings in the plane[END_REF][START_REF] Morrey | On the solutions of quasi-linear elliptic partial differential equations[END_REF].

An interesting example of a free boundary inverse problem involving an equation like (1) -albeit in doubly connected geometry-arises when trying to recover the surface of the plasma in plane sections of a toroidal tokamak from the so-called Grad-Shafranov equation [START_REF] Blum | Numerical Simulation and Optimal Control in Plasma Physics: With Applications to Tokamaks[END_REF].

In the present paper, we limit ourselves to the case when σ is Lipschitz-continuous. Moreover, we merely consider analogs H p ν to the classical Hardy spaces H p in the range 1 < p < ∞, on Dini-smooth simply connected domains. From the perspective of harmonic analysis, the main features of Hardy space theory in this range of exponents [START_REF] Duren | Theory of H p spaces[END_REF][START_REF] Garnett | Bounded analytic functions[END_REF] are perhaps the Fatou theorem on non-tangential boundary values, the p-summability of the non-tangential maximal function, the boundedness of the conjugation operator, which is the prototype of a convolution operator of Calderón-Zygmund type, and the fact that subsets of positive measure of the boundary are uniqueness sets (this is false in dimension greater than 2 [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension d ≥ 3[END_REF]). In this work, we show that Hardy solutions to (2) enjoy similar properties, and we use them to establish the density of the traces of such solutions in L p (Γ) whenever Γ is a subset of non-full measure of the boundary. This fact, whose proof is straightforward for classical Hardy spaces [START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF] and can be generalized to harmonic gradients in higher dimensions when Γ is closed [START_REF] Atfeh | Bounded extremal and Cauchy Laplace problems on the sphere and shell[END_REF], is of fundamental importance in extremal problems with incomplete boundary data and one of the main outcome of the paper. The generalized Hilbert transform H ν involved in [START_REF] Adams | Sobolev spaces[END_REF], that maps the boundary values of u to the boundary values of v, was introduced and studied in [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF][START_REF] Astala | A boundary integral equation for Calderón's inverse conductivity problem[END_REF] when p = 2 for less smooth (i.e. measurable bounded) σ but smoother (i.e. Sobolev W 1/2,2 ) arguments. Here, we shall prove its L p and W 1,p boundedness and compare it to the classical conjugation operator. In addition, studying its adjoint will lead us to a representation theorem for the dual of H p ν which generalizes the classical one. The latter is again of much importance when studying extremal problems. On our way to the proof of the density theorem, we establish regularity results for the solutions of (1) which are not entirely classical. For example, we obtain an analog of the Fatou theorem concerning solutions of the Dirichlet problem for [START_REF] Adamjan | Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem[END_REF] with L p boundary data, including L p -estimates for the nontangential maximal function. Also, the gradient of a solution to the Neumann problem with L p data has L p nontangential boundary values a.e. as well as L p -summable nontangential maximal function. For the ordinary Laplacian this is known to hold on C 1 domains in all dimensions [START_REF] Fabes | Potential techniques for boundary value problems on C 1 domains[END_REF], and on Lipschitz domains for restricted range of p [START_REF] Jerison | Boundary Value Problems on Lipschitz Domains[END_REF]. But for diffusion equations of the form (1), the authors could not locate such a result in the literature, even in dimension two; when p = 2 and σ is smooth, it follows from [50, Ch. II, Thms 7.3, 8.1] that this gradient converges radially in L 2 on (parallel transportations of) the boundary, and the result could be carried over to any p ∈ (1, ∞) using the methods of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], but no pointwise estimates are obtained this way 1 . The definition of generalized Hardy spaces that we use -see [START_REF] Auscher | Observations on W 1,p estimates for divergence elliptic equations with VMO coefficients[END_REF] below-dwells on the existence of harmonic majorants for |f | p , and also on the boundedness of L p norms of f on Jordan curves tending to the boundary of the domain [START_REF] Duren | Theory of H p spaces[END_REF]. As in the classical case, these two definitions of Hardy spaces coincide on Dini-smooth domains (the only case of study below) but not over non-smooth domains -where arclength on the boundary and harmonic measure are no longer mutually absolutely continuous. Although equation ( 2) is real linear only, our methods of investigation rely on complex analytic tools. In particular, we elaborate on ideas and techniques from [START_REF] Bers | On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications[END_REF] and we use standard facts from classical Hardy space theory together with well-known properties of the Beurling transform. This entails that higher dimensional analogs of our results, if true at all, require new ideas to be proven. We made no attempt at expounding the limiting cases p = 1, ∞. These have generated the deepest developments in the classical theory, centering around BMO and Fefferman duality, but trying to generalize them would have made the paper unbalanced and they are left here for further research. Finally, we did not consider Hardy spaces over doubly connected domains, in spite of the fact that the above-mentioned application to free boundary problems in plasma control takes place in an annular geometry. Including these would have made for a lengthy paper, but the results below lay ground for such a study.

Notations for function spaces

Throughout, D is the open unit disk and T the unit circle in the complex plane C. We let D r and T r stand for the open disk and the circle centered at 0 with radius r. For I an open subset of T, endowed with its natural differentiable structure, we put D(I) for the space of C ∞ complex functions supported on I.

If Ω ⊂ C is a smooth domain (the meaning of "smooth" will be clear from the context), we say that a sequence ξ n ∈ Ω approaches ξ ∈ ∂Ω non tangentially if it converges to ξ while no limit point of (ξ nξ)/|ξ n -ξ| is tangent to ∂Ω at ξ. A function f on Ω has non tangential limit ℓ at ξ if f (ξ n ) tends to ℓ for any sequence ξ n which approaches ξ non tangentially.

Hölder spaces

If Ω ⊂ R 2 is open, C k,γ (
Ω) indicates the subspace of complex functions whose derivatives are bounded and continuous up to order k, while those of order k satisfy a Hölder condition of exponent γ ∈ (0, 1]. Such functions extend continuously to Ω together with their derivatives of order at most k. A complete norm on C k,γ (Ω) is obtained by putting

f C k,γ (Ω) := sup 0≤|λ|≤k f (λ) L ∞ (Ω) + sup |λ|=k ξ =ζ |f (λ) (ξ) -f (λ) (ζ)| |ξ -ζ| γ , where λ = (λ 1 , λ 2 ) is a multi-index, |λ| = λ 1 +λ 2 , and f (λ) is the corresponding derivative. The space C ∞ (Ω) = ∩ k C k,1
(Ω) of smooth functions up to the boundary of Ω is topologized with the seminorms C k,1 (Ω) , where k ranges over N. We put C k,γ loc (Ω) for the functions whose restriction to any relatively compact open subset Ω 0 of Ω lies in C k,γ (Ω 0 ). A family of semi-norms making C k,γ loc (Ω) into a Fréchet space is given by C k,γ (Ωn) , with Ω n a sequence of relatively compact open subsets exhausting Ω.

Lebesgue and Sobolev spaces

We coordinatize R 2 ≃ C by ξ = x + iy and denote interchangeably the (differential of) planar Lebesgue measure by

dm(ξ) = dx dy = (i/2)dξ ∧ dξ ,
where dξ = dx + idy and dξ = dxidy. For 1 ≤ p ≤ +∞ and E a measurable subset of C, we put L p (E) for the familiar Lebesgue space with respect to dm. Let D(Ω) be the space of complex C ∞ functions with compact support in Ω, endowed with the inductive topology. Its dual D ′ (Ω) is the usual space of distributions on Ω. Whenever T ∈ D ′ (Ω), we use the standard notations:

∂T = ∂ z T = 1 2 (∂ x -i∂ y ) T and ∂T = ∂ z T = 1 2 (∂ x + i∂ y ) T,
and record the obvious identity: ∂T = ∂ T . We denote by W 1,p (Ω) the Sobolev space of those f ∈ L p (Ω) whose distributional derivatives ∂f and ∂f also belong to L p (Ω). The norm on W 1,p (Ω) is defined by

f p W 1,p (Ω) := f p L p (Ω) + ∂f p L p (Ω) + ∂f p L p (Ω) .
When Ω is smooth, any function f ∈ W 1,p (Ω) has a trace on ∂Ω (all the domains under consideration in the present paper are smooth enough for this to be true) which will be denoted by trf . The symbol W 1,p loc (Ω) refers to those distributions whose restriction to any relatively compact open subset Ω 0 of Ω lies in W 1,p (Ω 0 ). Equipped with the semi-norms

W 1,p (Ωn) , W 1,p loc (Ω) is a Fréchet space. The space W 1,∞ (Ω) is isomorphic to C 0,1 (Ω) [64, Ch.VI, Sec. 6.2]. In particular, every f ∈ W 1,∞ (Ω) extends (Lipschitz) continuously to the boundary ∂Ω of Ω. A C 1 -smooth Jordan curve is the injective image of T under a nonsingular continuously differentiable map from T into C.
For Γ an open subset of such a curve, we denote by L p (Γ) the Lebesgue space with respect to (normalized) arclength (there should be no confusion with our previous notation L p (E), as the context will always remain clear) and by W 1,p (Γ) the Sobolev space of those absolutely continuous ϕ ∈ L p (Γ) whose tangential derivative ∂ t ϕ with respect to arclength again lies in L p (Γ). A complete norm on W 1,p (Γ) is given by

ϕ p W 1,p (Γ) := ϕ p L p (Γ) + ∂ t ϕ p L p (Γ)
. Note that, on T, the tangential derivative ∂ t h coincides with ∂ θ h/(2π) where ∂ θ h indicates the derivative with respect to θ when ϕ is written as a function of e iθ .

We shall have an occasion to deal with W 2,p (Ω), comprised of W 1,p (Ω)-functions whose first derivatives again lie in W 1,p (Ω). A norm on W 2,p (Ω) is obtained by setting

f p W 2,p (Ω) = f p L p (Ω) + ∂f p W 1,p (Ω) + ∂f p W 1,p (Ω) .
As is customary, we indicate with a subscript "0" the closure of C ∞ compactly supported functions in an ambient space.

Finally, we indicate with a subscript "R", like in L p R (Ω), the real subspace of real-valued functions in a given space.

3 Definition of Hardy spaces

An elliptic equation

In the present paper, we investigate the L p boundary behaviour of solutions to a second order elliptic equation in divergence form on a planar domain. More precisely, let Ω ⊂ R 2 be a smooth simply connected domain (most of the time, we will take Ω = D, except in Section 6, where Ω will be assumed to be Dini-smooth) and σ ∈ W 1,∞ (Ω) be such that, for two constants c, C, one has 0

< c ≤ σ ≤ C . (3)
With the standard notation ∇u := (∂ x u , ∂ y u) T and div g = ∂ x g + ∂ y g, where the superscript "T " means "transpose", the elliptic equation that we will consider is div(σ∇u) = 0 a.e. in Ω,

Our approach to (4) proceeds via the study of a complex elliptic equation of the first order, namely the conjugate Beltrami equation:

∂f = ν ∂f a.e. in Ω , (5)
where ν ∈ W 1,∞ (Ω) is a real valued function that satisfies:

ν L ∞ (Ω) ≤ κ for some κ ∈ (0, 1) . (6)
Formally, equation ( 5) decomposes into a system of two real elliptic equations of the second order in divergence form. Indeed, for f = u + iv a solution to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] with real-valued u, v, we see on

putting σ = (1 -ν)/(1 + ν) that u satisfies equation (4) while v satisfies div 1 σ ∇v = 0 a.e. in Ω. ( 7
)
Note also, from the definition of σ, that (6) implies (3). Conversely, let u be a real-valued solution to (4). Then, since ∂ y (-σ∂ y u) = ∂ x (σ∂ x u) and Ω is simply connected, there is a real-valued function v, such that

∂ x v = -σ∂ y u, ∂ y v = σ∂ x u, (8)
hence f = u + iv satisfies (5) with ν = (1σ)/(1 + σ). Moreover, (3) implies [START_REF] Astala | Elliptic Partial Differential Equations and Quasiconformal mappings in the plane[END_REF].

In the present work, we consider several classes of solutions to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] for which the formal manipulations above will be given a precise meaning. All classes we shall deal with are embedded in L p (Ω) for some p ∈ (1, ∞), in which case the solutions to (4), [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF], and ( 7) can be understood in the distributional sense. This only requires defining distributions like σ∂ x u, which is done naturally using Leibniz's rule 2 when σ ∈ W 1,∞ (Ω) and u ∈ L p (Ω) [START_REF] Clop | Beltrami equations with coefficients in the Sobolev space W 1[END_REF].

It will turn out that our solutions actually lie in W 1,p loc (Ω) for some p ∈ (1, ∞), in which case (5) may as well be interpreted in the pointwise sense while (4) becomes equivalent to

Ω σ∇u.∇g dm = 0, g ∈ D R (Ω), (9)
where the dot indicates the Euclidean scalar product in R 2 . This follows easily from the fact that the product of a function in W 1,∞ (Ω) by a function in W 1,p loc (Ω) again lies in W 1,p loc (Ω) and its distributional derivative can be computed according to the Leibniz rule. We shall make use of these observations without further notice.

To find u with prescribed trace on ∂Ω is known as the Dirichlet problem for (4) in Ω. In light of the previous discussion, we slightly abuse terminology and still refer to the issue of finding f with prescribed Ref on ∂Ω as being the Dirichlet problem for [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF].

For simplicity, we shall work entirely over the unit disk D and only later, in Section 6, shall we indicate how one can carry our results over to Dini-smooth domains. As became customary in analysis, we tend to use the same symbol to mean possibly different constants, with subscripts indicating the dependence of the constant under examination.

When ν ∈ W 1,∞ (D), the solvability in W 1,p (D) of the Dirichlet problem for [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] with boundary data in the fractional Sobolev space W 1-1/p,p R (T) (an intrinsic definition of which can be found in [START_REF] Adams | Sobolev spaces[END_REF]Thm 7.48]) is a straightforward consequence of the known solvability of the corresponding Dirichlet problem for equation (4) [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF]. We shall however state and establish this fact which is our point of departure (see Theorem 4.1.1 in Section 4 below).

Below, we relax the assumptions on the boundary data, assuming only they belong to L p (T). Of course, the solution of the Dirichlet problem will no longer belong to W 1,p (D) in general, but rather to some generalized Hardy space H p ν (D) that we shall define and study throughout the paper, starting in the next section. , by our convention that arclength gets normalized, see [START_REF] Duren | Theory of H p spaces[END_REF][START_REF] Garnett | Bounded analytic functions[END_REF].

Definition of Hardy spaces

Of course H p can be introduced for p = 1, ∞ as well, but we do not consider such exponents here. We extend the previous definition to two classes of generalized analytic functions as follows.

The class

H p ν (D) If ν ∈ W 1,∞ R (D)
D) has a non-tangential limit a e. on T that we call the trace of f , denoted by tr f (see Section 2 for the definition of the non-tangential limit). This definition causes no discrepancy since, as we shall see in Proposition 4.3.3 below, any solution of (5) in W 1,p (D) belongs to H p ν (D) and, for an arbitrary function f ∈ W 1,p (D), the nontangential limit of f , when it exists, coincides with the trace of f in the Sobolev sense. It turns out that, for all f ∈ H p ν (D), tr f lies in L p (T) and tr f L p (T) defines an equivalent norm on H p ν (D). We single out the subspace H The explicit connection between H p ν and G p α is given by the following result, which relies on a transformation introduced in [START_REF] Bers | On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications[END_REF]:

Proposition 3.2.3.1. Let ν ∈ W 1,∞ R (D) satisfy (6) and define σ ∈ W 1,∞ R (D), α ∈ L ∞ (D) by σ = 1 -ν 1 + ν , α := - ∂ν 1 -ν 2 = ∂σ 2σ = ∂ log σ 1/2 . ( 16
)
Then f ∈ L p (D) solves [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] in the distributional sense if, and only if the function w, defined by

w := (f -νf )/ √ 1 -ν 2 = σ 1/2 u + i σ -1/2 v ( 17
)
does for [START_REF] Baratchart | How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?[END_REF]. Moreover, The proof is a straightforward computation, using that the distributional derivatives of (fνf )/ √ 1ν 2 can be computed by Leibniz's rule under our standing assumptions, and the fact that [START_REF] Baratchart | Robust identification in the disc algebra from band-limited data[END_REF] can also be rewritten as

(a) f = u + i v
f = w + νw √ 1 -ν 2 . ( 18
)
Observe that every constant c ∈ C is a solution to (5), the associated w via (17) being σ 1/2 Re c + i σ -1/2 Im c, which lies in W 1,∞ (D) and solves [START_REF] Baratchart | How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?[END_REF].

4 Statement of the results.

Throughout, we assume 1 < p < ∞, and we let

ν ∈ W 1,∞ R (D) satisfy ν L ∞ (D) ≤ κ < 1.

Solvability in Sobolev spaces

Our first result deals with the solvability of the Dirichlet problem for [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] with boundary data in W 1-1/p,p (T): [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] in D and such that Re (tr f ) = ϕ on T. Such an f is unique up to an additive pure imaginary constant.

Theorem 4.1.1. Let p ∈ (1, +∞) and ν ∈ W 1,∞ R (D) satisfy (6). (a) To each ϕ ∈ W 1-1/p,p R (T), there is f ∈ W 1,p (D) solving
(b) There exists C p,ν > 0 such that the function f in (a), when normalized by [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF],

satisfies f W 1,p (D) ≤ C p,ν ϕ W 1-1/p,p (T) . ( 19
)
Remark 4.1.1. Although we will not use it, let us point out that Theorem 4.1.1 still holds if we merely assume ν ∈ V MO(D), provided [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] is understood in the pointwise sense. The proof is similar, appealing to [START_REF] Auscher | Observations on W 1,p estimates for divergence elliptic equations with VMO coefficients[END_REF] rather than [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF] to solve the Dirichlet problem for (4).

Remark 4.1.2. When ϕ ∈ W 1-1/p,p R (T) and u ∈ W 1,p R (D)
is the solution to [START_REF] Atfeh | Bounded extremal and Cauchy Laplace problems on the sphere and shell[END_REF] such that tr u = ϕ granted by [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF], the normal derivative ∂ n u is classically defined as the unique member of the dual space

W -1/p,p R (T) = W 1-1/q,q R (T) * such that ∂ n u , σψ = D σ∇u.∇g dm , ψ ∈ W 1-1/q,q R (T), g ∈ W 1,q (D), tr g = ψ, ( 20
)
where g is any representative of the coset tr [START_REF] Bers | On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications[END_REF] depends on the fact that M σ , the multiplication by σ, is an isomorphism of W 1-1/q,q R (T); this follows by interpolation since M σ is an isomorphism both of L q R (T) and W 1,q R (T). Now, if f = u + iv ∈ W 1,p (D) is a solution to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] such that Re (tr f ) = ϕ as provided by Theorem 4.1.1, it is a straightforward consequence of (8) that ∂ n u = (∂ θ tr v)/σ.

-1 ψ in W 1,p R (D)/W 1,p 0,R (D). That ∂ n u is well- defined via
The results below generalize to H p ν and G p α , defined in Section 3.2, some fundamental properties of holomorphic Hardy classes [START_REF] Duren | Theory of H p spaces[END_REF][START_REF] Garnett | Bounded analytic functions[END_REF]. Observe that, on D, as the above definition shows (see Section 2), f has a non tangential ("n.t.") limit ℓ at e iθ ∈ T if, and only if for every 0 < β < π/2, f (z) tends to ℓ as z → e iθ inside any sector Γ e iθ ,β with vertex e iθ , of angle 2β, which is symmetric with respect to the ray (0, e iθ ). The non-tangential maximal function of f at ξ ∈ T is

M f (ξ) := sup z∈D∩Γ ξ,β |f (z)|, ( 21
)
where we dropped the dependence of M f on β.

We first mention properties of the class G p α , from which those of the class H p ν will be deduced using Proposition 3.2.3.1.

Properties of G p α

We fix α ∈ L ∞ (D). To proceed with the statements, we need to introduce two operators that will be of constant use in the paper. First, for ψ ∈ L 1 (T), we define a holomorphic function in D through the Cauchy operator:

Cψ(z) = 1 2πi T ψ(ξ) ξ -z dξ , z ∈ D .
It follows from a theorem of M. Riesz that C maps L p (T) onto H p , see the discussion after [START_REF] Garnett | Bounded analytic functions[END_REF]Ch. 3,Thm 1.5]; this would fail if p = 1, ∞.

Second, for p ∈ (1, +∞) and w ∈ L p (D), we define

T w(z) = 1 2πi D w(ξ) ξ -z dξ ∧ dξ , z ∈ D .
The following representation theorem for functions in G p α was implicit in [START_REF] Bers | On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications[END_REF] for continuous W 1,2 (D)-solutions to ( 14): Theorem 4.2.1. Let w ∈ L p (D) be a distributional solution to [START_REF] Baratchart | How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?[END_REF]. Then w can be represented as

w(z) = exp(s(z)) F (z) , z ∈ D , ( 22
)
where s ∈ W 1,l (D) for all l ∈ (1, +∞) and F is holomorphic in D. Moreover, s can be chosen such that its real part (or else its imaginary part) is 0 on T and

s L ∞ (D) ≤ 4 α L ∞ (D) . (23)
In particular w ∈ W 1,l loc (D) for all l ∈ (1, +∞), and w ∈ G p α if, and only if F ∈ H p in some, hence any factorization of the form [START_REF] Bojarski | Homeomorphic solutions of Beltrami systems[END_REF]. Moreover, w ∈ L p 1 (D), for all p 1 ∈ [p, 2p).

w(ξ) (24)
exists for almost every θ and

tr w L p (T) ≤ w G p α ≤ c α tr w L p (T) (25)
for some c α > 0. Moreover,

lim r→1 2π 0 w(re iθ ) -tr w(e iθ ) p dθ = 0 (26)
and, for any aperture β ∈ (0, π/2) of the sectors used in definition [START_REF] Blum | Numerical Simulation and Optimal Control in Plasma Physics: With Applications to Tokamaks[END_REF], there is a constant C p,α,β such that

M w L p (T) ≤ C p,α,β tr w L p (T) . ( 27
)
2. If w ∈ G p α and w ≡ 0, then log |tr w| ∈ L 1 (T); moreover the zeros of w are isolated in D, and numbering them as

α 1 , α 2 , • • • , counting repeated multiplicities, it holds that ∞ j=1 (1 -|α j |) < +∞. ( 28
)
3. Let w ∈ L p (D). Then w ∈ G p α if, and only if there is a function ϕ ∈ L p (T) such that

w = Cϕ + T (αw) , a.e. in D . ( 29
)
In this situation,

w G p α ≤ C p,α w L p (D) + ϕ L p (T) . (30)
A valid choice in ( 29) is ϕ = tr w.

4. If w ∈ G p α satisfies [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF] and Re tr w = 0 a.e. on T, then w ≡ 0 in D. When (16) holds, the same is true if w ∈ G p,0 α .

Remark 4.2.2. From (25) and the completeness of G p α , we see that tr G p α is a closed subspace of L p (T). We also observe, in view of the M. Riesz theorem, that assertion 3 can be recaped as: (a) If f ∈ H p ν , then f has a non-tangential limit a.e. on T, denoted by tr f , the L p (T)norm of which is equivalent to the H p ν -norm of f :

w ∈ G p α ⇐⇒ w -T (αw) ∈ H p .
tr f L p (T) ≤ f H p ν (D) ≤ c ν tr f L p (T) . (31)
Moreover

lim r→1 2π 0 f (re iθ ) -tr f (e iθ ) p dθ = 0,
and we have that

f ∈ L p 1 (D) for p ≤ p 1 < 2p. (b) The image space tr H p ν (resp. tr H p,0 ν ) is closed in L p (T). (c) Each f ∈ H p ν is such that log |tr f | ∈ L 1 (T) unless f ≡ 0. (d) If f ∈ H p ν and f ≡ 0, then its zeros are isolated in D; if we enumerate them as α 1 , α 2 , • • • , counting repeated multiplicities, then (28) holds.
(e) For any aperture β ∈ (0, π/2) of the sectors used in definition [START_REF] Blum | Numerical Simulation and Optimal Control in Plasma Physics: With Applications to Tokamaks[END_REF], there is a constant C p,ν,β such that 17) by [START_REF] Baratchart | 2-D inverse problems for the Laplacian: a meromorphic approximation approach[END_REF], so that items (a)-(e) follow at once from Proposition 4.2.1, Remark 4.2.2, and the fact that f and w share the same zeros because

M f L p (T) ≤ C p,ν,β tr f L p (T) . (f ) Each f ∈ H p ν satisfies
w = f -νf √ 1 -ν 2 .
To prove (f ), observe from Theorem 4.2.1 that w, thus also f belongs to

W 1,l loc (D) for each 1 < l < ∞. Moreover if we let ν f (z) := ν(z)∂f (z)/∂f (z)if ∂f (z) = 0 and ν f (z) = 0 otherwise, then f is a pointwise a.e. solution in D of the classical Beltrami equation: ∂f = ν f ∂f, |ν f | ≤ κ < 1. ( 32
)
Indeed, |ν(z)| ≤ κ for all z ∈ D and ∂f (z)/∂f (z) = 1 when ∂f (z) = 0. It is then a standard result [43, Thm 11.1.2] that f = G(h(z))
, where h is a quasi-conformal topological map D → C satisfying [START_REF] Duren | Theory of H p spaces[END_REF] and G a holomorphic function on h(D). The conclusion follows at once from the corresponding properties of holomorphic functions. The next result shows that H p ν contains all W 1,p (D) solutions to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. That this inclusion is a strict one follows at once from Theorem 4.4.2.1 to come. [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. Then f ∈ H p ν (D), and there exists C ν,p > 0 such that,

Proposition 4.3.3. Let f ∈ W 1,p (D) be a solution to
f H p ν (D) ≤ C ν,p f W 1,p (D) . ( 33
)
Moreover, the trace of f considered as an element of W 1,p (D) coincides with its trace seen as an element of H p ν (D). Note that [START_REF] Fabes | Potential techniques for boundary value problems on C 1 domains[END_REF] follows immediately from [START_REF] Doyle | Feedback Control Theory[END_REF] and the continuity of the trace operator from W 1,p (D) into L p (T), once it is known that f ∈ H p ν . The proof of Proposition 4.3.3 is given in Section 5.4.

P + ψ = tr (Cψ)
where the trace is a nontangential limit. As is well-known, P + ψ exists a.e. on T as soon as ψ ∈ L 1 (T), but it may not lie in L 1 (T). If, however, 1 < p < ∞, then P + is a continuous projection from L p (T) onto tr H p called the analytic projection [START_REF] Garnett | Bounded analytic functions[END_REF]Ch. III,Sec. 1]. It is an interesting variant of the Dirichlet problem to solve equation ( 14) while prescribing the analytic projection of the solution on T. As the next theorem shows, G p α is a natural space for this. Theorem 4.4.1.1. For α ∈ L ∞ (D) and g ∈ H p , there is a unique w ∈ G p α such that

P + (tr w) = tr g . ( 34
)
This solution satisfies

w = g + T (αw) , a.e. in D , (35)
and it holds that

w G p α ≤ C p,α g H p (D) . (36)
Here is now the solution of the (usual) Dirichlet problem for the class

G p α : Theorem 4.4.1.2. Let α ∈ L ∞ (D) and ψ ∈ L p R (T).
(b) When (16) holds, there uniquely exists w ∈ G p,0 α such that Re (tr w) = ψ a.e. on T. Furthermore, there is a constant c ′′ p,α such that

tr w L p (T) ≤ c ′′ p,α ψ L p (T) . (38)
The proofs of Theorems 4.4.1.1-4.4.1.2 are given in section 5.5.

Solvability of the Dirichlet problem in H p ν (D)

The following result shows that H p ν is the natural space to consider when handling L p boundary data in ( 5) and (4). Re

(tr f ) = ϕ . ( 39
)
Moreover, there exists c p,ν > 0 such that: Let us give at once the proof of Theorem 4.4.2.1, which is quite easy to deduce from previous statements. Define α through [START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF] and put ψ = ϕσ 1/2 ∈ L p R (T). Apply Theorem 4.4.1.2, point (b), to obtain w ∈ G p,0 α such that Re (tr w) = ψ. If we let f be given by ( 18), then f ∈ H p,0 ν by Proposition 3.2.3.1, point (a). Moreover, from [START_REF] Baratchart | Robust identification in the disc algebra from band-limited data[END_REF], we see that [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] holds. The uniqueness of f comes from Proposition 4.3.2. Inequality [START_REF] Gutlianskii | On the degenerate Beltrami equations[END_REF] follows from ( 18), ( 3), [START_REF] Goluzin | Generalized Carleman formula and its applications to analytic extension of functions[END_REF] and [START_REF] Carleson | Selected problems in exceptional sets[END_REF].

f H p ν (D) ≤ c p,ν ϕ L p (T) . (40)
Dwelling on Proposition 4.3.1 and Theorem 4.4.2.1, we are now able to derive an analog of the Fatou theory [35, Ch. I, Sec. 5] for (4), at least when 1 < p < ∞. It should be compared to classical results on the Dirichlet problem in Sobolev classes [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF][START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. For once, we recall all the assumptions to ease this comparison.

Theorem 4.4.2.2. Let 1 < p < ∞ and σ ∈ W 1,∞ R (D) satisfy (3). Any u ∈ L p R (D) satisfying (4) in the sense of distributions lies in W 1,l R,loc (D) for all l ∈ (1, ∞). If, moreover, u F,p := ess sup 0<r<1 u L p (Tr ) < +∞ , (41)
then u has a nontangential limit tr u a.e. on T which is also the limit of e iθ → u(re iθ ) in L p R (T) as r → 1 -. In this case, for M |u| the non tangential maximal function, we have

tr u L p R (T) ≤ u F,p ≤ c p,ν M |u| L p (T) ≤ C p,ν tr u L p R (T)
. Conversely, each member of L p R (T) is uniquely the non tangential limit of some distributional solution u ∈ L p (D) of (4) satisfying u F,p < +∞.

The proof of Theorem 4.4.2.2 is carried out in Section 5.6. Theorem 4.4.2.1 allows one to define a generalized conjugation operator H ν from L p (T) into itself, that was introduced on W 1/2,2 (T) in [START_REF] Astala | A boundary integral equation for Calderón's inverse conductivity problem[END_REF]

Improved regularity results for the Dirichlet problem

We turn to higher regularity for solutions to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. More precisely, we shall study the improvement in the conclusion of Theorem 4.4.2.1 when the boundary condition lies in

W 1,p R (T) ⊂ W 1-1/p,p R (T).
tr f W 1,p (T) ≤ C p,ν ϕ W 1,p (T) . (42)
(b) The functions ∂f and ∂f satisfy a Hardy condition of the form ess sup

0<r<1 ∂f L p (Tr ) ≤ C p,ν tr f W 1,p (T) , (43)
ess sup

0<r<1 ∂f L p (Tr) ≤ C p,ν tr f W 1,p (T) , (44)
and for the non tangential maximal function of ∇f , it holds that

M ∇f L p (T) ≤ C p,ν,β tr f W 1,p (T) , (45)
where ∇f (ξ) ∈ C 2 is the gradient of f and β the aperture of the sectors in [START_REF] Blum | Numerical Simulation and Optimal Control in Plasma Physics: With Applications to Tokamaks[END_REF].

(c) If we define Φ ∈ L p (T) by Φ(e iθ ) := -ie -iθ ∂ θ (tr f )(e iθ ) -ν(e iθ )∂ θ (tr f )(e iθ ) 1 -ν 2 (e iθ ) , (46)
then ∂f and ∂f have non tangential limit Φ and νΦ a.e. on T, and ∂f (re iθ ), ∂f (re iθ ) converge in L p (T) to their respective nontangential limits as r → 1.

Clearly, (a) is a rephrasing of Proposition 4.4.

ess sup 0<r<1 ∇u L p (Tr ) < +∞. ( 47
)
Then tr u ∈ W 1,p (T) and u ∈ W 1,p 1 R (D) for every p 1 ∈ (p, 2p). Moreover, M ∇u ∈ L p (T), and there exists a vector field Φ ∈ L p (T, R n ) such that ∇u → Φ n.t. almost everywhere on T. In particular, ∂ n u ∈ L p (T), and one has T σ∂ n u = 0.

Conversely, if g ∈ L p

R (T) satisfies T σg = 0, there exists a function u ∈ W 1,p R (D) solving div (σ∇u) = 0 in D such that ∇u satisfies a Hardy condition of the form [START_REF] Jerison | Boundary Value Problems on Lipschitz Domains[END_REF], M ∇u ∈ L p (T) and ∂ n u = g on T. Moreover, u is unique up to an additive constant.

All these results will be established in Section 5.7.

Density of traces

We come to some density properties of traces of solutions to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. Loosely speaking, they assert that if E ⊂ T is not too large, then every complex function on E can be approximated by the trace of a solution to (5) on D.

Density in Sobolev spaces

We say that an open subset I of T has the extension property if every function in W 1,p (I) is the restriction to I of some function in W 1,p (T). If I is a proper open subset of T, it decomposes into a countable union of disjoint open arcs (a j , b j ) and the extension property is equivalent to the fact that no a j (resp. b j ) is a limit point of the sequence (b k ) (resp. (a k )). We begin with a density property of Sobolev solutions to (5) on proper extension subsets: This should be held in contrast with the fact that the traces on T of W 1,p (D)-solutions to (5) form a proper closed subspace of W 1-1/p,p (T). The proof of Theorem 4.5.1.1 is given in Section 5.8.1.

Density in Lebesgue spaces

By the density of W 1-1/p,p (I) in L p (I), Theorem 4.5.1.1 easily implies that (tr H p ν ) | I is a dense subset of L p (I) for I a proper open subset of T having the extension property. The fact that this remains true as soon as I is not of full measure lies a little deeper: It is worth recasting Remark 4.5.2.1 in terms of ill-posedness of the inverse Dirichlet-Neumann problem from incomplete boundary data. Indeed, assume that u satisfies (4) and, say, tr u ∈ W 1,p (T). Observe that the normal derivative ∂ n u exists as a nontangential limit in L p (T) by Theorem 4.4.3.1. Thus, upon rewriting (8) on T in the form

∂ n v = -σ∂ θ u, ∂ θ v = σ∂ n u, (48)
we see that the knowledge of tr u and tr ∂ n u on some arc I ⊂ T is equivalent to the knowledge on I of tr f where f = u + iv meets (5) with boundary conditions Ref = u and, say, I v = 0. Note from Proposition 4.3.1 that this determines f completely. Now, if the knowledge of tr f gets corrupted by measurements and rounding off errors, as is the case in computational and engineering practice, it can still be approximated arbitrarily well in W 1-1/p,p (I) by a solution to (5) but the trace of the latter will grow large in L p (T \ I), a fortiori in W 1-1/p,p (T \ I) when the approximation error gets small. The proof of Theorem 4.5.2.1 is carried out in Section 5.8.2.

Duality

Keeping in mind that 1 < p < ∞ and 1/p + 1/q = 1, we introduce a duality pairing on L p (T) × L q (T), viewed as real vector spaces, by the formula:

f, g = Re 2π 0 f g dθ 2π . ( 49
)
Clearly this pairing isometrically identifies L q (T) with the dual of L p (T). The fact that H p is the orthogonal space to e iθ H q under (49) is basic to the dual approach of extremal problems in holomorphic Hardy spaces [START_REF] Duren | Theory of H p spaces[END_REF]Ch. 8]. In this section, we derive the corresponding results for the spaces H p ν . Recall that ∂ t = ∂ θ /2π on T. Proposition 4.6.1. The orthogonal to tr H p ν under the duality pairing defined in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF] is (i) Under the pairing [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF], the dual space (tr H p ν ) * of tr H p ν is naturally isometric to the quotient space L q (T)/(tr H p ν ) ⊥ , that is

(tr H p ν ) ⊥ = ∂ θ tr H q -ν ∩ W 1,q (T) .
(tr H p ν ) * ∼ L q (T)/ ∂ θ tr H q -ν ∩ W 1,q (T) .
(ii) For each Φ ∈ L q (T), it holds the duality relation

inf g∈∂ θ (tr H q -ν ∩W 1,q (T)) Φ -g L q (T) = sup f ∈H p ν tr f L p (T)=1 1 2π Re 2π 0 Φ tr f dθ. (50)
(iii) For each Ψ ∈ L p (T), it holds the duality relation [START_REF] Duren | Theory of H p spaces[END_REF]Thm 3.11]) that ∂ θ (tr H q ∩ W 1,q (T)) = e iθ H q , hence (50)-( 51) reduce to standard duality relations in Hardy spaces when ν = 0.

inf f ∈H p ν Ψ -tr f L p (T) = sup g∈tr H q -ν ∩W 1,q (T) ∂ θ g L q (T) =1 1 2π Re 2π 0 Ψ ∂ θ g dθ. (51)
The proofs of Proposition 4.6.1 and Theorem 4.6.1 will be given in Section 5.9.

5 Proofs

The Dirichlet problem in Sobolev spaces

Let us give first the proof of Theorem 4.1.1. Put σ = (1-ν)/(1+ν) , so that σ ∈ W 1,∞ (D) satisfies [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF]. By [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF], there uniquely exists u ∈ W 1,p (D) meeting tr u = ϕ for which [START_REF] Atfeh | Bounded extremal and Cauchy Laplace problems on the sphere and shell[END_REF] holds with Ω = D; moreover, by the open mapping theorem, one has

u W 1,p (D) ≤ c p,ν ϕ W 1-1/p,p (T) . (52)
Put G 0,p := {∇g; g ∈ W 1,p 0,R (D)} and

D q := {(∂ y h, -∂ x h) T ; h ∈ W 1,q R (D)}. ( 53
)
Proceeding by density on the divergence formula for smooth functions, we easily get

G, D := D G.D dm = 0, G ∈ G 0,p , D ∈ D q . ( 54
)
Now, by Hodge theory [43, Thm 10.5.1]5 , each vector field in L p (D) × L p (D) (resp. L q (D) × L q (D)) is uniquely the sum of a member of G 0,p (resp. G 0,q ) and a member of D p (resp. D q ). If we set accordingly σ∇u = G + D, we gather by density from ( 9) and ( 54) that G, V = 0 for every V ∈ L q (D) × L q (D), implying that σ∇u ∈ D p . In other words there is v ∈ W 1,p R (D) for which (8) holds, thus by inspection f = u + iv ∈ W 1,p (D) satisfies (5) pointwise a.e. on D. Then, f satisfies (5) in the distributional sense as well. To check f is unique, subject to Re tr f = ϕ, up to an additive pure imaginary constant, observe if f ∈ W 1,p (D) satifies ( 5) that u = Ref and v = Imf both lie in W 1,p R (D) and that (8) holds. Therefore σ∇u ∈ D p and, in view of (54), we see that [START_REF] Atfeh | Bounded extremal and Cauchy Laplace problems on the sphere and shell[END_REF] holds with Ω = D. As such a u is uniquely defined by tr u = ϕ, we conclude that v is uniquely defined by [START_REF] Astala | A boundary integral equation for Calderón's inverse conductivity problem[END_REF], up to an additive constant. Next,we observe from ( 8) and (3) that ∇v L p (D) ≤ C ∇u L p (D) . Therefore, if v gets normalized by [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF], it follows from [START_REF] Morrey | On the solutions of quasi-linear elliptic partial differential equations[END_REF] and the Poincaré inequality [START_REF] Ziemer | Weakly Differentiable Functions[END_REF]Ch. 4

, Ex. 4.10] that v W 1,p (D) ≤ C p ∇v L p (D) ≤ CC p c p,ν ϕ W 1-1/p,p (T)
so that (19) indeed holds.

Preliminaries on spaces and operators

In the present subsection, we recall some properties of the operators C and T , introduced in Section 4.2, and of the Beurling operator appearing in equation ( 55) below. For h ∈ L p (C), we define the operator T by

T h(z) = 1 2πi D h(ξ) ξ -z dξ ∧ dξ , z ∈ C . If w ∈ L p (D)
Su(z) = lim ε→0+ 1 2πi ξ∈C, |ξ-z|≥ε u(ξ) (ξ -z) 2 dξ ∧ dξ , a.e. z ∈ C . ( 55
)
The existence of Su a.e. follows from the Calderòn-Zygmund theory of singular integral operators [64, Ch. II, Thm 4]. Here are the properties of C, T , T and S that we use: Proposition 5.2.1. Let as usual 1 < p < +∞. Then the following assertions hold.

1. The Cauchy operator C is bounded from L p (T) onto H p (D) and from W 1-1/p,p (T) to W 1,p (D).

2. The Beurling operator S is bounded from L p (C) into itself. The next result will be of technical importance to establish the regularity properties of G p α -functions, compare Remark 4.2.2.

Lemma 5.2.1. Let p ∈ (1, +∞) as always, and α ∈ L ∞ (D).

1.

If g ∈ H p (D), then g ∈ L p 1 (D) for p 1 ∈ [p, 2p). 2. If w ∈ L p (D) and if w -T (αw) ∈ H p , then there is p * > 2 such that T (αw) ∈ W 1,p * (D) ⊂ C 0,1-2/p * (D) and T αw ∈ W 1,p * loc (C) ⊂ C 0,1-2/p * loc (C). Moreover, T (αw) W 1,p * (D) ≤ C p,α w L p (D) + w -T (αw) H p (D) . (56)
In order not to disrupt the reading, we postpone the proofs of Proposition 5.2.1 and Lemma 5.2.1 to Appendix A.

Factorization and boundary behaviour in G p α

This section is devoted to the proof of Theorem 4.2.1 and Proposition 4.2.1. In the proof of the former, we make use of the following Lemma.

Lemma 5.3.1. Let r ∈ L ∞ (C) be supported in D. Then, the function

u(z) = D zr(ζ) 1 -ζz dζ ∧ d ζ , z ∈ C ,
is holomorphic in D and belongs to W 1,l loc (C) for all l ∈ (1, +∞).

Proof of Lemma 5.3.1. The function u is clearly holomorphic in D, a fortiori u ∈ W 1,l loc (D). It is therefore enough to show, say, that u ∈ W 1,l loc (C \ D 1/2 ) for all l ∈ (1, +∞). In turn, it is sufficient to prove that

u(1/z) = - D r(ζ) z -ζ dζ ∧ d ζ,
lies in W 1,l (D 2 ). The conclusion now follows from assertion 3 in Proposition 5.2.1.

Proof of Theorem 4.2.1. put r(z) = α(z)w(z)/w(z) if w(z) = 0 and r(z

) = 0 if w(z) = 0 or z ∈ D. Then r ∈ L ∞ (C) and r L ∞ (D) ≤ α L ∞ (D) . Define s(z) = 1 2πi D r(ζ) ζ -z + zr(ζ) 1 -ζz dζ ∧ d ζ , for z ∈ C . (57)
and observe, from Lemma 5.3.1 and assertion 3 in Proposition 5.2.1, that s ∈ W 1,l loc (C) for all l ∈ (1, +∞). In particular s is continuous and, since 1/z = z for z ∈ T, we see from (57) that Im s(z) = 0 there. Also, assertion 4 of Proposition 5.2.1 and Lemma 5.3.1 show that ∂s = r in D. Furthermore, a straightforward majorization gives us for z ∈ C since w ∂s = wr = αw, where we used in the fourth equality that e -ψn ψ ∈ D(D). This proves the claim and provides us with [START_REF] Bojarski | Homeomorphic solutions of Beltrami systems[END_REF] where Im tr s = 0. Now, by the Sobolev imbedding theorem, s is bounded, and since exp is locally Lipschitz on C it follows that w ∈ W 1,l loc (D) for all l ∈ (1, ∞). Finally, by the boundedness of s, it is clear from the definitions that w ∈ G p α if, and only it F ∈ H p . In this case, it follows from Lemma 5.2.1, point 1, that w = e s F ∈ L p 1 (D) for all p 1 ∈ [p, 2p). To obtain from [START_REF] Bojarski | Homeomorphic solutions of Beltrami systems[END_REF] another factorization w = e s 1 F 1 , where this time Re tr s 1 = 0, it is enough to change the "+" sign into a "-" one in the definition (57) of s .

|s(z)| ≤ α L ∞ (D) π D 1 |ζ -z| + 1 |ζ -1/z| dm ≤ 4 α L ∞ (D) , (58) thus
For the proof of Proposition 4.2.1, we need the following version of the Cauchy-Green formula.

Lemma 5.3.2. When ψ ∈ W 1,p (D), it holds for almost every z ∈ D that

ψ(z) = C (tr ψ) (z) + 1 2πi D ∂ψ(ξ) ξ -z d ξ ∧ d ξ. ( 59
)
Proof. Note that (59) means ψ = C(tr ψ) + T ( ∂ψ). For ψ ∈ D(R 2 ), this is standard [41, thm. 1.2.1]. In general ψ is the limit in

W 1,p (D) of a sequence (ψ n ) n∈N ∈ D(R 2 )| D .
By continuity of the trace and Proposition 5.2.1, items 1, 4, the conclusion follows from taking a pointwise convergent subsequence of the L p (D) convergent sequence T ( ∂ψ n ).

Proof of Proposition 4.2.1. Let w ∈ G p α . By Theorem 4.2.1, we have w = e s F where s ∈ W 1,l (D), 1 < l < ∞ and F ∈ H p . As s is continuous on D, hence the existence of the non tangential limit [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF] and the majorization [START_REF] Chalendar | Approximation in reflexive Banach spaces and applications to the invariant subspace problem[END_REF] follow from the corresponding properties of H p functions [35, thm 3.1]. From Fatou's lemma, we then get the first half of (25), and since F L p (Tr ) ≤ tr F L p (T) for F ∈ H p [32, thm 1.5], we obtain by [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension d ≥ 3[END_REF] w L p (Tr) ≤ e 2 s L ∞ (D) tr w L p (T) ≤ e 8 α L ∞ (D) tr w L p (T) , which yields the second half of [START_REF] Carleson | Selected problems in exceptional sets[END_REF]. Finally, [START_REF] Chalendar | Interpolation between Hardy spaces on circular domains with applications to approximation[END_REF] follows from the continuity of s and the corresponding property for H p -functions [32, thm. 2.6]. This demonstrates assertion 1. Since e s is continuous and never zero on D, as noticed in Remark 4.2.1, assertion 2 is a consequence of ( 22) and of the corresponding properties for H p -functions [32, thms. 2.2, 2.3]. We turn to the proof of assertion 3. Assume first that w ∈ L p (D) satisfies w = Cϕ+T (αw) for some ϕ ∈ L p (T). As ∂Cϕ = 0 on D because Cϕ is holomorphic there, we know from Proposition 5.2.1, point 4, that ∂w = α w on D. Further, the M. Riesz theorem yields

Cϕ H p (D) ≤ C p ϕ L p (T)
hence w -T (αw) ∈ H p . Lemma 5.2.1 now provides us with the chain of inequalities:

T (αw) L p (Tr) ≤ T (αw) L ∞ (D) ≤ C ′ p T (αw) W 1,p * (D) ≤ C ′′ p w L p (D) + Cϕ H p (D) ,
where we used the Sobolev imbedding theorem. Therefore w ∈ G p α and ( 30

Comparison between Sobolev and Hardy solutions

In this section, we prove Proposition 4.3.3.

Proof of Proposition 4.3.3. We first check that W 1,p (D) ⊂ H p ν (D) and that (33) holds. Define α through [START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF]. In view of (18) and Proposition 3.2.3.1, point (b), it is enough to check the corresponding property for G p α . But if w ∈ W 1,p (D) meets ( 14), then Lemma 5.3.2 yields w = C(tr w) + T (αw), and since tr w ∈ W 1-1/p,p (T) ⊂ L p (T) we get from Proposition 5.2.1, point 1, that w -T (αw) = C(tr w) lies in H p , implying that w ∈ G p α by Remark 4.2.2. Moreover, it follows from [START_REF] Demengel | Espaces fonctionnels pour la théorie des équations[END_REF] and the trace theorem that w G p α ≤ C p,α w W 1,p (D) , as desired.

To end the proof of Proposition 4.3.3, we establish the more general fact that, if f ∈ W 1,p (D) has a nontangential limit almost everywhere on T, then this nontangential limit coincides with the trace of f in the Sobolev sense. We provide an argument because we could not locate this "elementary" result in the literature. Note that, when p > 2, each f ∈ W 1,p (Ω) (has a representative which) extends continuouly to Ω by the Sobolev imbedding theorem, so the nontangential limit exists everywhere on ∂Ω and is in fact an unrestricted limit. Assume now that p ≤ 2. Notice that f is the restriction to D of some f ∈ W 1,p (R 2 ) with compact support [START_REF] Adams | Sobolev spaces[END_REF]Thm 4.26]. By Hölder's inequality f ∈ W 1,s (R 2 ) with some s ∈ (1, 2), hence the non-Lebesgue points of f have Hausdorff 1-dimension zero [67, Thms 3.3.3, 2.6.16]. In particular, for each r > 0 we have that re iθ is a Lebesgue point of f for a.e. θ. Then, regularizing f yields a sequence of D(R 2 )-functions converging to f both in W 1,p (R 2 ) and pointwise a.e. on every circle T r [67, Thm 1.6.1]. Consequently if we put f r (ξ) := f (rξ) for 0 < r < 1 and ξ ∈ D, we deduce that tr f r (ζ) = f (rζ) for a.e. ζ ∈ T, and since f r tends to f in W 1,p (D) as r → 1 we get that f (rζ) → tr f (ζ) in W 1-1/p,p (T). Finally, as any W 1-1/p,p (T)-converging sequence has a pointwise a.e. converging subsequence, there is r n → 1 such that f (r n ζ) → tr f (ζ) for a.e. ζ ∈ T, hence tr f must be the radial (a fortiori nontangential) limit of f when the latter exists.

Remark 5.4.1. Since Sobolev functions can be redefined on a set of zero measure so as to be absolutely continuous on almost every line [START_REF] Ziemer | Weakly Differentiable Functions[END_REF]Rem. 2.1.5]), it is easy to see (use polar coordinates) that any function in W 1,p (D) has a radial limit almost everywhere on T. Note, however, that a function in W 1,2 (D) may have no nontangential limit at all (see [START_REF] Carleson | Selected problems in exceptional sets[END_REF]). Note also that a more precise result involving capacity was proven for continuous functions in W 1,p (D), see [START_REF] Mizuta | Existence of various boundary limits of Beppo Levi functions of higher order[END_REF][START_REF] Yu | Boundary behaviour of functions with generalized derivatives[END_REF].

w L p (D) ≤ C p,α g L p (D) ≤ C p,α g H p ,
hence [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] holds in view of [START_REF] Demengel | Espaces fonctionnels pour la théorie des équations[END_REF]. Observe from the M. Riesz theorem that ϕ → ϕ + iH 0 ϕ is continuous from L p R (T) into tr H p,0 ⊂ L p (T), hence A is well-defined and continuous from L p R (T) × R into itself by [START_REF] Carleson | Selected problems in exceptional sets[END_REF] and Theorem 4.4.1.1. Put for simplicity T α (w) = T (αw). In view of ( 35

→ T α (w ϕ,c ) is continuous from L p (T) × R into W 1,p (D), so the first component of B is continuous from L p (T) × R into W 1-1/p,p (T). Therefore it is compact from L p (T) × R into L p (T)
Since, v has mean c on T, we get from ( 23) and the M. Riesz theorem that

|b| ≤ e 4 α L ∞ (D) c p e 8 α L ∞ (D) ψ L p (T) + |c| ,
where c p is the norm of H 0 on L p (T). Plugging this in (61) implies now

v L p (T) ≤ e 8 α L ∞ (D) |c| + 1 + e 8 α L ∞ (D) c p ψ L p (T)
from which [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF] follows immediately. This concludes the proof of (a). Assume next that ( 16) holds and let w 1 ∈ G p α , satisfy Re(tr w 1 ) = ψ a.e. on T. Such a w 1 exists by (a). From the observation made after Proposition 3.2.3.1, we see that the function w defined as

w := w 1 -iσ -1/2 1 2π 2π 0 σ 1/2 Im tr w 1 (e iθ ) dθ
lies in G p α , and since it readily satisfies [START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF] we deduce that w ∈ G p,0 α . Clearly w has the same real part as w 1 , and by Proposition 4.2.1 point 4 it is the only member of G p,0 α with this property. This settles the existence and uniqueness part in (b). The reasoning leading to [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF] is easily adapted to yield [START_REF] Goluzin | Generalized Carleman formula and its applications to analytic extension of functions[END_REF], upon trading the meanequal-to-c condition for [START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF] and taking (3) into account. This completes the proof.

A Fatou theorem for Re H p ν

As a preparation for the proof of Theorem 4.4.2.2, we establish the following Hodge type lemma.

Lemma 5.6.1. Define two subspaces G 0,∞ and D

0,∞ of C ∞ R (D) × C ∞ R (D) by G 0,∞ := {∇g; g ∈ C ∞ R (D), tr g = 0}, D 0,∞ := {(∂ y h, -∂ x h) T ; h ∈ C ∞ R (D), tr h = 0}. Then, each V ∈ C ∞ R (D) × C ∞ R (D)
that vanishes on T can be written uniquely in the form V = G + D, where G ∈ G 0,∞ and D ∈ D 0,∞ . Moreover, it holds for some constant C p that

G L p (D) + D L p (D) ≤ C p V L p (D) , ( 62
)
where the subscript L p (D) refers here to the norm of an R 2 -valued mapping. Then, all we have to show is that h | T is constant, because substracting this constant will produce a new h with vanishing trace on T, as desired. Now, because tr g = 0 we deduce that G is normal to T there, and since V | T = 0 it follows that D is also a normal vector field on T. Consequently x∂ y h(x, y) -y∂ x h(x, y) = 0 for x + iy ∈ T, which means exactly that h is constant on T. Proof of Theorem 4.4.2.2. Since ∂ x (σ∂ x u) = ∂ y (-σ∂ y u) by ( 4), there is a distribution v on D such that ( 8) holds [62, Ch. II, Sec. 6, Thm VI]. Then, for Φ ∈ D R (D), we obtain

= (∂ y h, -∂ x h) T with g ∈ W 1,p 0,R (D) and h ∈ W 1,p R (D). Since V ∈ C ∞ R (D) × C ∞ R (D),
v, ∂ x Φ = σ∂ y u, Φ = -u, σ∂ y Φ + Φ∂ y σ , v, ∂ y Φ = -σ∂ x u, Φ = u, σ∂ x Φ + Φ∂ x σ ,
which entails by ( 3), Hölder's inequality, and the Poincaré inequality that

| v, ∂ x Φ | ≤ u L p (D) C p,σ ∇Φ L q (D) , | v, ∂ y Φ | ≤ u L p (D) C p,σ ∇Φ L q (D) . (63)
Next, we observe that any g ∈ C ∞ R (D) satisfying tr g = 0 lies in W 1,q 0,R (D), therefore it is the limit in W 1,q R (D) of some sequence Φ n ∈ D R (D). In particular ∇Φ n converges to ∇g in L q (D), implying in view of ( 63) that (Φ 1 , Φ 2 ) → v, Φ 1 + Φ 2 is a bounded functional on both G 0,∞ and D 0,∞ when endowed with the L q (D)-norm. By Lemma 5.6.1, this functional is L q (D) × L q (D) bounded on the subspace of C ∞ R (D) × C ∞ R (D) comprising those vector fields that vanish on T. Therefore, by density, (Φ

1 , Φ 2 ) → v, Φ 1 + Φ 2 is a bounded functional on L q R (D) × L q R (D), so that in fact v ∈ L p R (D). If we put f = u + iv ∈ L p (D) and ν = (1-σ)/(1+σ) ∈ W 1,∞ R (D)
, it is now a mechanical consequence of (8) that equation ( 5) is satisfied in the distributional sense. Defining w ∈ L p (D) and α ∈ L ∞ (D) through ( 17) and ( 16), we see from Proposition 3.2.3.1 that w solves [START_REF] Baratchart | How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?[END_REF]. Hence Theorem 4.2.1 applies to the effect that w, thus also f and a fortiori u, lie in W 1,l loc for l ∈ (1, ∞). Moreover, we may write [START_REF] Bojarski | Homeomorphic solutions of Beltrami systems[END_REF] with Im tr s = 0 and, say, F = a + ib a holomorphic function in D. As s lies in C 0,γ (D), for each ε > 0 we can pick r 0 such that |Im exp(s(z))| < ε| exp(s(z))| as soon as r 0 < |z| ≤ 1, and for such z we deduce from [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension d ≥ 3[END_REF] that

Re w(z) ≥ e -4 α L ∞ (D) ((1 -ε 2 ) 1/2 |a(z)| -ε|b(z)|).
Since b r = H 0 (a r ) ( recall that b r (z) = b(rz), resp. a r (z) = a(rz), for all z ∈ T), we obtain from the M. Riesz theorem when r 0 < r < 1 that Re w(z

) L p (Tr) ≥ e -4 α L ∞ (D) ((1 -ε 2 ) 1/2 -εc p ) a(z) L p (Tr ) ,
and picking ε so small that ((1

-ε 2 ) 1/2 -εc p ) = C > 0 we conclude by (17) that b L p r (T) ≤ c p a L p r (T) ≤ C p,σ,u Re w(z) L p (Tr) ≤ C ′ p,σ,u u L p (Tr ) , r 0 < r < 1.
Consequently, if u F,p < ∞, then F = a + ib ∈ H p so that w ∈ G p α by Theorem 4.2.1, whence f ∈ H p ν by Proposition 3.2.3.1. All the assertions now readily follow from Theorems 4.2.1 and 4.4.2.1.

Higher regularity

In order to prove Theorem 4.4.3.1, we shall make use of the following observation:

W = (1 -ν 2 ) 1/2 ∂f ∈ L p (D) is a solution to (14) with α = ∂ν/(1 -ν 2 ), that is to say ∂W = ∂ν 1 -ν 2 W ( 64
)
in the sense of distributions. Moreover, it holds that

∂f = e s F ( 65
)
where F is holomorphic in D, s ∈ C 0,γ (D) for every 0 < γ < 1, and for some constant

c κ we have s L ∞ (D) ≤ c κ ν W 1,∞ (D) .
Proof. As ν ∈ W 1,∞ (D), observe that the distributional derivative of ν∂f ∈ L p (D) can be computed according to Leibniz's rule:

∂ ν∂f = ∂ν∂f + ν∂ ∂f , (66)
where we emphasize that the second summand in the right-hand side of (66) is to be interpreted as indicated in the footnote before Theorem 4.1.1. Indeed, pick a function ϕ ∈ D(D). By definition, ν∂f , ∂ϕ =νf , ∂∂ϕ -∂νf , ∂ϕ .

By the Leibniz rule

-νf , ∂∂ϕ = -f , ν∂∂ϕ = -f , ∂(ν∂ϕ) -∂ν∂ϕ . It follows that ν∂f , ∂ϕ = -f , ∂(ν∂ϕ) = ∂f , ν∂ϕ = ∂f , ∂(νϕ) -ϕ∂ν = -∂∂f , νϕ -∂ν∂f , ϕ = -ν∂∂f , ϕ -∂ν∂f , ϕ ,
which is the desired conclusion.

Setting G := ∂f and applying ∂ to (5), we thus obtain, since ∂ and ∂ commute, that ∂G = ν∂G + (∂ν)G. As ν is real, conjugating this last equation provides us with an expression for ∂G, and solving for ∂G after substituting back yields

∂G = ν∂ν 1 -ν 2 G + ∂ν 1 -ν 2 G
from which we deduce that W = (1ν 2 ) 1/2 G satisfies (64), as ∂W can in turn be computed by the chain rule because (1ν 2 ) 1/2 ∈ W 1,∞ (D). The remaining assertions follow from Theorem 4.2.1 upon setting α

= ∂ν/(1 -ν 2 ) ∈ L ∞ (D). Proof of Proposition 4.4.3.1. Let u ∈ W 1,p R (T) ⊂ W 1-1/p,p R (T) and put v = H ν u. We must show that v ∈ W 1,p R (T)
, and for this we may assume that u has zero mean on T for adding a constant to u does not affect v. Then, v = H ν u becomes equivalent to u = -H -ν v as follows immediately from the fact that f satisfies (5) if, and only if if satisfies a similar equation with ν replaced by -ν. From Theorem 4.1.1 we know that v ∈ W 1-1/p,p R (T), so let v n be a sequence of C ∞ (T)-functions converging to v there and set u n = -H -ν v n . Since v n converges to v in W 1-1/p,p (T), we get from Corollary 4.4.2.1 that u n converges to u there. By the definition of H ν , we have that u n + iv n is the trace on T of the solution to (5) in W 1,p (D) whose real part on T is u n . With a slight abuse of notation, we still designate by u n , v n the real and imaginary parts of that solution in W 1,p R (D). By inspection, the generalized Cauchy-Riemann equations ( 8) do hold with u replaced by u n and v by v n , so that u n , v n may as well be characterized respectively as the unique solutions in D to (4), [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF] whose traces on T are our previous u n and v n [START_REF] Campanato | Elliptic systems in divergence form, Interior regularity[END_REF]. Now, we know that u n , v n lie in W 1,p (D); however, since u n is smooth on T, it follows from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm 9.15] that in fact u n ∈ W 2,r R (D) for all r ∈ (1, ∞). In view of ( 8), we deduce that the same is true of v n as σ ∈ W 1,∞ R (D). Pick any ϕ ∈ C ∞ (T) and put ψ = H ν ϕ with ν-conjugate ψ. The W 2,r (D) regularity just mentioned allows us by density to apply the divergence formula so as to obtain

T v n ∂ θ ϕ dθ = -2π T v n ∂ n ψ/σ dθ = -2π T ∂ n v n ψ/σ dθ = T ∂ θ u n ψ dθ,
where we used [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimumenergy transfer functions of linear systems[END_REF]. As u n , v n converge to u, v in W 1-1/p,p (T), we get in the limit, since differentiation is continuous W 1-1/p,p (T) → W -1/p,p (T), that

T v∂ θ ϕ dθ = T ∂ θ uψ dθ,
where ∂ θ u is to be understood as a member of W -1/p,p . However, we have by assumption that in fact ∂ θ u ∈ L p (T), therefore from Hölder's inequality

T v∂ θ ϕ dθ ≤ ∂ θ u L p (T) ψ L q (T) .
But from Corollary 4.4.2.1 we know that ψ L q (T) ≤ C ν ϕ L q (T) , so the distribution ∂ θ v in fact lies in L p (T) and the conclusion holds.

Proof of Theorem 4.4.3.1. Write f = u + iv to indicate the real and imaginary parts of f . By Proposition 4.4.3.1, tr f ∈ W 1,p (T). First we shall prove that the left hand sides of ( 43) and ( 44) are finite, that is, we show ∂f and ∂f satisfy a Hardy condition of order p. To this effect, we consider w = (fν f )/ √ 1ν 2 and we establish the equivalent fact that both ∂w and ∂w satisfy a Hardy condition of order p. We first notice:

Lemma 5.7.2. If p 1 ∈ [p, 2p) then f, w ∈ W 1,p 1 (D); in particular f, w ∈ C 0,1-2/p 0 (D) for some p 0 > 2.
Proof. Note that tr w ∈ W 1,p (T) since tr f does. As we will see in the proof of Proposition 5.2.1 in Appendix A (cf. (85)), it entails ∂(Ctr w) ∈ H p , and since ∂(Ctr w) = 0 we see from the Poincaré inequality and Lemma 5.2.1, point 1, that (Ctr w) lies in W 1,p 1 (D). As w, thus αw belongs to L p 1 (D) by Theorem 4.2.1, we get from Proposition 5.2.1, point 4, that T α w ∈ W 1,p 1 (D). Since w = C(tr w) + T α w, we see that w, therefore also f is in w ∈ W 1,p 1 (D). As 2p > 2, the last assertion now follows from the Sobolev imbedding theorem.

Back to proof of Theorem 4.4.3.1 we observe that, to prove the finiteness of the left hand sides in ( 43) and ( 44), we may as well add a real constant to f . Since the latter is (even Hölder) continuous on D by Lemma 5.7.2, we may thus assume that its real part is larger than a positive constant. Then, the same is true of w = (fνf )/ √ 1ν 2 , say, Re w(z) ≥ c 0 > 0 for z ∈ D. This results in w/w being Hölder continuous of exponent 1 -2/p 0 in D. Now, consider the function s introduced in Theorem 4.2.1. Letting B(z, ε) indicate the ball of center z with radius ε, we gather from (57) that, for a.e. z ∈ D,

∂s(z) = lim ε→0 1 2πi D\B(z,ε) r(ζ) (ζ -z) 2 dζ ∧ d ζ + 1 2πi D r(ζ) (1 -ζz) 2 dζ ∧ d ζ , (67)
where the function r was defined as r = αw/w and the existence of the limit a.e. comes from the existence of the Beurling transform as a singular integral operator of Calderòn-Zygmund type. To evaluate the first integral in (67), we establish a lemma which is best stated in terms of the space BMOA(D), comprised of those H 2 -functions whose trace on T has bounded mean oscillation, see e.g. [32, p. 240]. To us, the important fact will be that BMOA(D) ⊂ H p for all p < ∞.

lim ε→0 1 2πi D\B(z,ε) r(ζ) (ζ -z) 2 dζ ∧ d ζ = b(z) + ψ(z). (68)
Proof. We may rewrite the first integral in the right hand side of (67) as

1 2πi D\B(z,ε) r(ζ) (ζ -z) 2 dζ ∧ d ζ = 1 2πi D\B(z,ε) α(ζ)((w/w)(ζ) -(w/w)(z)) (ζ -z) 2 dζ ∧ d ζ + (w/w(z)) 2πi D\B(z,ε) α(ζ) (ζ -z) 2 dζ ∧ d ζ .
Since α is bounded and w/w is Hölder continuous of order 1 -2/p 0 , the first integral in the right hand side is majorized by

C ν,w D |ζ -z| -1-2/p 0 dζ ∧ d ζ ≤ C ν,w |ξ|≤2 |ξ| -1-2/p 0 dm(ξ) < +∞.
As to the second integral, we put for simplicity Φ := log(σ 1/2 ) and we recall from ( 16) that α = ∂Φ, whence by Stoke's theorem Besides, ∂w = α w is bounded, being the product of a L ∞ (D) function and a Hölder continuous one. Thus both ∂w and ∂w satisfy a Hardy condition of order p, and since w is bounded it follows that ∂f and ∂f also satisfy a Hardy condition of order p, that is, the left-hand sides of ( 43) and ( 44) are finite, as announced.

1 2πi D\B(z,ε) α(ζ) (ζ -z) 2 dζ ∧ d ζ = - 1 2iπ T Φ(ζ)dζ (ζ -z) 2 + 1 2iπ ∂B(z,ε) Φ(ζ)dζ (ζ -z) 2 . ( 69
From this, in view of Lemma 5.7.1, we deduce that W = (1ν) 1/2 ∂f lies in G p α 1 with α 1 = ∂ν/(1ν 2 ). Clearly α 1 lies in L ∞ (D), hence by [START_REF] Astala | Elliptic Partial Differential Equations and Quasiconformal mappings in the plane[END_REF] and the relation ∂f = ν∂f we deduce from Proposition 4.2.1 that ∂f and ∂f have nontangential limits, say, Φ 1 and Φ 2 = ν | T Φ 1 to which ∂f (re iθ ) and ∂f (re iθ ) converge in L p (T) as r → 1. It only remains for us to establish the explicit expression [START_REF] Jaoua | Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains[END_REF] for tr ∂f , because the latter readily implies that tr ∂f L p (T) ≤ C ν tr f W 1,p (T) , hence assertions (b) and (c) of Theorem 4.4.3.1 will follow from Proposition 4.2.1 as applied to W , and since we already pointed out that (a) is a rephrasing of Proposition 4.4.3.1 the proof will be complete. To establish [START_REF] Jaoua | Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains[END_REF], observe by the absolute continuity on a.e. coordinate line characterizing Sobolev functions (choose polar coordinates) that

f (re iθ 1 ) -f (re iθ 2 ) = - θ 2 θ 1 r sin θ(∂f + ∂f ) -ir cos θ(∂f -∂f ) (re iθ ) dθ
for a.e. r ∈ (0, 1) and all θ 1 , θ 2 . Letting r → 1, we obtain by the continuity of f and the L p (T)-convergence of ∂f (re iθ ) and ∂f (re iθ ) to Φ 1 and Φ 2 that

f (e iθ 1 ) -f (e iθ 2 ) = - θ 2 θ 1 (sin θ(Φ 1 + Φ 2 ) -i cos θ(Φ 1 -Φ 2 )) (e iθ ) dθ, thereby showing that ∂ θ f (e iθ ) = ie iθ Φ 1 (e iθ ) -ie -iθ Φ 2 (e iθ ) = ie iθ Φ 1 (e iθ ) -ie -iθ ν(e iθ )Φ 1 (e iθ ).
Conjugating this identity, we obtain since ν is real-valued that

∂ θ f (e iθ ) = -ie -iθ Φ 1 (e iθ ) + ie iθ ν(e iθ )Φ 1 (e iθ ), whence ∂ θ f (e iθ ) -ν(e iθ )∂ θ f (e iθ ) = (1 -ν 2 )ie iθ Φ 1 (e iθ ).
which is [START_REF] Jaoua | Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains[END_REF]. 71), hence it belongs both to W 1,l loc (D) for 1 < l < ∞ and to L p 1 (D), for p 1 ∈ (p, 2p) in view of Theorem 4.2.1. Clearly the same is true of ∂f and ∂f = ν∂f , in particular f ∈ W 1,p 2 (D) for some p 2 > 2, hence is Hölder continuous on D. By the Sobolev embedding theorem as applied to its derivatives, f is moreover continuously differentiable in D, so we get for every re iθ ′ ∈ D that:

f (re iθ ′ ) -f (r) = θ ′ 0 (∂ θ f )(re iθ )dθ (72)
where (∂ θ f )(z) = i(z ∂f (z) -zν∂f (z)). Now, by Proposition 4. Passing to the limit as r → 1 in (72) yields that tr f is absolutely continuous on T with tangential derivative i(e iθ he -iθ ν h) ∈ L p (T), proving that tr f ∈ W 1,p (T). Since ∂f ∈ G p α (D), Proposition 4.2.1 implies M ∂f ∈ L p (T). The same is true of M ∂f because of (5) and the boundedness of ν, therefore M ∇f ∈ L p (T). Moreover, since ∂f and ∂f both have non tangential limits in L p (T), so does ∇f . The same conclusions then hold for u = Ref . In addition, the Green-Riemann formula ensures that

Proofs of the density results

Density for Sobolev traces

For the proof of Theorem 4.5.1.1, we need two lemmas. Below, for I an open subset of T, the symbol •, • I stands for the duality bracket between W -1/p,p R (I) and W 1-1/q,q 0,R (I). We drop the subscript when I = T. Lemma 5.8.1.1. For ϕ ∈ W 1-1/p,p R (T) and ψ ∈ W 1-1/q,q R (T), one has

∂ θ (H ν ϕ), ψ = ϕ, ∂ θ (H ν ψ) . ( 73
)
Proof. Assume first that ϕ and ψ lie in C ∞ (T). Let f and g be the solutions to (5) on D, normalized as in [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF], such that Re (tr f ) = ϕ and Re (tr g) = ψ respectively. By definition of H ν , it holds that Im (tr f ) = H ν ϕ and Im (tr g) = H ν ψ. As pointed out in the proof of Proposition 4.4.3.1, the functions f and g lie in W 2,r (D) for all r ∈ (1, ∞) by [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm 9.15]. Put u = Re f and u 1 = Re g. By the divergence formula

0 = D (div(σ∇u)u 1 -div(σ∇u 1 )u) dm = T σ ((∂ n u)u 1 -(∂ n u 1 )u) dθ,
and from [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimumenergy transfer functions of linear systems[END_REF] we see that σ∂ n u = ∂ θ H ν ϕ while σ∂ n u 1 = ∂ θ H ν ψ. This yields the desired conclusion for smooth ϕ and ψ. In the general case, pick two sequences (ϕ k ) k≥1 , (ψ k ) k≥1 of smooth functions converging respectively to ϕ in W 1-1/p,p (T) and to ψ in W 1-1/q,q (T). By Corollary 4.4.2.1, we have that

H ν ϕ k → H ν ϕ in W 1-1/p,p (T) and H ν ψ k → H ν ψ in W 1-1/q,q (T), therefore ∂ θ (H ν ϕ k ) → ∂ θ (H n uϕ) in W -1/p,p (T) and ∂ θ (H ν ψ k ) → ∂ θ (H ν ψ)
in W -1/q,q (T). Identity (73) now follows from the first part of the proof by a limiting argument.

To proceed with the second lemma, we introduce the following piece of notation that will be of use in the next section as well: whenever I ⊂ T and J = T \ I is the complementary subset, then for u I (resp. u J ) a function on I (resp. J) we put u I ∨u J for the concatenated function on T which is u I (resp. u J ) on I (resp. J).

Let now I, J be proper open subsets of T such that J = T \ I = ∅. For any function u J ∈ W 1-1/p,p 0,R (J), we form the concatenated function 0 ∨ u J and we set

A u J = ∂ θ H ν (0 ∨ u J ) | I . ( 74
) Note that 0 ∨ u J ∈ W 1-1/p,p R (T), so that A : W 1-1/p,p 0,R (
* : W 1-1/q,q 0,R (I) → W -1/q,q R (J)
is one-to-one. Now, for u I ∈ W 1-1/q,q 0,R (I) and u J ∈ W 1-1/p,p 0,R (J), we get by Lemma 5.8.1.1

A * u I , u J J = u I , Au J I = u I , ∂ θ H ν (0 ∨ u J ) | I I = (u I ∨ 0), ∂ θ H ν (0 ∨ u J ) = (0 ∨ u J ), ∂ θ H ν (u I ∨ 0) = u J , ∂ θ H ν (u I ∨ 0) | J J , hence A * u I = ∂ θ H ν (u I ∨ 0) | J
. Thus the relation A * u I = 0 means that H ν (u I ∨ 0) is constant on each component of J. Let J 0 be such a component and f be the solution of (5) in W 1,p (D) such that Re (tr f ) = u I ∨ 0, normalized as in [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF] so that Im (tr f ) = H ν (u I ∨ 0). Since u I ∨ 0 vanishes on J, there exists c ∈ R such that tr f = ic on J 0 . Therefore f ≡ ic in D by Proposition 4.3.3 and Proposition 4.3.1 point (c). In particular Re (tr f ) = 0 on T, hence u I = 0 as desired.

Proof of Theorem 4.5.1.1. By definition of an extension set, we can write I = ∪ j (a j , b j ) where each a j lies at positive distance from the b k , therefore also from the a k for k = j. This implies there is an open arc I 1 with I 1 = T such that I ⊂ I 1 . By the extension property and a smooth partition of unity argument, each function in W 1-1/p (I) extends to a function in W 1-1/p 0 (I 1 ). Thus, upon trading I for I 1 , it is enough to prove that any function in W 1-1/p,p 0 (I) can be approximated in W 1-1/p,p (I) by the trace of a solution of (5). Let ε > 0 and ϕ I ∈ W

v = (v I -H ν (u I ∨ 0)) |I ∈ W 1-1/p,p R (I). By Lemma 5.8.1.2, there exists u J ∈ W 1-1/p,p 0,R (J) satisfying ∂ θ v -∂ θ H ν (0 ∨ u J ) W -1/p,p (I) ≤ ε, (75)
from which it follows by elementary integration and the Poincaré inequality on I that, for some c v ∈ R and some

C I independent of v, v -H ν (0 ∨ u J ) -c v W 1-1/p,p (I) ≤ C I ε. (76)
Consider now the concatenated function (u I ∨ u J ) ∈ W 1-1/p (T), and define

ψ = (u I ∨ u J ) + iH ν (u I ∨ u J ) + ic v .
Then ψ ∈ W 1-1/p,p (T) by Corollary 4.4.2.1, and by construction it is the trace on T of a solution to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. Since

ψ = (u I ∨ 0) + (0 ∨ u J ) + iH ν (u I ∨ 0) + iH ν (0 ∨ u J ) + ic v ,
it follows from (76) that ϕ Iψ W 1-1/p,p (I) ≤ C I ε.

Density for Hardy traces

The proof of Theorem 4.5.2.1 resembles that of Theorem 4.5.1.1, but makes use of different operators. Herafter, the symbol •, • indicates not only the duality between W -1/p,p R (T) and W 1-1/q,q 0,R (T), as in Lemma 5.8.1.1, but also the L p R -L q R and the W 1,p R -W -1,q R duality. This should cause no confusion. First, we will need the following version of Lemma 5.8.1.1: Lemma 5.8.2.1. For all ϕ ∈ W 1,p R (T) and all ψ ∈ L q R (T), one has

∂ θ (H ν ϕ), ψ = ϕ, ∂ θ (H ν ψ) .
where

Ψ ∈ W 1,q R (T) is such that ∂ θ Ψ = ϕ I ∨ 0. ( 80
)
Consider the solution f ∈ W 1,p (D) to (5) satisfying Re (tr f ) = Ψ on T, normalized so that (12) holds. By (80) it holds that ∂ θ Ψ = 0 a.e. on J, and if B * ϕ I = 0 we get in addition from (79) that ∂ θ H ν Ψ = 0 a.e. on J. As tr f = Ψ + iH ν Ψ, it entails altogether that ∂ θ (tr f ) = 0 a.e. on J. But from Theorem 4.4.3.1 point (c) we know that ∂f admits a non tangential limit a.e. on T, and by [START_REF] Jaoua | Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains[END_REF] we now see that tr ∂f = 0 a.e. on J. But Theorem 4.4.

I , v I ∈ L p R (I). Set v = v I -H ν (u I ∨ 0) and observe from (78) that v -H ν (0 ∨ u J ) -c v L p (I) ≤ ε. (81)
for some u J ∈ L p R (J) and c v ∈ R. Define

ψ = (u I ∨ u J ) + iH ν (u I ∨ u J ) + ic v .
By construction ψ is the trace on T of a H p ν -function, and since

ψ = (u I ∨ 0) + (0 ∨ u J ) + iH ν (u I ∨ 0) + iH ν (0 ∨ u J ) + ic v , it follows from (81) that ϕ I -ψ L p (I) ≤ ε.
The proof is now complete.

5.9 A characterization of (tr H p ν ) ⊥

The expression for H * ν obtained in Lemma 5.8.2.2 enables us to characterize the orthogonal space (tr H p ν ) ⊥ of tr H p ν for the duality product [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF], hereafter denoted by ., . . Proof of Proposition 4.6.1. Let ϕ = ϕ 1 + iϕ 2 , with ϕ k ∈ L q R (T). In view of Theorem 4.4.2.1, we get that ϕ ∈ (tr H p ν ) ⊥ if, and only if

Re

ϕ 1 + iϕ 2 , u + iH ν u + ic = 0 , u ∈ L p R (T), c ∈ R . (82)
Picking for u an arbitrary constant shows that ϕ 1 and ϕ 2 have zero mean on T, hence there are Φ 1 , Φ 2 ∈ W 1,q R (T) such that Φ := Φ 1 + iΦ 2 satisfies ∂ θ Φ = ϕ; we may impose in addition that Φ itself has zero mean. Now, from (82) we get for every u ∈ L p R (T) that

0 = Re u + iH ν u, ϕ 1 + iϕ 2 = u, ϕ 1 -u, H * ν ϕ 2 , which means ϕ 1 = H * ν ϕ 2 or else Φ 1 = -H ν Φ 2
, by elementary integration and using Lemma 5.8.2.2. Therefore Φ = i (Φ 2 + i H ν Φ 2 ) lies in itr H q ν = tr H q -ν , and since ∂ θ Φ = ϕ we conclude that (tr H p ν ) ⊥ ⊂ ∂ θ tr H q -ν ∩ W 1,q (T) .

Conversely let ϕ = ∂ θ Φ for some Φ ∈ (tr H q -ν ∩ W 1,q (T)). We can write Φ = γ + i H -ν γ for some γ ∈ W 1,q R (T), and from Lemma 5.8.2.2 applied with -ν rather than ν we obtain for

u ∈ L p R (T) Re ϕ, u + iH ν u = ∂ θ γ, u -∂ θ (H -ν γ), H ν u = ∂ θ γ, u + H * -ν (∂ θ γ), H ν u = ∂ θ γ, u + ∂ θ γ, H -ν (H ν u) = ∂ θ γ, 2π 0 udθ = 0 , since H -ν (H ν u) = -u + 2π
0 udθ as follows immediately from the fact that iH p ν = H p -ν .

Remark 5.9.1. In order to extend the definition of H ν to complex valued functions, it is natural to set (cf. [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF][START_REF] Astala | A boundary integral equation for Calderón's inverse conductivity problem[END_REF])

H ν (i u) = i H -ν (u) ,
for we then have

i u + i H ν (i u) = i (u + i H -ν (u)) ,
which is indeed a solution to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. With this definition, we recap Proposition 4.6.1 by saying that ϕ ∈ L q (T) belongs to (tr

H p ν ) ⊥ if and only if ϕ = ∂ θ Φ, where Φ ∈ W 1,q (T) satisfies H -ν Φ = -i Φ, . (83)
Note that if ν = 0 then (83) characterizes traces of holomorphic functions with zero mean. In this case (82) follows readily from the Cauchy formula.

Hardy spaces over Dini-smooth domains

In this final section, we indicate how the spaces H p ν that we studied on the disk can be defined more generally on Dini-smooth domains.

Recall that a function h is called Dini-continuous if ε 0 (ω h (t)/t)dt < +∞ for some, hence any ε > 0, where ω h is the modulus of continuity of h. A function is said to be Dini-smooth if it has Dini-continuous derivative. A bounded planar domain in C is termed Dini-smooth if its boundary is a Jordan curve with nonsingular Dini-smooth parametrization. Such domains Ω have the property that any conformal map from D onto Ω extends continuously from D onto Ω together with its derivative, in such a way that the latter is never zero [58, thm 3.5], and that is why we are able to generalize our previous results to this class of domains.

Let Ω ⊂ C be a simply connected Dini-smooth domain, as defined in the introduction, and assume that ν ∈ W 1,∞ (Ω) with ν L ∞ (Ω) ≤ κ < 1. Any conformal transformation ψ from D onto Ω has a C 1 (D) conformal extension onto Ω that we still denote by ψ [58, thm 3.5]. Introduce the Hardy classes H p ν (Ω) as the space of functions f on Ω such that f • ψ ∈ H p ν•ψ (D). A straightforward computation [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF]Ch. 1,C] shows that this class does not depend on the particular choice of ψ and consists of distributional solutions to [START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF]. Note that ν • ψ ∈ W 1,∞ (D) with ν • ψ L ∞ (D) ≤ κ < 1. Similarly, the class G p α (Ω) consists of those functions g on Ω such that g • ψ ∈ G p α•ψ (D). As in the classical case of holomorphic Hardy spaces over simply connected domains [START_REF] Duren | Theory of H p spaces[END_REF], it appears that w ∈ G p α (Ω) if and only if it is a solution to [START_REF] Baratchart | How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?[END_REF] such that |w| p has a harmonic majorant (we insist that harmonic is understood here in the usual sense). Indeed, it is enough to check this property in D, since it is preserved by composition with conformal maps. Then, it is a consequence of Theorem 4.2.1 that functions in G p α (D) possess this property. For the converse, observe that any solution w to ( 14) can be factorized as w = e s F , where s ∈ C(D) and F is holomorphic in D. Since |F | p = e -pRe s |w| p , it admits a harmonic majorant, and therefore it belongs to H p (D), which immediately yields that w ∈ G p α (D). Another possibility to generalize H p ν to a Dini-smooth simply connected domain Ω is to parallel the definition of the so-called Smirnov classes [START_REF] Duren | Theory of H p spaces[END_REF]Ch. 10]. Namely, one requires the uniform boundedness of the L p norm on a sequence of rectifiable contours eventually encompassing every compact subset of Ω. For Dini-smooth domains, the two generalizations turn out to be equivalent. The results of Sections 4.2, 4.3, 4.4, and 4.5 remain valid in this framework, as can be seen easily by composing with ψ and appealing to the regularity of ψ -1 .

Conclusion

This paper took a few steps towards a theory of two-dimensional Hardy spaces for the conjugate Beltrami equation on simply connected domains. Conspicuously missing is a factorization theory, whose starting point should be a characterization of those pairs (s, F ) for which [START_REF] Bojarski | Homeomorphic solutions of Beltrami systems[END_REF] holds. Also, we did not pursue the solution to the extremal problems stated in Theorem 4.6.1, points (ii) -(iii). Finally, the case of multiply connected domains, that motivated in part the present study (cf. the introduction), was not touched upon. It is to be hoped that suitable deepenings of the present techniques will enable one to approach such issues.

A Appendix. Proof of technical results

We establish here Proposition 5.2.1 and Lemma 5.2.1.

Proof of Proposition 5.2.1: The boundedness of C from L p (T) onto H p (D) follows from the M. Riesz theorem, mentioned already. To establish the second half of 1, let ϕ ∈ L p (T). From the definition of the H p -norm and the M. Riesz theorem, we get for each r ∈ (0, 1) and every ǫ > 0 that

Cϕ L p (Tr ) ≤ C p ϕ L p (T) ≤ C p (1 -r) ǫ ϕ L p (T) ,
where the last inequality is trivial. It then follows from [32, (1r) (1+ǫ)/p ϕ W 1-1/p,p (T) .

Choosing ǫ so small that p > 1+ǫ, the above right hand side is integrable w.r.t. r ∈ (0, 1), so by Fubini's theorem ∂Cϕ ∈ L p (D) as soon as ϕ ∈ W 1-1/p,p (T). Since ∂Cϕ = 0, this establishes 1. Assertion 2 is a consequence of the fact that S is a L Finally, set T α w = T (αw). By a theorem of F. Riesz, to see that I -T α is an isomorphism of L p (D), it is enough since T α is compact to check that I -T α is one-to-one. Let w ∈ L p (D) be such that (I -T α )w = 0 and set u = T ( αw) ∈ W 1,p loc (C). Observe there is p 1 > 2 such that u ∈ L p 1 loc (C). Indeed, by the Sobolev imbedding theorem, if p < 2 then W 1,p loc (C) is embedded in L p * loc (C), with p * = 2p 2-p , whereas if p ≥ 2 then W 1,p loc (C) is embedded in L λ loc (C) for every λ ∈ (2, ∞). Now, since w = T α w, we have u = w in D and so w ∈ L p 1 (D) hence αw ∈ L p 1 (C). Thus, by assertion 3, we get that u ∈ W 1,p 1 loc (C). Moreover, from (86) and since u = w in D, it holds in the sense of distributions that ∂u = αw = α u a.e. in C.

(88)

In addition, u(z) clearly goes to 0 when |z| goes to +∞. It now follows from the generalized Liouville theorem [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF]Prop. 3.3] that u = 0, therefore w = 0.

Proof of Lemma 5. which is [START_REF] Patil | Representation of H p functions[END_REF].

Aknowledgments.

For 1 10 ) 2

 1102 < p < ∞, we denote by H p (D) the classical Hardy space of holomorphic functions f on D such that f H p (D) := ess sup 0<r<1 f L p (Tr) < +∞ , (For instance if f ∈ L p (Ω) and ν ∈ W 1,∞ R (Ω), we define by ν∂f to be the distribution ν∂f , φ = -Ω (νf ∂φ + ∂νf φ)dm , ∀φ ∈ D(Ω) .

Remark 4 . 2 . 1 .

 421 By the Sobolev imbedding theorem ([2, Thm 5.4, Part II]), s ∈ C 0,γ (D) and w ∈ C 0,γ loc (D) for all γ ∈ (0, 1). Theorem 4.2.1 will allow for us to carry over to G p α the essentials of the boundary behaviour of holomorphic Hardy functions: Proposition 4.2.1. 1. If w ∈ G p α , then tr w(e iθ ) := lim ξ∈D, ξ→e iθ n.t.

Theorem 4 . 2 . 1 and 4 . 3 Proposition 4 . 3 . 1 .

 42143431 Proposition 4.2.1 are proven in Section 5.3. Properties of the Hardy class H p ν The following statements hold true.

Remark 4 . 3 . 1 .Remark 4 . 3 . 2 .

 431432 When ν = 0, that is, when dealing with holomorphic Hardy spaces, the best constant in[START_REF] Doyle | Feedback Control Theory[END_REF] is c 0 = 1 because f L p (Tr ) increases with r, and then equality holds throughout. For general ν, a bound on c ν depending solely on ν W 1,∞ (D) is easily derived from Proposition 3.2.3.1 and Theorem 4.2.1, but the authors do not know of a sharp estimate. Assertion (c) in Proposition 4.3.1 implies that a function f ∈ H p ν (D) whose trace is zero on a subset of T having positive Lebesgue measure must vanish identically. As in the holomorphic case, a function in H p,0 ν is uniquely defined by its real part on T: Proposition 4.3.2. Let f ∈ H p,0 ν be such that Re (tr f ) = 0 a.e. on T. Then f ≡ 0. Proof of Proposition 4.3.2. Let f ∈ H p,0 ν (D) satisfy Re tr f = 0 a.e. on T. If we define w through (17), then w ∈ G p,0 α by Proposition 3.2.3.1 and clearly Re tr w = 0 a.e. on T. Therefore w ≡ 0 in view of Proposition 4.2.1, assertion 4, whence f ≡ 0 in D.

4. 4

 4 Regularity of the Dirichlet problem 4.4.1 Solvability of the Dirichlet problem in G p α (D) We first focus on a slight variation of the Dirichlet problem for the G p α (D) class. Let us introduce one more piece of notation by letting

0

  (a) To each c ∈ R, there uniquely exists w ∈ G p α such that Re (tr w) = ψ a.e. on T and 2π Im tr w(e iθ ) dθ = 2πc. Moreover there are constants c p,α and c ′ p,α such that tr w L p (T) ≤ c p,α ψ L p (T) + c ′ p,α |c| .

Theorem 4 . 4 . 2 . 1 .

 4421 For all ϕ ∈ L p R (T), there uniquely exists f ∈ H p,0 ν such that, a.e. on T:

From Proposition 4 . 3 . 3 ,

 433 Theorem 4.4.2.1 clearly extends Theorem 4.1.1 when the boundary data belong to L p (T).

  First, the generalized conjugation operator preserves this smoothness class (compare Corollary 4.4.2.1): Proposition 4.4.3.1. The operator H ν is bounded on W 1,p R (T). Next, assuming in Theorem 4.4.2.1 that ϕ ∈ W 1,p R (T), not only does f belong to W 1,p (D), as predicted by Theorem 4.1.1, but the derivatives of f satisfy a condition of Hardy type: Theorem 4.4.3.1. Let ϕ ∈ W 1,p R (T) and f ∈ W 1,p (D) be the unique solution to (5) on D satisfying Re (tr f ) = ϕ and such that (12) holds. Then, (a) tr f ∈ W 1,p (T), and it holds that

Theorem 4 .

 4 5.1.1. Let I = T be an open subset of T having the extension property.Then, the restrictions to I of traces of W 1,p (D)-solutions to (5) form a dense subspace of W 1-1/p,p (I).

Theorem 4 .

 4 5.2.1. Let I ⊂ T be a measurable subset such that T \ I has positive Lebesgue measure. The restrictions to I of traces of H p ν -functions are dense in L p (I).Remark 4.5.2.1. When I ⊂ T is not of full measure and φ ∈ L p (I), Theorem 4.5.2.1 entails there is a sequence of functions f k ∈ H p ν whose trace on I converges to φ in L p (I). Now, since balls in tr H p ν are weakly compact by Proposition 4.3.1 point (b), it must be that either φ is the trace on I of a H p ν -function or tr f k L p (T\I) → +∞ with k. In view of Theorem 4.5.1.1, the corresponding remark applies when I is an open subset of T with the extension property and ϕ ∈ W 1-1/p,p (I) gets approximated in this space by a sequence of traces of W 1,p (D)-solutions to (5).

Proposition 4 . 6 .

 46 1 and the Hahn-Banach theorem now team up to yield: Theorem 4.6.1.

  and w is the extension of w by 0 outside D, then obviously ( T w) | D = T w. Next, for u ∈ L p (C), we denote by S the Beurling operator:

3 . 4 .

 34 The operator T maps L p (C) continuously into W 1,p loc (C). The operator T is bounded from L p (D) into W 1,p (D), and is compact from L p (D) to L p (D). Moreover ∂T w = w and ∂T w = (S w) | D for all w ∈ L p (D). For any α ∈ L ∞ (D) the operator w → w -T (αw) is an isomorphism of L p (D).

  [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension d ≥ 3[END_REF] holds. Next, we put F = e -s w and claim that F is holomorphic in D. Indeed, F ∈ L p (D) hence, by Weyl's lemma[START_REF] Forster | Lectures on Riemann surfaces[END_REF] Thm 24.9], it is enough to check that ∂F = 0 on D in the sense of distributions. Let ψ ∈ D(D) and ψ n a sequence in D(R 2 ) | D converging to s in W 1,l (D) for some l > max(q, 2). Thus ψ n converges boundedly to s in W 1,q (D) by the Sobolev imbedding theorem and Hölder's inequality. Then, by dominated convergence, ∂F, ψ =e -s w, ∂ψ =lim n w, e -ψn ∂ψ =lim n w, ∂(e -ψn ψ) + ψe -ψn ∂ψ n = lim n αw, e -ψn ψlim n w, ψe -ψn ∂ψ n = e -s (αww ∂s), ψ = 0

  ) holds. Conversely, let w ∈ G p α . Then ∂w = α w ∈ L p (D) and Proposition 5.2.1, assertion 4, tells us that the L p (D)-function w -T (α w) anihilates ∂ in the distributional sense, hence is holomorphic on D by Weyl's lemma. From Proposition 5.2.1, point 4 again, this entails w ∈ W 1,p loc (D). Appealing to Lemma 5.3.2, with rD in place of D, we obtain dζ + T (αwχ Dr )(z) , |z| < r < 1, and letting r → 1 we get by dominated convergence, (26), and Proposition 5.2.1, point 4, that w = C(tr w) + T (αw) a. e. in D . (60) Assertion 3 is now completely proven. Finally, assume that w ∈ G p α satisfies Re tr w = 0. By Theorem 4.2.1 we can write w = e s F , where s is continuous on D and real on T, while F ∈ H p (D). Thus Re tr F is zero on T, and by the Poisson representation of H p -functions it follows that Re F ≡ 0 on D thus F is a pure imaginary constant, say, c [32, Thm 3.1]. Since w = c e s , it has zero mean on T if and only if c = 0. When (16) holds, condition (15) likewise implies that c = 0.

5. 5

 5 The Dirichlet problem in the class G p α Proof of Theorem 4.4.1.1. Let g ∈ H p . By assertion 3 of Proposition 4.2.1 and Remark 4.2.2, together with the Cauchy formula for H p -functions, we see that w ∈ L p (D) belongs to G p α and satisfies P + (tr w) = tr g if, and only if w -T (αw) = g. But since g a fortiori lies in L p (D), there is a unique w ∈ L p (D) to meet the latter equation as follows from Proposition 5.2.1, assertion 4. Moreover, by the same assertion, it holds that

Proof of Theorem 4 . 4 . 1 . 2 .

 4412 For each pair (ϕ, c) ∈ L p R (T) × R, set A(ϕ, c) := Re tr w ϕ.c , Im 1 2π 2π 0 tr w ϕ.c (e iθ ) dθ ∈ L p R (T) × R ,where w ϕ,c is the unique function in G p α such that P + (tr w ϕ,c ) = ϕ + i(H 0 ϕ + c).

  ), we have that (I -T α )w ϕ,c = g where g ∈ H p satisfies tr g = ϕ + i(H 0 ϕ + c), hence A(ϕ, c) = (ϕ, c) + B(ϕ, c) where B(ϕ, c) := Re tr T α (w ϕ,c ) , Im 1 2π 2π 0 tr w ϕ,c (e iθ ) dθc . Since the G p α -norm is finer than the L p (D)-norm, we see from Theorem 4.4.1.1 and Proposition 5.2.1, point 4, that (ϕ, c)

  and, since the second component is R-valued and continuous, B is compact from L p (T) × R into itself. Moreover, A is injective by Proposition 4.2.1, point 4, consequently it is an isomorphism of L p (T) × R (see e.g. [66, Ch. XVII, prop. 2.3]) thereby establishing the existence and uniqueness part of (a). To establish[START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF], put tr w = ψ + iv, where v ∈ L p R (T) in view of Proposition 4.2.1. Thanks to Theorem 4.2.1, we can write ψ + iv = e tr s F where F ∈ H p (D) and tr s is real-valued, s ∈ C 0,γ (D) for 0 ≤ γ < 1 (cf. Remark 4.2.1). By definition tr F = h + iH 0 h + ib, where h ∈ L p R (T) and b is a real constant. Thus v = e tr s (H 0 h + b) and ψ = e tr s h, which gives us v = e tr s H 0 (e -tr s ψ) + be tr s .

Proof.

  Recall from (53) the Hodge decomposition V = G + D, with G ∈ G 0,p and D ∈ D p , for which (62) is known to hold [43, Thm 10.5.1]. Put G = (∂ x g, ∂ y g) T and D

  the same is true of G and D [43, Sec. 10.5], hence g, h ∈ C ∞ R (D) and tr g = 0.

Lemma 5 . 7 . 1 .

 571 Let ϕ ∈ W 1-1/p,p R (T), and f ∈ W 1,p (D) be the unique solution to (5) satisfying Re(tr f ) = ϕ and (12), cf. Theorem 4.1.1. Then

Lemma 5 . 7 . 3 .

 573 There exist a function b ∈ L ∞ (D) and a function ψ ∈ BMOA(D) such that, for a.e. z ∈ D,

Proof of Corollary 4 . 4 . 3 . 1 .

 4431 Let f = u + iv ∈ W 1,p (D) be the solution to[START_REF] Astala | Area distortion of quasi-conformal mappings[END_REF] satisfying[START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF] granted by Theorem 4.1.1. By (47) and (8), f meets ess sup0<r<1 ∇f L p (Tr) < +∞. (71)If W := (1ν 2 ) 1/2 ∂f , we saw in the proof of Lemma 5.7.1 that ∂W = αW for some α ∈ L ∞ (D). Thus W ∈ G p α (D) by (

2 . 1

 21 applied to W , the derivatives ∂f and ∂f have non tangential limits h and νh respectively, where h ∈ L p (iθ )h(e iθ ) p dθ = lim r→1 2π 0 ∂f (re iθ )ν h(e iθ ) p dθ = 0 .

  )(x)dx = T σ∂ n tr u . This establishes point 1. As to point 2, let v ∈ W 1,p R (T) be such that ∂ θ v = σg. By Theorem 4.1.1, there exists f ∈ W 1,p (D) such that (5) holds and Im tr f = v on T. Then, Theorem 4.4.3.1 implies that u := Re f fulfills all the requirements. To prove uniqueness up to an additive constant, consider u ∈ W 1,p R (D) with div(σ∇u) = 0 in D and ∂ n u = 0 on T. Put f = u + iv for a solution to (5) in W 1,p (D) such that u = Re f . By Remark 4.1.2 ∂ θ v = σ∂ n u = 0 on T, which means that tr v is constant, i.e. Im tr f is constant. By Theorem 4.1.1, u is constant in D.

  J) → W -1/p,p R(I) is well-defined and bounded by Corollary 4.4.2.1 and the boundedness of ∂ θ from W 1-1/p,p (I) into W -1/p,p (I).

Lemma 5 . 8 . 1 . 2 .

 5812 The operator A defined in (74) has dense range.Proof. It is equivalent to show that the adjoint operator A

  with ϕ I = u I + iv I , where u I and v I are real valued. Set

  3.1 point (b) and Lemma 5.7.1 equation (65) together imply that ∂f = e s F , where s is continuous on D while F ∈ H p . Necessarily then, tr F = 0 on J which has positive measure, hence F ≡ 0 implying that ∂f = ∂f ≡ 0. Therefore f is constant, in particular 0 = ∂ θ Re(tr f ) = ∂ θ Ψ = ϕ I ∨ 0 on T. Therefore ϕ I ≡ 0 thus B * is injective, as desired. Proof of Theorem 4.5.2.1. Let ε > 0 and ϕ I ∈ L p (I) with ϕ I = u I + iv I where u

  thm 5.5] that ∂Cϕ L p (Tr) ≤ C ǫ,p (1r) 1+ǫ ϕ L p (T) .(84)If moreover ϕ ∈ W 1,p (T), integrating by parts the Cauchy formula gives us∂Cϕ(z) = 1 2iπ T ϕ(ξ) (ξz) 2 dξ = 1 i T ∂ t ϕ(ξ) ξ(ξz) dξ,from which it follows by the M. Riesz theorem again that∂Cϕ L p (Tr) ≤ C p ϕ W 1,p (T) .(85)Introduce the operators A 1 r : W 1,p (T) → L p (T r ) and A 0 r : L p (T) → L p (T r ), where in both cases A i r ϕ = (∂Cϕ) | Tr . We gather from (85) and (84) that|||A 1 r ||| ≤ C p while |||A 0 r ||| ≤ C ǫ,p (1r) 1+ǫ .Since W 1-1/p,p (T) = [W 1,p (T), L p (T)] 1/p , interpolating between A 1 r and A 0 r yields∂Cϕ L p (Tr) ≤ C ′ ǫ,p

  2 (C)-isometric Calderón-Zygmund operator, see[START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF] Ch. V, Sec. D] or [64, Ch. II,Thm 3]. Next, observe that if K ⊂ C and w ∈ L p (C), we have forz ∈ K T w(z) = 1 π ((χ D w) * g K ) (z), with g K (ξ) := χ K+D (ξ) ξ ,where χ E denotes the characteristic function of a set E.Clearly g K ∈ L 1 (C) when K is compact, showing that T is bounded from L p (C) into L p loc(C). We claim that ∂ T w = χ D w and ∂ T w = S(χ D w) in the sense of distributions. When w is C 2 -smooth and compactly supported in D, whence χ D w = w, this is a simple computation [3, Ch. V, Lem. 2]. In the general case, pick a sequence of functions v n ∈ D(D) converging to χ D w in L p (C). By what precedes T v n converges to T (χ D w) = T w in L p loc (C), hence as distributions∂( T w) = lim n ∂( T v n ) = lim n v n = χ D w,(86)where the last limit holds in L p (C) thus a fortiori in the distributional sense. Using the L p (C) boundedness of S, a similar argument yields∂( T w) = lim n ∂( T v n ) = lim n S(v n ) = S(χ D w),(87)proving the claim. Since ∂ T wL p (C) = S(χ D w) L p (C) ≤ C w L p (D) , assertion 3 holds. As to assertion 4, the boundedness of T from L p (D) to W 1,p (D) follows from 3 and the identity ( T w) | D = T w, an so do the relations ∂T w = w and ∂T w = (S w) | D in view of (86) and (87). The compactness of T : L p (D) → L p (D) then follows from the compactness of the injection W 1,p (D) → L p (D) [2, Ch. VI, Thm 6.2].

2 . 1 :

 21 It is a result by Hardy and Littlewood[32, thm 5.9] that for any g ∈ H pg L p 1 (Tr) ≤ C p g L p (Tr ) (1r) 1/p 1 -1/p , 0 < r < 1 , p ≤ p 1 ≤ ∞ . (89)Taking p 1 ∈ [p, 2p) and raising (89) to the power p 1 , we obtain upon integrating with respect to r gp 1 L p 1 (D) ≤ C p,p 1 tr g p 1 L p (T)thereby proving 1. For the rest of the proof, we fix p 1 ∈ (2, 2p). Assume now that w -T (αw) = g ∈ H p . By Proposition 5.2.1, point 4, we get T (αw) ∈ W 1,p (D). Moreover, as already stressed in the proof of that proposition, we know from the Sobolev imbedding theorem that W 1,p (D) ⊂ L p 2 (D) for some p 2 > 2. Thus, by point 1 and Hölder's inequality, w = g + T (αw) belongs to L p * (D) with p * = min{p 1 , p 2 } > 2, and consequently T (αw) ∈ W 1,p * (D) ⊂ C 0,1-2/p * (D) by the Sobolev imbedding theorem again. Using Proposition 5.2.1, point 3, we establish similarly that T αw ∈ W 1,p * loc (C) ⊂ C 0,1-2/p * loc (C). Recaping what we just did, we obtain T (αw) W 1,p * (D) ≤ c p,α w L p * (D) = c p,α g+T (αw) L p * (D) ≤ c ′ p,α T (αw) L p 2 (D) + g L p 1 (D) ≤ c ′′ pα T (αw) W 1,p (D) + g H p (D) ≤ c ′′′ p,α w L p (D) + g H p (D)

  -∂ν/(1ν 2 ), as shown in Proposition 3.2.3.1 below. From equation (16) below, we see that α has this form for some ν ∈ W 1,∞

	in the sense of distributions on D. Note that . H p ν (D) and . G p α (D) formally coincide, but the equations (5) and (14) are different. Again . G p α (D) makes G p α into a real Banach space. The reason why we introduce G p α is the tight connection it has with H p ν when
	we set α := R R (D). Making such an assumption in the definition would be artificial, meeting (6) if, and only if α = ∂h for some h ∈ W 1,∞ since most of the properties of G p α to come are valid as soon as α ∈ L ∞ (D). However, if (16) holds and only in this case (see section 3.2.3 below), we shall find it convenient to
	introduce the space G p,0 α of those w ∈ G p α normalized by
		1 2π	0	2π	σ 1/2 Im tr w (e iθ ) dθ = 0.	(15)
	3.2.3 The link between H p ν and G p α
		p,0 ν	of H p ν consisting of those f for which
				2π
					Im (tr f (e iθ )) dθ = 0	(12)
				0
	holds. We further let H p,00 ν	be the subspace of functions f ∈ H p,0 ν	such that
					2π
					tr f (e iθ ) dθ = 0 .	(13)
				0
	Remark 3.2.1. In what follows, we make use of both H p ν (D) and H p (D). For simplicity, we drop the dependence on D and denote them by H p ν and H p , respectively. In particular, H p (no subscript) always stands for the classical holomorphic Hardy space of the disk.
	3.2.2 The class G p α (D)
	For α ∈ L ∞ (D) and 1 < p < ∞ (note that α may be complex-valued here), we define another space G p α (D) = G p α , consisting of those Lebesgue measurable functions w on D
	such that:
		w G p α := ess sup 0<r<1	w L p (Tr) < +∞
	and
					∂w = αw	(14)

  the maximum principle, i.e. |f | cannot assume a relative maximum in D unless it is constant. More generally, a non constant function in H p ν is open and discrete 3 . It is rather easy to deduce Proposition 4.3.1 from the corresponding properties for the G p α (D) class. Indeed, by (6) and Proposition 3.2.3.1 we can invert (

  That the operator H 0 is continuous from L p R (T) into itself is the well-known M. Riesz theorem [35, Ch. III, thm. 2.3]. Corollary 4.4.2.1 thus generalizes the latter.

as the ν-Hilbert transform. More precisely, to each ϕ ∈ L p R (T), we associate the unique function f ∈ H p,0 ν such that Re tr f = ϕ, and we set H ν ϕ = Im trf ∈ L p (T). It now follows from Theorems 4.1.1 and 4.4.2.1 that: Corollary 4.4.2.1. The operator H ν is bounded both on L p R (T) and on W 1-1/p,p R (T).

When ν = 0, we observe that H 0 ϕ is just the harmonic conjugate 4 of ϕ normalized to have zero mean on T.

)

  Proof. Observe, since Re s = 0 on T, that |F | = |w| there. Moreover, by Lemma 5.7.2 and Remark 4.2.1, F = e -s w is (Hölder) continuous on D and it does not vanish there by our assumption on w. Therefore F can have no inner factor in its inner-outer decomposition [35, Ch. II, Cor. 5.7, Thms 6.2, 6.3] thus it is an outer function: , and by (70) so does the derivative of F since the latter is bounded. That is, we have proven that F ′ ∈ H p . Now, since ∂w = e s ∂s F + e s F ′ , we see by the boundedness of F , s, and the Hardy character of ∂s, F ′ just established that ∂w satisfies a Hardy condition of order p.

				F (z) = ξ 0 exp	1 2π	0	2π	e iθ + z e iθ -z	log |w(e iθ )| dθ	(70)
	with ξ 0 a unimodular constant. As tr w ∈ W 1,p (T) is bounded in modulus from above and below by strictly positive constants, log |w(e iθ )| also lies in W 1,p (T). Hence, in view of (85) in Appendix A below , the derivative of the holomorphic function of z defined by
	1 2π	0	2π	e iθ + z e iθ -z	log |w(e iθ )| dθ = -	1 2π T	log |w(e iθ )| dθ +	1 iπ T	log |w(ξ)| ξ -z	dξ
	lies in H p

In other respects the results of[START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] are of course much more general since they deal with arbitrary non-homogeneous elliptic equations in any dimension and can handle distributional boundary conditions.

A map is discrete if the preimage of any value is a discrete subset of its domain.

Though it has the same behaviour, it is distinct from the Hilbert transform, see[35, Chap. III, So. 1].

The result is stated there using the language of differential forms that we did not introduce here.

We wish to thank K. Astala, S. Hofmann, F. Murat and L. Päivärinta for helpful discussions and advice. We are indebted to the referee whose remarks helped us to improve the presentation of this paper. This work was partially supported by the ANR project "AHPI" (ANR-07-BLAN-0247-01).

and Φ is Lipschitz continuous with constant, say, K (because σ ∈ W 1,∞ (D) and in view of (3)) we get

Thus the second integral in the right hand side of (69) is uniformly bounded. As to the first, we observe by the Lipschitz character of Φ that Φ | T is absolutely continuous on T with derivative ϕ := ∂ θ Φ(e iθ ) which is bounded in modulus by K for a.e. θ. Therefore, integrating by parts, we get 67) is, by our very construction, the derivative of the holomorphic function

that vanishes at 0 and whose real part on T is Υ = -Re B | T , see the discussion after [START_REF] Peller | Hankel Operators and their Applications[END_REF].

Writing -B = -(Ba)a, we have that Υ = Υ 1 + Υ 2 where Υ 1 is Lipschitz continuous and thus absolutely continuous with bounded derivative on T, while Υ 2 = -Re (tr a).

Consequently tr (H + a) = Υ 1 + iH 0 (Υ 1 ) where we recall that H 0 denotes the usual conjugation operator. Now, Υ 1 is a fortiori in W 1,l (T) for all 1 < l < ∞, therefore the same is true of H 0 (Υ 1 ) by Proposition 4.4.3.1 applied with ν = 0. Consequently (H + a) ′ = H ′ + ψ lies in H l for all 1 < l < ∞, and finally so does H ′ since it is the case of ψ ∈ BMOA(D). Altogether, considering separately the summands in the right-hand side of (67) and recalling Lemma 5.7.

Lemma 5.8.2.2. For any function u ∈ L q R (T), it holds that

where U ∈ W 1,q R (T) is any function such that

, and consider U ∈ W 1,q R (T) such that (77) holds. Then, since H ν v has zero mean on T, we get from Lemma 5.8.2.1

Remark 5.8.2.1. If we restrict H ν to the space of L q R (T)-functions with zero mean, which is mapped into itself by H ν , we may summarize the content of Lemma 5.8.2.2 as

Let I ⊂ T be as in Theorem 4.5.2.1 and put J = T \ I. For u J ∈ L p R (J), let us define

Corollary 5.8.2.1. The operator B is bounded from L p R (J) to L p R (I)/R and has dense range.

Note that we consider B as a mapping from L p R (J) into the quotient space L p R (I)/R rather than L p R (I). That B has dense range means: given v ∈ L p R (I) and ε > 0, there exist u

Proof. Clearly B is well-defined and bounded by Corollary 4.4.2.1. To prove it has dense range, it is enough to check that B * : L q,0 R (I) → L q (J) is one-to-one; here, by L q,0 R (I), we mean the subspace of L q R (I) comprised of functions with zero mean. For ϕ I ∈ L q,0 R (I), we deduce from Lemma 5.8.2.2 that