I Bailleul
email: ismael.bailleul@univ-rennes1.fr

F Bernicot

D Frey
email: d.frey@tudelft.nl

Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations

Keywords: Stochastic singular PDEs, semigroups, paraproducts, paracontrolled calculus, Parabolic Anderson Model equation, multiplicative stochastic Burgers equation

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a 3-dimensional Riemannian setting, in either bounded or unbounded domains. Aiming that, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity. This constitutes to a first step in building a higher order paracontrolled calculus via semigroup methods.

Introduction

It is probably understated to say that the work [START_REF] Hairer | A theory of regularity structures[END_REF] of Hairer has opened a new era in the study of stochastic singular parabolic partial differential equations. It provides a setting where one can make sense of a product of a distribution with parabolic non-positive Hölder regularity index, say a, with a function with nonnegative regularity index, say b, even in the case where a `b is non-positive, and where one can make sense of and solve a large class of parabolic stochastic singular partial differential equations by fixed point methods. The parabolic Anderson model equation (PAM) pB t `Lqu " uζ, (1.1) studied in Section 5 in a 3-dimensional unbounded background, is an example of such an equation, as it makes sense in that setting to work with a distribution ζ of Hölder exponent α ´2, for some α ă 1 2 , while one expects the solution u to the equation to be of parabolic Hölder regularity α, making the product uζ undefined since α `pα ´2q ď 0.

The way out of this quandary found by Hairer has its roots in Lyons' theory of rough paths, which already faced the same problem. Lyons' theory addresses the question of making sense of, and solving, controlled differential equations dz t " V i pz t q dX i t (1.2) in R d say, driven by an R -valued 1 p -Hölder control X " `X1 , . . . , X ˘, with p ě 2, and where V i are sufficiently regular vector fields on R d . Typical realizations of a Brownian path are 1 p -Hölder continuous, with p ą 2, for instance. One expects a solution path to equation (1.2) to be 1 p -Hölder continuous as well, in which case the product V i pz t q dX i t , or the integral ş t 0 V i pz s q dX i s , cannot be given an intrinsic meaning since 1 p ``1 p ´1˘ď 0. Lyons' deep insight was to realize that one can make sense of, and solve, equation (1.2) if one assumes one is given an enriched version of the driving signal X that formally consists of X together with its non-existing iterated integrals. The theory of regularity structures rests on the same philosophy, and the idea that the enriched noise should be used to give a local description of the unknown u, in the same way as polynomials are used to define and describe locally C k functions.

At the very same time that Hairer built his theory, Gubinelli, Imkeller and Perkowski proposed in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] another implementation of that philosophy building on a different notion of local description of a distribution, using paraproducts on the torus. The machinery of paracontrolled distributions introduced in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] rests on a first order Taylor expansion of a distribution that happened to be sufficient to deal with the stochastic parabolic Anderson equation (1.1) on the 2-dimensional torus, the stochastic additive Burgers equation in one space dimension [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], the Φ 4 3 equation on the 3-dimensional torus [START_REF] Catellier | Paracontrolled distributions and the 3-dimensional stochastic quantization equation[END_REF][START_REF] Zhu | Lattice approximation to the dynamical Φ 4 3 model[END_REF] and the stochastic Navier-Stokes equation with additive noise [START_REF] Zhu | Three-dimensional Navier-Stokes equations driven by space-time white noise[END_REF][START_REF] Zhu | Approximating three-dimensional Navier-Stokes equations driven by space-time white noise[END_REF]. The KPZ equation can also be dealt with using this setting [START_REF] Gubinelli | KPZ equation reloaded[END_REF]. Following Bony's approach [START_REF] Bony | Calcul symbolique et propagation des singulariés pour les équations aux dérivées partielles non linéaires[END_REF], the paraproduct used in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] is defined in terms of Fourier analysis and does not allow for the treatment of equations outside the flat background of the torus or the Euclidean space, if one is ready to work with weighted functional spaces. The geometric restriction on the background was greatly relaxed in our previous work [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] by building paraproducts from the heat semigroup associated with the operator L in the semilinear equation. A theory of paracontrolled distributions can then be considered in doubling metric measure spaces where one has small time Gaussian estimates on the heat kernel and its 'gradient' -see [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]. This setting already offers situations where the theory of regularity structures is not known to be working. The stochastic parabolic Anderson model equation in a 2dimensional doubling manifold was considered in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] as an example. The first order 'Taylor expansion' approach of paracontrolled calculus seems however to restrict a priori its range of application, compared to the theory of regularity structures, and it seems clear that a kind of higher order paracontrolled calculus (see the last paragraph of Section 2) is needed to extend its scope. We tackle in the present work the first difficulty that shows off in this program, which is related to the crucial use of commutator estimates between the heat operator and a paraproduct, which is one of the three workhorses of the paracontrolled calculus method, together with Schauder estimates and another continuity result on some commutator.

Working in unbounded spaces with weighted functional spaces requires a careful treatment which was not done so far. We shall illustrate the use of our machinery on two examples: The parabolic Anderson model (PAM) equation (1.1) in a (eventually unbounded) 3-dimensional Riemannian manifold, and Burgers equation with multiplicative noise in the 3-dimensional bounded Riemannian manifold. Hairer and Labbé have very recently studied the (PAM) equation in R 3 from the point of view of regularity structures [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] -see also the work [START_REF] Hairer | A Wong-Zakai theorem for stochastic PDEs[END_REF] of Hairer and Pardoux. They had to introduce some weights to get a control on the growth at spatial infinity of quantities of interest. A non-trivial part of their work consists into tracking the time-behavior of their estimates, with respect to the time, which requires the use of time-dependent weights -see Section 3.4 for details about them. We also need to use weighted spaces, for the same reasons, and working with the weights of [START_REF] Hairer | A Wong-Zakai theorem for stochastic PDEs[END_REF][START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] happens to be convenient. Our treatment is however substantially easier, as we do not need to travel backwards in time such as required in the analysis of the reconstruction operator in the theory of regularity structures. As a matter of fact, our results on the (PAM) equation give an alternative approach, and provide a non-trivial extension, of the results of [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] to a non-flat setting, with a possibly wider range of operators L than can be treated presently in the theory of regularity structures. As for Burgers equation with multiplicative noise, it provides a description of the random evolution of a velocity field subject to a random rough multiplicative forcing, and whose dynamics reads pB t `Lqu `pu ¨∇qu " M ζ u, (1.3) where ζ is a 3-dimensional white noise with independent coordinates, and

M ζ u :" `ζ1 u 1 , ζ 2 u 2 , ζ 3 u 3 ˘,
for the velocity field u " `u1 , u 2 , u 3 ˘: M 3 Ñ R 3 . With zero noise ζ, this 3dimensional Burgers system plays a very important role in the theory of PDEs coming from fluid mechanics, and later from condensed matter physics and statistical physics. It has been proposed by Burgers in the 30's as a simplified model of dynamics for Navier-Stokes equations. A change of variables, called after Cole and Hopf, can be used to reduce the deterministic quasilinear parabolic equation to the heat equation, thus allowing the derivation of exact solutions in closed form. Despite this fact, the study of Burgers system is still very fashionable as a benchmark model that can be used to understand the basic features of the interaction between nonlinearity and dissipation. Motivated by the will to turn Burgers equation into a model for turbulence, stochastic variants have been the topic of numerous recent works [START_REF] Bertini | The stochastic Burgers equation[END_REF][START_REF] Hairer | Approximations to the Stochastic Burgers equation[END_REF][START_REF] Hairer | Rough Burgers-like equations with multiplicative noise[END_REF][START_REF] Hairer | A theory of regularity structures[END_REF][START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Gubinelli | KPZ equation reloaded[END_REF], where a random forcing term is added in the equation, mainly in one space dimension, with an additive space-time white noise -that is with a space-time white noise instead of M ζ u with ζ space white noise. The Cole-Hopf transformation can formally be used again, and turns a solution to the 1-dimensional stochastic Burgers equation with additive space-time noise to the heat equation with multiplicative space-time noise, with a very singular noise, such as detailed in [START_REF] Gubinelli | KPZ equation reloaded[END_REF]. A similar change of variable trick can be used for the study of the above multidimensional stochastic Burgers system with multiplicative noise (1.3); we shall analyse it in Section 5.4. Also, one can consider the study of this example as a first step to understanding the dynamics of the 3-dimensional stochastic incompressible Navier-Stokes equation, with multiplicative noise, where the incompressibility brings the additional difficulty to deal with the Leray projector to keep the vanishing divergence property. In any case, equation (1.3) seems not to have been studied so far, to the best of our knowledge. Contrary to the theory of regularity structures, whose introduction requires to set up a whole new algebraic-analytic setting, the analytic part of paracontrolled calculus is based only on classical ingredients, and its use in solving some singular stochastic partial differential equation involves an elementary reasoning. This machinery is described in simple terms in Section 2, which serves as a baseline for the study of the parabolic Anderson and Burgers equations in Section 5.

The geometric and functional settings in which we lay down our study are described in Section 3. In short, we work on a doubling metric measure manifold pM, d, µq, equipped with a Riemannian operator L given by the finite sum of squares of vector fields. The heat semigroup of the operator L is assumed to have a kernel that satisfies Gaussian pointwise bounds, together with its iterated derivatives; precise conditions are given in the item Conditions in the beginning of Section 3.1. Such a setting covers a number of interesting cases. One can use the semigroup to construct in an intrinsic way the scale of spatial Hölder spaces C α pM q on M and a scale of parabolic Hölder spaces C α `r0, T s ˆM ˘in which the (PAM) and Burgers equations will eventually be solved. Some Schauder-type regularity estimates for the heat semigroup, proved in Section 3.4, will be instrumental for that purpose. We call resolution map of the heat semigroup the map that associates to a distribution f the solution to the equation pB t `Lqv " f , with zero initial condition. One of our main contributions is the introduction of a pair of paraproducts built from the heat semigroup, intertwined via the resolution map, that are used to get exact formulas where formulas with a remainder were used previously [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Gubinelli | KPZ equation reloaded[END_REF][START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]. These two paraproducts share the same algebraic structure and the same analytic properties, most importantly a cancellation property that we introduce in Section 3.2. It allows in particular to set the stage in a more natural function space than previously done. They consist in some sense of space-time paraproducts in the parabolic variable. The powerful relies on a suitable combination of the Schauder estimates with these space-time new paraproducts, which allows us to obtain the expected and desired estimates in L 8

T C α pM q spaces (see Remark 17 for more details). The technical core of the paracontrolled calculus, such as defined by Gubinelli, Imkeller and Perkowski, is a continuity estimate for a corrector that allows to make sense of an a priori undefined term by compensating it by another potentially undefined term with a simpler structure, and to separate analytic from probabilistic considerations. We prove in Section 4.2 that this result holds in our general setting as well. As a result, we are able to prove the following kind of results on the (PAM) in a 3-dimensional (unbounded) measured manifold pM, d, µq that is Ahlfors regular, working with a second order differential operator L that satisfies some mild assumptions stated in Section 3.1. We also study the multiplicative Burgers equations in bounded ambiant space. In statements below, ξ stands for a space white noise on pM, µq, and ξ ε :" `e´εL ˘ξ stands for its regularization via the heat semigroup. Full details on the mathematical objects involved in the statements will be given along the way. The notion of solution to the (PAM) equation (1.1) depends on a notion of (PAM)-enhancement p ζ of a distribution ζ P C α´2 pM q. To every such enhancement of ζ is associated a Banach space D `p ζ ˘of distributions within which one can make sense of the equation and look for the solution to it -this is the space of paracontrolled distributions; see Sections 2 and 5.

Theorem 1. Given α P `2 5 , 1 2 ˘, and a (PAM)-enhancement of a distribution ζ P C α´2 , the parabolic Anderson model equation on M (a 3d space) has a unique paracontrolled solution in D `p ζ ˘. Moreover, the space white noise ξ has a natural (PAM)-enhancement, and there exists a sequence `λε ˘0ăεď1 of time-independent and deterministic functions such that for every finite positive time horizon T and every initial data u 0 P C 4α w 0 pM q, the solution u ε of the renormalized equation

B t u ε `Lu ε " u ε `ξε ´λε ˘, u ε p0q " u 0
converges in probability to the solution u P C α w `r0, T s ˆM ˘of the parabolic Anderson model equation on M associated with the natural enhancement of ξ. The result holds with w " 1 and T " 8 if µpM q is finite.

Let emphasize that uniqueness has to be understood as the unique solution in the suitable class of paracontrolled distributions, in which we solve the problem. Note also that we use weighted spatial and parabolic Hölder spaces to deal with the unbounded nature of the ambient space M . In R 3 , one can typically work with the weights wpx, τ q " e τ p1`|x|q and w 0 pxq " wpx, 0q a constant -these weights were already used by Hairer and Labbé in [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF]; see section 3.3. Hairer and Labbé [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] are able to work in the range ´1 2 ă α ď 0, in the setting of regularity structures; we do not know how to deal with such a situation in our setting. Note on the other hand that we described in the appendix of [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] how to extend the paracontrolled calculus to a Sobolev setting. Together with the present work, this allows to solve the (PAM) equation in Sobolev spaces W α,p for a large enough finite positive exponent p. The above Hölder setting corresponds to working with p " 8. The robustness of our framework in terms of the operator L or the ambient geometry is useful, at least insofar as the tools of regularity structures have not been adapted so far in a non-flat setting. Moreover, as explained before, it is easier to deal with the time-dependent weight through the current paracontrolled approach than via the regularity structures theory, as done in [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF].

As we shall see, the computations needed to handle the (PAM) equation and multiplicative Burgers system involve almost the same quantities. As far as the latter is concerned, we can prove the following result, under the same conditions on the ambient geometry and the operator L as above. We state the result here in the same framework as (PAM) with the additional assumption that the ambiant space M is bounded (a prototype of such framework is given by the torus or the sphere). We identify in the renormalized equation (1.4) below a symmetric matrix d with its associated quadratic form. We can work in such a bounded domain with the weight w : px, τ q Þ Ñ e κτ , for a large enough positive numerical constant κ. Note here that the above mentioned notion of enhancement p ζ of a distribution ζ P C α´2 depends on the equation under study, which is why we called it (PAM)-enhancement above there.

Theorem 2. Given α P `2 5 , 1 2 ˘, and a (Burgers)-enhancement of ζ P C α´2 , the multiplicative Burgers equation (1.3) on M (a bounded 3d space) has a unique local in time paracontrolled solution in D `p ζ ˘. Moreover, the space white noise ξ has a natural (Burgers)-enhancement, and there exists sequences of time-independent and deterministic R 3 -valued functions `λε ˘0ăεď1 and p3 ˆ3q-symmetric-matrix-valued functions `dε ˘0ăεď1 on M , such that if one denotes by u ε the solution of the renormalized equation

B t u ε `Lu ε ``u ε ¨∇˘u ε " M ξ ε ´λε u ε ´dε `uε , u ε ˘uε p0q " u 0 (1.4)
with initial condition u 0 P C 4α , then u ε converges in probability to the solution u P C α of the multiplicative Burgers equation, locally in time.

All details on Theorems 1 and 2 can be found in Section 5. These statements are two-sided, with the well-posedness of the paracontrolled version of the equations on the one hand, and the link between this notion of solution and the convergence of solutions to a renormalized regularized version of the initial equation on the other hand. A full proof of these statements requires a renormalization step that will be done in a forthcoming work.

Notations. Let us fix here some notations that will be used throughout the work.

' Given a metric measure space pM, d, µq, we shall denote its parabolic version by pM, ρ, νq, where M :" M ˆR is equipped with the parabolic metric ρ `px, τ q, py, σq ˘" dpx, yq `a|τ ´σ| and the parabolic measure ν " µ b dt. Note that for px, τ q P M and small radii r ą 0, the parabolic ball B M `px, τ q, r ˘has volume

ν ´BM `px, τ q, r ˘¯« r 2 µ `Bpx, rq ˘.
We shall denote by e a generic element of the parabolic space M. ' Given an unbounded linear operator L on L 2 pM q, we denote by D 2 pLq its domain. We give here the definition of a distribution, as it is understood in the present work. The definition will always be associated with the operator L described in Subsection 3.1 below.

Fix a point o P M and then define a Fréchet space S o of test functions f on M requiring that

› › › `1 `|t| ˘a1 `1 `dpo, ¨q˘a 2 B a 3 τ L a 4 f › › › 2,dν ă 8,
for all integers a 1 , . . . , a 4 ; we equip S o with the metric }f } :" sup

a 1 ,a 2 ,a 3 ,a 4 PN 1 ^› › › `1 `|t| ˘a1 `1 `dpo, ¨q˘a 2 B a 3 τ L a 4 f › › › 2,dν .
A distribution is a continuous linear functional on S o ; we write S 1 o for the set of all distributions.

(Let us point out that the arbitrary choice of the point o P M is only relevant in the case of an unbounded ambient space M ; even in that case, the space S o does not depend on o, for o ranging inside a bounded subset of M .) ' As a last bit of notation, we shall always denote by K Q the kernel of an operator Q, and write À T for an inequality that holds up to a positive multiplicative constant that depends only on T .

' Spatial Hölder spaces C γ and parabolic Space-time Hölder spaces C γ will be rigorously defined in Section 3.3, and the weights and p a will be introduced in Section 3.4. To deal with remainder terms in some paracontrolled expansions, we shall use the following notation. For γ P R and c a non negative integer, we shall denote by γ c an element of C γ pca , and by γ 7 c an element of L 8 T C γ pca .

2

Paracontrolled calculus in a nutshell

The theories of regularity structures and paracontrolled calculus aim at giving a framework for the study of a class of classically ill-posed stochastic parabolic partial differential equations (PDEs), insofar as they involve illicit operations on the objects at hand. This is typically the case in the above parabolic Anderson model and Burgers equations, where the products uζ and M ζ u are a priori meaningless, given the expected regularity properties of the solutions to the equations. So a regularization of the noise does not give a family of solutions to a regularized problem that converge in any reasonable functional space to a limit that could be defined as a solution to the original equation. To bypass this obstacle, both the theory of regularity structures and paracontrolled calculus adopt a point of view similar to the point of view of rough paths analysis, according to which a good notion of solution requires the enhancement of the notion of noise into a finite collection of objects/distributions, built by purely probabilistic means, and that a solution to the equation should locally be entirely described in terms of these objects. This collection of reference objects depends on the equation under study, and plays in the setting of regularity structures the role played by polynomials in the world of C k maps, where they provide local descriptions of a function in the form of a Taylor expansion. Something similar holds in paracontrolled calculus. In both approaches, the use of an ansatz for the solution space allows to make sense of the equation and get its well-posed character by deterministic fixed point methods, and provides as a consequence solutions that depend continuously on all the parameters in the problem.

To be more concrete, let us take as an introduction to these theories the example of the 2-dimensional (PAM) equation, fully studied in [START_REF] Hairer | A theory of regularity structures[END_REF][START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Hairer | A simple construction of the continuum parabolic Anderson model on R 2[END_REF][START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]. The space white noise ζ is in that case p´1 ´q-Hölder continuous, and the intuition suggests that the solution u to the (PAM) equation should be p1 ´q-Hölder continuous, as a consequence of the regularizing effect of the heat semigroup. So at small time-space scales, u should essentially be constant, as a first approximation. This could suggest to try a perturbative approach in which, if one denotes by Z the solution to the equation pB t `∆qZ " ζ, with null initial condition, one looks for a distribution/function v :" u ´Z with better regularity than the expected regularity of u, so as to get a well-posed equation for v. Such an attempt is bound to fail as v needs to satisfy the same equation as u. The same trick invented by Da Prato-Debbusche in their study of the 2-dimensional stochastic quantization equation [START_REF] Da Prato | Strong solutions to the stochas-tic quantization equations[END_REF], also fails in the study of 3-dimensional scalar Φ 4 3 equation, but a local 'version' of this idea is at the heart of the theory of regularity structures, while a tilted version of that point of view is also the starting point of paracontrolled calculus. Both make sense, with different tools, of the fact that a solution should locally "look like" Z. Whereas 'usual' Taylor expansions are used in the theory of regularity structures to compare a distribution to a linear combination of some given model distributions constructed by purely probabilistic means, such as the a priori undefined product Zζ, the paracontrolled approach uses paraproducts as a means of making sense of the sentence "u looks like Z at small scales", such as given in the definition below. For readers unfamiliar with paraproducts, recall that any distribution f can be described as an infinite sum of smooth functions f i with the Fourier transform p f i of f i essentially equal to the restriction of p f on a compact annulus depending on i. A product of two distributions f and g can thus always be written formally as

f g " ÿ f i g j " ÿ iďj´2 f i g j `ÿ |i´j|ď1 f i g j `ÿ jďi´2 f i g j ": Π f pgq `Πpf, gq `Πg pf q.
(2.1)

The term Π f pgq is called the paraproduct of f and g, and the term Πpf, gq is called the resonant term. The paraproduct is always well-defined for f and g in Hölder spaces, with possibly negative indices α and β respectively, while the resonant term only makes sense if α `β ą 0. (The book [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] provides a gentle introduction to paraproducts and their use in the study of some classes of PDEs.) This result of Bony on paraproducts [START_REF] Bony | Calcul symbolique et propagation des singulariés pour les équations aux dérivées partielles non linéaires[END_REF] already offers a setting that extends Schwartz operation of multiplication of a distribution by a smooth function; it is not sufficient however for our needs, even for the (PAM) equation in dimension 2, as u is expected there to be 1 ´-Hölder and ζ is p´1 ´q-Hölder in that case. Needless to say, things are even worse in dimension 3 and for Burgers system. However, the point is that we do not want to multiply any two distributions but rather very special pairs of distributions.

A reference distribution Z in some parabolic Hölder space C α , defined later, is given here.

Definition. Let β ą 0 be given. A pair of distributions pf, gq P C α ˆCβ is said to be paracontrolled by Z if pf, gq 7 :" f ´Πg pZq P C α`β .

The distribution g is called the derivative of f with respect to Z. The following elementary remark gives credit to this choice of name. It also partly explains why we shall solve the (PAM) equation in the way we do it here -using some kind of Cole-Hopf transform. Assume α is positive, and write p2αq for a function in C 2α that may change from line to line. For a pair pf, f q paracontrolled by Z, one can write f " e Z g, for some function g in C 2α . It suffices indeed to notice that Bony's decomposition gives e ´Z f " Π e ´Z pf q `Πf `e´Z ˘`p2αq

" Π e ´Z `Πf pZq ˘`Π f `Π´e ´Z pZq ˘`p2αq " Π e ´Z f pZq ´Πe ´Z f pZq `p2αq " p2αq.

We used in the second and third equalities two elementary results on paraproducts which are well-known in the classical setting, and proved below in the more general setting of the present work.

The twist offered by this definition, as far as the multiplication problem of u by ζ is concerned, is the following. Take for Z the solution to the equation pB t `LqZ " ζ, with null initial condition; the noise ζ is thus here pα ´2q-Hölder. From purely analytic data, the product uζ is meaningful only if α `pα ´2q ą 0, that is α ą 1.

For a distribution pu, u 1 q controlled by Z, with β " α say, the formal manipulation uζ " Π u pζq `Πζ puq `Πpu, ζq

" Π u pζq `Πζ puq `Π`Π u 1 pZq, ζ ˘`Π ` 2α , ζ ": Π u pζq `Πζ puq `CpZ, u 1 , ζq `u1 ΠpZ, ζq `Π` 2α , ζ ˘,
gives a decomposition of uζ where the first two terms are always well-defined, with known regularity, and where the last term makes sense provided 2α `pα ´2q ą 0, that is α ą 2 3 . It happens that the corrector CpZ, u 1 , ζq :" Π `Πu 1 pZq, ζ ˘´u 1 ΠpZ, ζq can be proved to define an `α`α`pα´2q ˘-Hölder distribution if α ą 2 3 , although the resonant term Π `Πu 1 pZq, ζ ˘is only well-defined on its own if α ą 1. So we see that the only undefined term in the decomposition of uζ is the product u 1 ΠpZ, ζq, where the resonant term ΠpZ, ζq does not make sense so far. This is where probability comes into play, to show that one can define a random distribution ΠpZ, ζq as a limit in probability of renormalized quantities of the form ΠpZ ε , ζ ε q ´cε , where ζ ε is a regularized noise, with associated Z ε , and c ε is a deterministic function, a constant in some cases. The convergence can be proved to hold in C α`pα´2q , so the product u 1 ΠpZ, ζq eventually makes perfect sense if α `p2α ´2q ą 0, that is α ą 2 3 . This combination of analytic and probabilistic ingredients shows that one can define the product uζ, or more properly pu, u 1 q ζ, for α ą 2 3 , which is definitely beyond the scope of Bony's paradigm. Once the distribution ζ has been enhanced into a pair p ζ :" `ζ, ΠpZ, ζq ˘with good analytic properties, one can define the product pu, u 1 q p ζ as above for a generic distribution paracontrolled by Z, and reformulate a singular PDE such as the (PAM) equation in dimension 2 as a fixed point problem in some space of paracontrolled distribution, and solve it uniquely by a fixed point method. Note that the very notion of product, and hence the meaning of the equation, depends on the choice of enhancement of ζ into p ζ.

The above reasoning will not be sufficient, however, to deal with the (PAM) and multiplicative Burgers equations in dimension 3, for which α ă 1 2 , and one needs first to reformulate the equation differently to make it accessible to this first order expansion calculus. In analogy with Lyons' rough paths theory, and parallelly to the logical structure of the theory of regularity structures, one may also consider developing a higher order paracontrolled calculus where a collection of reference functions pZ 1 , .., Z k q, with increasing regularity (for example Z i of regularity iα for some α ą 0), are given, and used to give some sort of Taylor expansion of a function f P C α of the form pf, g 1 , .., g k q 7 :" f ´pΠ g 1 pZ 1 q `... `Πg k pZ k qq P C kα`β .

for some tuple pg 1 , .., g k q of C α functions with similar expansions at lower order. We shall develop this framework in a forthcoming work.

3

Geometric and functional settings

We describe in this section the geometric and functional settings in which we shall construct our space-time paraproducts in Section 4, and provide a number of tools. We shall work in a Riemannian setting under fairly general conditions; parabolic Hölder spaces are defined Section 3.3 purely in terms of the semigroup generated by L. In Section 3.4 we prove some fundamental Schauder-type regularity estimates. The cancellation properties put forward in Section 3.2 are fundamental for proving in Section 4 some continuity results for some iterated commutators and correctors.

3.1. Riemannian framework Our basic setting in this work will be a complete volume doubling measured Riemannian manifold pM, d, µq; all kernels mentioned in the sequel are with respect to the fixed measure µ. We are going to introduce in the sequel a number of tools to analyze singular partial differential equations involving a parabolic operator on R `ˆM L :" B t `L, with L built from first order differential operators pV i q i"1.. 0 on M . That is, L " ´ř 0 i"1 V 2 i , with V i satisfying the Leibniz rule

V i pf gq " f V i pgq `gV i pf q (3.1)
for all functions f, g in the domain of L. Given a tuple I " pi 1 , . . . , i k q in t1, . . . , 0 u k , we shall set |I| :" k and

V I :" V i k ¨¨¨V i 1 .
Conditions. We shall assume throughout that ' the operator L is a sectorial operator in L 2 pM q, L is injective on L 2 pM q (or the quotient space of L 2 pM q by the space of constant functions if µ is finite), it has a bounded H 8 -calculus on L 2 pM q, and ´L generates a holomorphic semigroup pe ´tL q tą0 on L 2 pM q, ' DpLq Ď DpV 2 i q, V i satisfies the Leibniz rule (3.1) on DpLq, and L is given by

L " ´ 0 ÿ i"1 V 2 i on DpLq,
' the heat semigroup is conservative, that is `e´tL ˘p1 M q " 1 M for every t ą 0, where 1 M stands for the constant function on M -or in a weak sense that Lp1 M q " 0, (a) Euclidean domains. In the particular case of the Euclidean space, all of the current work can be reformulated in terms of Fourier transform rather than in terms of the heat semigroup; which may make some reasoning a bit more familiar but does not really simplify anything. The case of a bounded domain with its Laplacian associated with Neumann boundary conditions fits our framework if the boundary is sufficiently regular. We may also consider other kind of second order operator, like L " ´divpA∇q for some smooth enough matrix-valued map satisfying the ellipticity (or accretivity) condition.

'
(b) Riemannian manifolds. Assume M is a parallelizable d-dimensional manifold with a smooth global frame field V " pV 1 , . . . , V d q. One endows M with a Riemannian structure by turning V into orthonormal frames. The above assumption on the heat kernel holds true if M has bounded geometry, that is if (i) the curvature tensor and all its covariant derivatives are bounded in the frame field V , (ii) Ricci curvature is bounded from below, (iii) and M has a positive injectivity radius; see for instance [START_REF] Coulhon | Sobolev algebras on Lie groups and Riemannian manifolds[END_REF] or [START_REF] Triebel | Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds[END_REF]. One can actually include the Laplace operator in this setting by working with its canonical lift to the orthonormal frame bundle, given by 1 2

ř d i"1 H 2 i `1 2 ř 1ďjăkďd V jk ,
where the H i are the canonical horizontal vector fields of the Levi-Civita connection, and the V jk are the canonical vertical vector fields on the orthonormal frame bundle, inherited from its SOpR d q-principal bundle structure. The bundle OM is parallelizable and satisfies the assumptions Conditions if the Riemannian base manifold M satisfies the above three conditions (i-iii).

Approximation operators and cancellation property

We introduce in this section a fundamental notion of approximation operators that will be the building blocks for the definition and study of the paraproducts, commutators and correctors, used in our analysis of singular PDEs. Some of them enjoy some kind of orthogonality, or cancellation, property quantified by condition (3.10) below. Note that we shall be working in a parabolic setting with mixed cancellation effects in time and space.

All computations below make sense for a choice of large enough integers b, 1 that will definitely be fixed at the end of Section 4.1 to ensure some continuity properties for some useful operators. Recall that generic elements of the parabolic space M " R ˆM are denoted by e " px, τ q or e 1 " py, σq, and that t stands for a scaling parameter. The following parabolic Gaussian-like kernels pG t q 0ătď1 will be used as reference kernels in this work. For 0 ă t ď 1 and σ ď τ , if dpx, yq ď 1, set G t `px, τ q, py, σq ˘:" 1 ν ´BM `px, τ q, ? t ˘¯˜1 `ρ`p x, τ q, py, σq

˘2 t ¸´ 1 ,
otherwise we set G t `px, τ q, py, σq ˘:"

1 ν ´BM `px, τ q, 1 ˘¯ˆ1 `|τ ´σ| t ˙´ 1 ˆ1 `dpx, yq 2 t ˙´ 1 exp ˆ´c dpx, yq 2 t
ḟor dpx, yq ě 1, and G t " 0 if τ ď σ. We do not emphasize the dependence of G on the positive constant c in the notation for the 'Gaussian' kernel, and we shall allow ourselves to abuse notations and write G t for two functions corresponding to two different values of that constant. This will in particular be the case in the proof of Lemma 4. We have for instance, for two scaling parameters s, t P p0, 1q, the estimate

ż M G t `e, e 1 ˘Gs `e1 , e 2 ˘νpde 1 q À G t`s `e, e 2 ˘. (3.4)
(Indeed, the space variables and the time variables are separated in the kernel G t . Then both in space and time variables, the previous inequality comes from classical estimates for convolution of functions with fast decay at infinity, such as done in [4, Lemma A.5] for example.) This somewhat unnatural definition of a Gaussian-like kernel is justified by the fact that we shall mainly be interested in local regularity matters; the definition of G t in the domain dpx, yq ě 1 ( is only technical and will allow us to obtain global estimates with weights. Presently, note that a large enough choice of constant 1 ensures that we have sup tPp0,1s

sup ePM ż M G t pe, e 1 q νpde 1 q ă 8, (3.5)
so any linear operator on a function space over M, with a kernel pointwisely bounded by some G t is bounded in L p pνq for every p P r1, 8s.

Definition. We shall denote throughout by G the set of families pP t q 0ătď1 of linear operators on M with kernels pointwisely bounded by

ˇˇK Pt pe, e 1 q ˇˇÀ G t pe, e 1 q.
The letter G is chosen for 'Gaussian'. A last bit of notation is needed before we introduce the cancellation property for a family of operators in a parabolic setting. Given a real-valued integrable function m on R, define its rescaled version as

m t p¨q :" 1 t m ´ẗ ¯;
the family pm t q 0ătď1 is uniformly bounded in L 1 pRq. We also define the "convolution" operator m ‹ associated with m via the formula

m ‹ pf qpτ q :" ż 8 0 mpτ ´σqf pσqdσ.
Note that if m has support in R `, then the operator m ‹ has a kernel supported on the same set pσ, τ q ; σ ď τ ( as our Gaussian-like kernel. Moreover, we let the reader check that if m 1 , m 2 are two L 1 -functions with m 2 supported on r0, 8q, with convolution m 1 ˚m2 , then we have

`m1 ˚m2 ˘‹ " m ‹ 1 ˝m‹ 2 .
Given an integer b ě 1, we define a special family of operators on L 2 pM q setting γ b :" pb ´1q! and

Q pbq t :" γ ´1 b ptLq b e
V I R px, yq ˇˇˇ_ ˇˇˇK t |I| 2 RV I px, yq ˇˇˇÀ 1 µ `Bpx, ? tq ˘e´c dpx,yq 2 t , (3.6)
with R standing here for

P pbq t or Q pbq t .
The parameters b and 1 will be chosen large enough, and fixed throughout the paper. See Proposition 15 and the remark after Proposition 16 for the precise choice of b and 1 .

Definition. Let an integer a P 0, 2b be given. The following collection of families of operators is called the standard collection of operators with cancellation of order a, denoted by StGC a . It is made up of all the space-time operators

´`t

|J| 2 V J ˘ptLq a´|J|´2k 2 P pcq t b m ‹ t ¯0ătď1
where k is an integer with 2k `|J| ď a, and c P 1, b , and m is any smooth function supported on

" 1 2 , 2 ‰ such that ż τ i mpτ q dτ " 0, (3.7)
for all 0 ď i ď k ´1, with the first b derivatives bounded by 1. These operators are uniformly bounded in L p pMq for every p P r1, 8s, as functions of the scaling parameter t. So a standard collection of operators Q can be seen as a bounded map Q : t Ñ Q t from p0, 1s to the set BpL p q of bounded linear operators on L p pMq. We also set StGC r0,2bs :"

ď 0ďaď2b StGC a .
The cancellation effect of such operators is quantified in Proposition 3 below; note here that it makes sense at an intuitive level to say that L a´|J|´2k 2 encodes cancellation in the space-variable of order a´|J|´2k, that V J encodes a cancellation in space of order |J| and that the moment condition (3.7) encodes a cancellation property in the time-variable of order k for the convolution operator m ‹ t . Since we are in the parabolic scaling, a cancellation of order k in time corresponds to a cancellation of order 2k in space, so that V J L a´|J|´2k 2 P pcq t b m ‹ t has a space-time cancellation property of order a. We invite the reader to check that each operator

`t |J| 2 V J ˘ptLq a´|J|´2k 2 P pcq t
b m ‹ t in the standard collection has a kernel pointwisely bounded from above by some G t . This justifies the choice of name StGC a for this space, where St stands for 'standard', G for 'Gaussian' and C for 'cancellation'. The paracontrolled analysis, that we are going to explain, is based on these specific operators. We emphasize that because of the Gaussian kernel G t and the function m, all of these operators have a support in time included in tpτ, σq, τ ě σu.

In particular, that means that we never travel backwards in time through these operators. This fact will be very important, to deal further with the weight , which will depend on time. We give one more definition before stating the cancellation property.

Definition. Given an operator Q :" V I φpLq, with |I| ě 1, defined by functional calculus from some appropriate function φ, we write Q ' for the formal dual operator

Q ' :" φpLqV I .
For I " H, and Q " φpLq, we set Q ' :" Q. For an operator Q as above we set

`Q b m ‹ ˘' :" Q ' b m ‹ .
Note that the above definition is not related to any classical notion of duality, and let us emphasize that we do not assume that L is self-adjoint in L 2 pµq. This notation is only used to indicate that an operator Q, resp. Q ' , can be composed on the right, resp. on the left, by another operator ψpLq, for a suitable function ψ, due to the functional calculus on L. In the setting of analysis on a finite dimensional torus, the operators Q pbq t are given in Fourier coordinates λ, as the multiplication operators by pt|λ| 2 q b e ´t|λ| 2 ; as this function is almost localized in an annulus |λ| " t ´1 The frequency analysis of the operators Q pbq s is not very relevant in the non-homogeneous parabolic space M. We keep however from the preceeding analysis the idea that relation (3.8) encodes some kind of orthogonality, or cancellation effect. Proposition 3. Consider Q 1 P StGC a 1 and Q 2 P StGC a 2 two standard collections with cancellation, and set a :" minpa 1 , a 2 q. Then for every s, t P p0, 1s, the composition

Q 1 s ˝Q2'
t has a kernel pointwisely bounded by

ˇˇK Q 1 s ˝Q2' t pe, e 1 q ˇˇÀ ˆts ps `tq 2 ˙a 2 G t`s pe, e 1 q. (3.9) Proof -Given Q 1 s " s j 1 2 V J 1 psLq a 1 ´j1 ´2k 1 2 P pc 1 q s b m 1‹ s and Q 2' t " ptLq a 2 ´j2 ´2k 2 2 P pc 2 q t t j 2 2 V J 2 b m 2‹
t a standard operator and the dual of another, we have

Q 1 s ˝Q2' t " s a 1 ´2k 1 2 t a 2 ´2k 2 2 V J 1 L a 1 ´j1 ´2k 1 `a2 ´j2 ´2k 2 2 P pc 1 q s P pc 2 q t V J 2 b `m1 s ˚m2 t ˘‹.
Assume, without loss of generality, that 0 ă s ď t. Then the kernel of the time-convolution operator m p1q s ˚mp2q

t is given by

K m 1 s ˚m2 t pτ ´σq " ż m 1 ˆτ ´λ s ˙m2 ˆλ ´σ t ˙dλ st .
Since m 1 has vanishing k 1 first moments, we can perform k 1 integration by parts and obtain that

ˇˇK m 1 s ˚m2 t pτ, σq ˇˇÀ ´s t ¯k1 ż B ´k1 m 1 ˆτ ´λ s ˙Bk 1 m 2 ˆλ ´σ t ˙dλ st ,
where we slightly abuse notations and write B ´k1 m 1 for the k th 1 primitive of m 1 null at 0. Then we get

ˇˇK m 1 s ˚m2 t pτ, σq ˇˇÀ ´s t ¯k1 ż ˆ1 `|τ ´λ| s ˙´ 1 `2 ˆ1 `λ ´σ t ˙´ 1 `2 dλ st À ´s t ¯k1 ˆ1 `|τ ´σ| s `t ˙´ 1 ps `tq ´1.
In the space variable, the kernel of

V J 1 L a 1 ´j1 ´2k 1 `a2 ´j2 ´2k 2 2 P pc 1 q s P pc 2 q t V J 2 is bounded above by ps `tq ´a1 `2k 1 ´a2 `2k 2 2 µ ´Bpx, ? s `tq ¯´1 exp ˆ´c dpx, yq 2 s `t ˙,
as a consequence of the property (3.6). Altogether, this gives

ˇˇK Q 1 s ˝Q2' t pe, e 1 q ˇˇÀ ´s t ¯k1 s a 1 ´2k 1 2 t a 2 ´2k 2 2 ps `tq ´a1 `2k 1 ´a2 `2k 2 2 G t`s pe, e 1 q À ´s t ¯a1 2 G t`s pe, e 1 q À ´s t ¯a 2 G t`s pe, e 1 q,
where we used that s ď t and a ď a 1 .

Definition. Let 0 ď a ď 2b be an integer. We define the subset GC a of G of families of operators with the cancellation property of order a as the set of elements Q of G with the following cancellation property. For every 0 ă s, t ď 1 and every standard family S P StGC a 1 , with a 1 P a, 2b , the operator Q t ˝S' s has a kernel pointwisely bounded by ' The families

ˇˇK Qt˝S ' s pe, e 1 q ˇˇÀ ˆst ps `tq 2 ˙a 2 G t`s pe, e 1 q. ( 3
´Qp a 2 q t b m ‹ t ¯0ătď1 and ´t |I| 2 V I P pcq t b m ‹ t ¯0ătď1 belong to GC a if |I| ě a;
' If ş τ k mpτ q dτ " 0 for all integer k " 0, ..., a´1, then we can see by integration by parts along the time-variable that `P pcq t b m ‹ t ˘0ătď1 P GC a . ' If ş τ k mpτ q dτ " 0 for all integer k " 0, ..., a 2 with a 1 `a2 " a, then the families

´Q a 1 2 t b m ‹ t ¯0ătď1 and ´t |I| 2 V I P pcq t b m ‹ t ¯0ătď1
, where |I| ě a 1 , both belong to GC a .

We see on these examples that cancellation in the parabolic setting can encode some cancellations in the space variable, the time-variable or both at a time.

We introduced above the operators Q pbq t and P pbq t acting on the base manifold M . We end this section by introducing their parabolic counterpart. Choose arbitrarily a smooth real-valued function ϕ on R, with support in " 1 2 , 2 ‰ , unit integral and such that for every integer k " 1, . . . , b, we have ż τ k ϕpτ q dτ " 0. [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF], it would be convenient to work with ϕ ˚ϕ rather than ϕ.) Note that, from its very definition, a parabolic operator

Set

Q pbq t belongs at least to GC 2 , for b ě 2. Remark that if ζ is a time-independent distribution then Q pbq t ζ " Q pbq t ζ.
Note also that due to the normalization of ϕ, then for every f P L p pRq supported on r0, 8q then

ϕ ‹ t pf q ÝÝÑ tÑ0 f in L p .
So, the operators P t tend to the identity as t goes to 0, on the set of functions f P L p pMq with time-support included in r0, 8q, whenever p P r1, 8q, and on the set of functions f P C 0 pMq with time-support included in r0, 8q. The following Calderón reproducing formula follows as a consequence. For every continuous function f P L 8 pMq with time-support in r0, 8q, we have f "

ż 1 0 Q pbq t pf q dt t `Ppbq 1 pf q. (3.11)
This formula will play a fundamental role for us. Noting that the measure dt t gives unit mass to intervals of the form " 2 ´pi`1q , 2 ´i‰ , and considering the operator Q pbq t as a kind of multiplier roughly localized at frequencies of size t ´1 2 , Calderón's formula appears as nothing else than a continuous time analogue of the Paley-Littlewood decomposition of f , with dt t in the role of the counting measure.

Parabolic Hölder spaces

We define in this section space and space-time weighted Hölder spaces, with possibly negative regularity index, and give a few basic facts about them. The setting of weighted function spaces is needed for the applications to the parabolic Anderson model and multiplicative Burgers equations on unbounded domains studied in Section 5. The weights we use were first introduced in [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF].

Let us start recalling the following well-known facts about Hölder spaces on M , and single out a good class of weights on M . A function w : M Ñ r1, 8q will be called a spatial weight if one can associate to any positive constant c 1 a positive constant c 2 such that one has wpxq e ´c1 dpx,yq ď c 2 wpyq, (3.12) for all x, y in M . Given 0 ă α ď 1, the classical metric Hölder space H α w is defined as the set of real-valued functions f on M with finite H α w -norm, defined by the formula

}f } H α ω :" › › w ´1f › › L 8 pM q `sup 0ădpx,yqď1
ˇˇf pxq ´f pyq ˇwpxq dpx, yq α ă 8.

Distributions on M were defined in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] using a very similar definition as in the end of Section 1, where their parabolic counterpart is defined.

Definition. For α P p´3, 3q and w a spatial weight, define C α w :" C α w pM q as the set of distributions on M with finite C α w -norm, defined by the formula

}f } C α ω :" › › ›w ´1e ´Lf › › › L 8 pM q `sup 0ătď1 t ´α 2 › › ›w ´1Q paq t f › › › L 8 pM q
, and equip that space with the induced norm. The latter does not depend on the integer a ą |α| 2 , and one can prove that the two spaces H α w and C α w coincide and have equivalent norms when 0 ă α ă 1 -see [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF].

These notions have parabolic counterparts which we now introduce. A spacetime weight is a function ω : M Ñ r1, 8q with ωpx, ¨q non-decreasing function of time, for every x P M , and such that there exists two constants c 1 and c 2 with ωpx, τ q e ´c1 dpx,yq ď c 2 ωpy, τ q, (3.13) for all pairs of points of M of the form `px, τ q, py, τ q ˘. The function w τ :" ωp¨, τ q is in particular a spatial weight for every time τ . For 0 ă α ď 1 and a space-time weight ω, the metric parabolic Hölder space H α ω " H α ω pMq is defined as the set of all functions on M with finite H α ω -norm, defined by the formula

}f } H α ω :" › › ω ´1f › › L 8 pMq
`sup 0ăρ `px,τq,py,σq ˘ď1; τ ěσ |f px, τ q ´f py, σq| ωpx, τ q ρ `px, τ q, py, σq ˘α .

As in the above spatial setting, one can recast this definition in a functional setting, using the parabolic standard operators. This requires the use of the following elementary result. Recall that the kernels G t depend implicitly on a constant c that may take different values with no further mention of it. We make this little abuse of notation in the proof of this statement.

Lemma 4. Let A be a linear operator on M with a kernel K A pointwisely bounded by a Gaussian kernel G t , for some t P p0, 1s. Then for every space-time weight ω, we have

› › ω ´1Af › › L 8 pMq À › › ω ´1f › › L 8 pMq .
Proof -Indeed, for every px, τ q P M we have

1 ωpx, τ q ˇˇpAf qpx, τ q ˇˇÀ ż M G t `px, τ q, py
À › › ω ´1f › › 8
, where ' we used in the second inequality the fact that the function ωpx, ¨q of time is non-decreasing, and G t is null if σ ě τ , ' the implicit constant in G t was changed in the right hand side of the third inequality, and we used the growth condition (3.13) on ω as a function of its first argument here, ' we used the uniform bound (3.5) on a Gaussian integral in the last line.

Recall that distributions were introduced in the end of Section 1.

Definition. For α P p´3, 3q and a space-time weight ω, we define the parabolic Hölder space C α ω :" C α ω pMq as the set of distributions with finite

C α ω -norm, defined by }f } C α ω :" sup QPStGC k 0ďkď2b › › ω ´1Q 1 pf q › › L 8 pMq `sup QPStGC k |α|ăkď2b sup 0ătď1 t ´α 2 › › ω ´1Q t pf q › › L 8 pMq ,
equipped with the induced norm.

The restriction α P p´3, 3q is irrelevant and will be sufficient for our purpose in this work; taking b large enough we can allow regularity of as large an order as we want. Building on Calderón's formula (3.11), one can prove as in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] that the two spaces H α ω and C α ω coincide and have equivalent norms, when 0 ă α ă 1. Proposition 5. For α P p0, 1q and every space-time weight ω, the two spaces H α ω and C α ω coincide and have equivalent norms. Proof -We first check that H α ω is continuously embedded into C α ω . So fix a function f P H α ω , then by Lemma 4 we easily deduce that sup

QPStGC k 0ďkď2b › › ω ´1Q 1 pf q › › L 8 pMq À › › ω ´1f › › L 8 pMq .
For the high frequency part, we consider t P p0, 1s and Q P StGC k with α ă k ď 2b. Then Q t has at least a cancellation of order 1, hence

Q t `f ˘peq " Q t `f ´f peq ˘peq " ż K Qt `
e, e 1 ˘`f pe 1 q ´f peq ˘νpde 1 q.

Due to the kernel support of Q t , the integrated quantity is non-vanishing (and so relevant) only for τ ě σ, with e " px, τ q and e 1 " py, σq. If ρpe, e 1 q ď 1, then by definition ˇˇf pe 1 q ´f peq ˇˇď ωpeqρpe, e 1 q α }f } H α ω and if ρpe 1 , eq ě 1, then by the property of the weight we have ˇˇf pe 1 q ´f peq ˇˇď `ωpeq `ωpe

1 q ˘› › ω ´1f › › L 8 pMq . Hence ˇˇQ t pf qpeq ˇˇÀ ωpeq "ż ρď1 G t pe, e 1 qρpe, e 1 q α νpde 1 q `żρě1 G t pe, e 1 q ˆ1 `ωpe 1 q ωpeq ˙νpde 1 q * }f } H α ω À ωpeqt α 2 }f } H α ω ,
uniformly in e P M and t P p0, 1q; this concludes the proof of the continuous embedding of H α ω into C α ω .

To prove the converse embedding, let us start by fixing a function f P C α ω . The low frequency part of f is easily bounded, using Calderón's reproducing formula

› › ω ´1f › › L 8 pMq À › › ›ω ´1P p1q 1 f › › › L 8 pMq `ż 1 0 › › ›ω ´1Q p1q t f › › › L 8 pMq dt t À }f } C α
ω , since α ą 0. Now fix e " px, τ q and e 1 " py, σq in M, with ρ :" ρpe, e 1 q ď 1 and τ ě σ. We again decompose

f " P p1q 1 f `ż 1 0 Q p1q t f dt t .
For t ă ρ 2 , we have

ˇˇQ p1q t f peq ˇˇÀ t α 2 ωpeq}f } C α ω and ˇˇQ p1q t f pe 1 q ˇˇÀ t α 2 ωpe 1 q}f } C α ω À t α 2 ωpeq}f } C α ω
where we used that the weight is increasing in time and then that dpx, yq ď ρ ď 1 with the property of the weight. So we may integrate over t ă ρ 2 and we have

ż ρ 2 0 ˇˇQ p1q t f peq ´Qp1q t f pe 1 q ˇˇd t t À ˜ż ρ 2 0 t α 2 dt t ¸ωpeq}f } C α ω À ρ α ωpeq}f } C α ω .
For the low frequency parts, Q p1q t with ρ 2 ď t ď 1 or P p1q 1 , we use that

ˇˇQ p1q t f px, τ q ´Qp1q t f px, σq ˇˇÀ |τ ´σ| 1 2 ˜sup ςPpσ,τ q ˇˇB τ Q p1q t f px, ςq ˇˇ¸˜sup ςPpσ,τ q ˇˇQ p1q t f px, ςq ˇˇÀ ρ ωpx, τ q t α´1 2 }f } C α ω
where we used that ρ ď 1 with the fact that the two collections of operators `tB τ Q p1q t ˘0ătď1 and `Qp1q t ˘0ătď1 are of type StGC 1 , that is have cancellation of order at least 1, and that the weight is non-decreasing in time. Similarly we can estimate the variation in space with the assumed finite-increment representation (3.3), where one considers a local frame field pX j q in a neighbourhood of a geodesic px, yq from x to y. This gives

ˇˇQ p1q t f px, σq ´Qp1q t f py, σq ˇˇÀ dpx, yq sup zPpx,yq ;j ˇˇX j Q p1q t f pz, ςq ˇÀ ρ ωpx, τ q t α´1 2 }f } C α ω . So we get ż 1 ρ 2 ˇˇQ p1q t f peq ´Qp1q t f pe 1 q ˇˇd t t À ρ ˆż 1 ρ 2 t α´1 2 dt t ˙ωpeq}f } C α ω À ρ α ωpeq }f } C α ω ,
because α ă 1. A similar estimate for P p1q 1 ends the proof of continuous embedding of C α ω into H α ω .

The next proposition introduces an intermediate space whose unweighted version was first introduced in the setting of paracontrolled calculus in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], and used in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF].

To fix notations, and given a space-time weight ω, we denote by

´C α 2 τ L 8 x ¯pωq " ´L8 x C α 2
τ ¯pωq the set of parabolic distributions such that

sup xPM › › f px, ¨q› › C α 2
ωpx,¨q pR `q ă 8.

Also ´L8

τ C α x ¯pωq stands for the set of parabolic distributions such that

sup τ › › f p¨, τ q › › C α
ωτ pM q ă 8. Proposition 6. Given α P p0, 2q and a space-time weight ω, set

E α ω :" ´C α 2 τ L 8 x ¯pωq X ´L8 τ C α x ¯pωq.
Then E α ω is continuously embedded into C α ω . Furthermore, if α P p0, 1q, the spaces E α ω , C α ω and H α ω are equal, with equivalent norms. Proof -We first check that E α ω is continuously embedded into C α ω , and fix for that purpose a function f P E α ω . As done in [4, Proposition 2.12], we know that for all integers k, j with k `j 2 ą α 2 and every space function g P C α pM q, we have

› › ›t j 2 V J ptLq k e ´tL g › › › L 8 pM q À t α 2 }g} C α pM q ,
for any subset of indices J with |J| " j. So consider a generic standard family

´t j 2 V j ptLq a´j 2 ´kP pcq t b m ‹ t ¯0ătď1
in StGC a , with 3 ď a ď b, and a smooth function m with vanishing first k moments. If k " 0 we have seen that we have

› › ›ω ´1 τ t j 2 V J ptLq a´j 2 P pcq t f p¨, τ q › › › L 8 pM q À t α 2 › › f pτ q › › C α ωτ
for every τ , so

› › ›ω ´1t j 2 V J ptLq a´j 2 P pcq t b m ‹ t pf q › › › L 8 pMq À t α 2 }f } L 8 τ C α x pωq
since m ‹ t is a L 8 pRq-bounded operator as a convolution with an L 1 -normalized function.

If k " 1 (or k ě 1), the same reasoning shows that we have

› › ›ωpx, ¨q´1 m ‹ t pf qpx, ¨q› › › L 8 pR `q À t α 2 › › f px, ¨q› › C α 2 ωpx,¨q pR `q,
for every x P M , since α 2 P p0, 1q, and m encodes a cancellation at order 1 in time as it has a vanishing first moment. Hence

› › ›ω ´1t j 2 V J ptLq a´j 2 P pcq t b m ‹ t pf q › › › L 8 pMq À t α 2 }f } C α 2 τ L 8 x pωq
, which concludes the proof of the embedding E α ω ãÑ C α ω . The remainder of the statement is elementary since

C α ω " H α ω is embedded in E α ω .
Before turning to the definition of an intertwined pair of parabolic paraproducts, we close this section with two other useful continuity properties involving the Hölder spaces C σ ω .

Proposition 7. Given α P p0, 1q, a space-time weight ω, some integer a ě 0 and a standard family P P StGC a , there exists a constant c depending only on the weight ω, such that ωpeq ´1ˇ`P t f ˘peq ´`P s f ˘pe 1 q ˇˇÀ `s `t `ρpe, e 1 q 2 ˘α 2 e cdpx,yq › › f

› › C α ω ,
uniformly in s, t P p0, 1s and e " px, τ q and e 1 " py, σq P M, with τ ě σ.

Proof -We explain in detail the most difficult case corresponding to P P StGC 0 , so P encodes a priori no cancellation. Then P t takes the form

P t " P pcq t b m ‹ t
for some integer c ě 1 and some smooth function m. There is no loss of generality in assuming that ş mpτ q dτ is equal to 1, as P is actually an element of StGC 1 if m has zero mean -this case is treated at the end of the proof.

In this setting, since f is bounded and continuous, we have the pointwise identity f " lim tÑ0 P t pf q.

(i) Consider first the case where ρpe, e 1 q ď 1,, with e " px, τ q and e 1 " py, σq. Decompose ωpeq ´1ˇ`P t f ˘peq ´`P s f ˘pe 1 q ˇď ωpeq ´1ˇf peq ´f pe 1 q ˇˇ`ωpeq

´1ˇ`P t f ˘peq ´f peq ˇω peq ´1ˇ`P t f ˘pe 1 q ´f pe 1 q ˇÀ ωpeq ´1ˇf peq ´f pe 1 q ˇˇ`› › ω ´1 pP t f ´f q › › L 8 pMq `› › ω ´1 pP s f ´f q › › L 8 pMq .
We have ωpeq ´1ˇf peq ´f pe 1 q ˇˇď ρpe, e 1 q α }f } H α ω À ρpe, e 1 q α }f } C α ω . For the two other terms, we use that

› › ω ´1`P t f ´f ˘› › L 8 pMq ď ż t 0 › › ›ω ´1 u B u P u f › › › L 8 pMq du u ,
and note that uB u P u " Q pcq u b m u `P pcq u b k u with kpτ q " B τ `τ mpτ q ˘, is actually the sum of two terms in StGC ě1 since it is clear for the first one and the function k has a vanishing first moment. It follows by definition of the Hölder spaces with α ă 1, that we have

› › ω ´1 pP t f ´f q › › L 8 pMq À ˆż t 0 u α 2 du u ˙}f } C α ω À t α 2 }f } C α ω .
A similar estimate holds by replacing t by s, which then concludes the proof in this case.

(ii) In the case where ρpe, e 1 q ě 1, we do not use the difference and use condition (3.13) on the weight ω to write ωpx, τ q ´1 ď ωpx, σq ´1 À ωpy, σq ´1.

and obtain as a consequence the estimate ωpeq ´1ˇ`P t f ˘peq ´`P s f ˘pe 1 q ˇˇď ωpeq ´1ˇ`P t f ˘peq ˇˇ`ωpeq

´1ˇ`P s f ˘pe 1 q ˇď › › ω ´1P t f › › L 8 pMq `ωpx, σq ´1 ˇˇ`P t f ˘pe 1 q ˇÀ › › ω ´1P t f › › L 8 pMq `ecdpx,yq › › ω ´1P s f › › L 8 pMq
, for some positive constant c. Since we know by Lemma 4 that P t and P s are bounded in L 8 pωq, we deduce that ωpeq

´1ˇ`P t f ˘peq ´`P s f ˘pe 1 q ˇˇÀ e cdpx,yq › › ω ´1f › › L 8 pMq À e cdpx,yq }f } C α ω , since C α ω Ă L 8 ω ,
given that α ą 0. The expected estimate follows from that point.

' In the easier situation where P P StGC a for some integer a ě 1, we can perform the same reasoning and use in addition the fact that lim tÑ0 P t pf q " 0, which makes the case easier since we do not have to deal with the first term f peq ´f pe 1 q.

With an analogous reasoning (indeed simpler) we may prove the following. Proposition 8. Given α P p´3, 0q, a space-time weight ω and a standard family P P StGC 0 , one has

}P t f } L 8 pMq À t α 2 }f } C α ω , uniformly in t P p0, 1s.
Proof -The proof follows the same idea as the the proof of Proposition 7. Indeed, we use the fact that since P is a standard family then

P t f " ż 1 t p´sB s P s qf ds s `P1 f.
The key point is that p´sB s P s q s can be split into a finite sum of families of StGC ě1 , which allows us to conclude as previously.

Schauder estimates

We provide in this subsection a Schauder estimate for the heat semigroup in the scale of weighted parabolic Hölder spaces. This quantitative regularization effect of the heat semigroup will be instrumental in the proof of the well-posedness of the parabolic Anderson model (PAM) and multiplicative Burgers equations studied in Section 5. Define here formally the linear resolution operator for the heat equation by the formula Rpvq τ :"

ż τ 0 e ´pτ ´σqL v σ dσ. (3.14)
We fix in this section a finite positive time horizon T and consider the space

M T :" M ˆr0, T s,
equipped with its parabolic structure. Denote by L 8 T the corresponding function space over r0, T s. We first state a Schauder estimates that was more or less proved in the unweighted case in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] -see Lemma A.9 in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] and Proposition 3.10 in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]. Proposition 9. Given β P p´2, 0q and a space-time weight ω, we have

› › Rpvq › › C β`2 ω À T }v} `L8 T C β x ˘pωq .
We shall actually prove a refinement of this continuity estimate in the specific case where ω has a special structure motivated by the study of the (PAM) and multiplicative Burgers equations done in Section 5. These special weights were first introduced by Hairer and Labbé in their study of the (PAM) equation in R 2 and R 3 , via regularity structures [START_REF] Hairer | A simple construction of the continuum parabolic Anderson model on R 2[END_REF][START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF]. Let o " o ref be the reference point in M fixed and used in the definition of S o at the end of Section 1, and set p a pxq :" `1 `dpo ref , xq ˘a, px, τ q :" e κτ e p1`τ q `1`dpo ref ,xq ˘,

for 0 ă a ă 1 and a positive constant κ. (The introduction of an extra exponential factor e κτ in our space-time weight will allow us to get around an iterative step in the forthcoming application of the fixed point theorem used to solve the (PAM) and multiplicative Burgers equations, as done in [START_REF] Hairer | A simple construction of the continuum parabolic Anderson model on R 2[END_REF][START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF].) For τ ě 0, we use the notation τ : x P M Þ Ñ px, τ q for the spatial weight. The space-time weight satisfies condition (3.13) on r0, T s M , uniformly with respect to κ ą 0. The above special weights satisfy in addition the following crucial property, already used in [START_REF] Hairer | A simple construction of the continuum parabolic Anderson model on R 2[END_REF][START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF]. We have

p a pxq px, σq À κ ´εpτ ´σq ´a´ε px, τ q, (3.16)
for every non-negative real number ε small enough, uniformly with respect to x P M, κ ą 0 and 0 ă σ ă τ ď T . The next improved Schauder-type continuity estimate shows how one can use the above inequality for the specific weights to compensate a gain on the weight by a loss of regularity.

Proposition 10. Given β P R, a P p0, 1q and ε P r0, 1q small enough such that a`e ă 1, we have the continuity estimate

› › Rpvq › ›`L 8 T C β`2p1´a´εq x ˘p q À κ ´ε› › v › ›`L 8 T C β x ˘p paq . Moreover if ´2 `2pa `εq ă β ă 0, then }Rpvq} C β`2´2a´2ε À κ ´ε› › v › ›`L 8 T C β x ˘p paq .
Proof -Let us first check the regularity in space. So consider an integer c ě |β| 2 `1 and a parameter r P p0, 1s. Then for every fixed time τ P r0, T s we have

Q pcq r `Rpvq τ ˘" ż τ 0 Q pcq r e ´pτ ´σqL v σ dσ.
By using the specific property (3.16) of the weights p a and , one has

› › › ´1 τ Q pcq r e ´pτ ´σqL v σ › › › L 8 pM q À ˆr r `τ ´σ ˙c › › ´1 τ Q pcq r`τ ´σv σ › › L 8 pM q À κ ´ε ˆr r `τ ´σ ˙c pr `τ ´σq β 2 pτ ´σq ´a´ε › › v σ › › C β pa σ
.

So by integrating and using that c is taken large enough, we see that

› › › ´1 τ Q pcq r `Rpvq τ ˘› › › L 8 pM q À κ ´ε "ż τ 0 ˆr r `τ ´σ ˙c pr `τ ´σq β 2 pτ ´σq ´a´ε dσ * › › v › ›`L 8 T C β x ˘ppa q À κ ´ετ β 2 `1´a´ε › › v › ›`L 8 T C β x ˘ppa q .
This holds uniformly in r P p0, 1s and τ P r0, T s and so one concludes the proof of the first statement with the global inequality

› › › ´1 τ Rpvq τ › › › L 8 pM q À κ ´ε "ż τ 0 pτ ´σq ´a´ε dσ * }v} `L8 T C β x ˘ppa q À κ ´ετ 1´a´ε }v} `L8 T C β x ˘ppa q .
For the second statement, we note that for 0 ď σ ă τ ď T we have

Rpvq τ ´Rpvq σ " ´e´pτ´σqL ´Id ¯Rpvq σ `ż τ σ e ´pτ ´rqL v r dr " ż τ ´σ 0 Q p1q r Rpvq σ dr r `ż τ σ e ´pτ ´rqL v r dr.
We have by the previous estimate

› › › › ´1 τ ż τ ´σ 0 Q p1q r Rpvq σ dr r › › › › L 8 pM q À κ ´ε ˆż τ ´σ 0 r β 2 `1´a´ε dr r ˙› › Rpvq σ › › C β`2´2a´2ε σ À κ ´εpτ ´σq β 2 `1´a´ε › › v › ›`L 8 T C β x ˘p q
where we used that τ ě σ for σ ď τ . Moreover, since β is negative, we also have

› › › › ´1 τ ż τ σ e ´pτ ´rqL v r dr › › › › L 8 pM q À κ ´ε ż τ σ ˆż 1 τ ´r pτ ´rq ´a´ε › › ›p ´a ´1 r Q p1q s v r › › › L 8 pM q ds s `› › › ´1 r e ´L`v r ˘› › › L 8 pM q ˙dr κ ´ε À ż τ σ ˆ› › v r › › C β pa r pτ ´rq ´a´ε ż 1 τ ´r s β 2 ds s `pτ ´rq ´a´ε › › ›e ´L`v r ˘› › › C β pa r ˙dr À κ ´εpτ ´σq β 2 `1´a´ε › › v › ›`L 8 T C β x ˘ppa q ,
where we used (3.16) and β 2 `1 ´a ´ε ą 0.

The following result comes as a consequence of the proof, combined with Lemma 4; we single it out here for future reference. Lemma 11. Let A be a linear operator on M with a kernel pointwisely bounded by G t for some t P p0, 1s. Then for every a `ε P p0, 1q, we have }A} L 8 pa pMqÑL 8 pMq À κ ´εt ´a´ε .

Schauder estimates can also be extended to spaces of positive regularity.

Proposition 12. Given β P p0, 2q, a P p0, 1q and ε P r0, 1q small enough such that a `e ă 1, we have the continuity estimate

}Rpvq} C β`2´2a´2ε À κ ´ε› › v › › C β pa .
Proof -This follows from Proposition 10. For v P C β pa Ă `L8 T C β x qp p a ˘, it is known that Lv P `L8

T C β´2

x ˘p p a q, to which Proposition 10 can be applied since β ´2 ă 0. Now use that R and L commute to deduce that LpRvq P C β´2a´2ε , hence Rv P `L8

T C β`2´2a´2ε

x ˘p q. On the other hand, B t pRvq " v ´LRv, from which follows that B t Rv P C β{2 T L 8

x , and consequently Rv P C β{2`1 T L 8

x .

The constraint β ă 2 is not relevant. Indeed, by iteration the previous Schauder estimates can be proved for an arbitrary exponent β ą 0.

4

Time-space paraproducts

We introduce in this section the machinery of paraproducts which we shall use in our analysis of the singular PDEs of Anderson (1.1) and Burgers (1.3). In the classical setting of analysis on the torus, the elementary definition of a paraproduct given in Section 2 in terms of Fourier analysis should make convincing, for those who are not familiar with this tool, the fact that Π f pgq is a kind of "modulation" of g, insofar as each mode g j of g, in its Paley-Littlewood decomposition, is modulated by a signal which oscillates at frequencies much smaller -the finite sum ř 0ďiďj´2 f i . So it makes sense to talk of a distribution/function of the form Π f pgq as a distribution/function that "locally looks like" g. This is exactly how we shall use paraproducts, as a tool that can be used to provide some kind of Taylor expansion of a distribution/function, in terms of some other 'model' distributions/functions. This will be used crucially to bypass the ill-posed character of some operations involved in the (PAM) and Burgers equations, along the line of what was written in Section 2.

Working in a geometric setting where Fourier analysis does not make sense, we shall define our paraproduct entirely in terms of the semigroup generated by the operator L " B t `L on the parabolic space. The definition of a paraproduct comes together with the definition of a resonant operator Πp¨, ¨q, tailor-made to provide the decomposition f g " Π f pgq `Πpf, gq `Πg pf q of the product operation, and with Π f pgq and Πpf, gq with good continuity properties in terms of f and g in the scales of Hölder spaces. Such a construction was already done in our previous work [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF], where the generic form of the operator L, given by its first order carré du champ operator, imposed some restrictions on the range of the method and allowed only a first order machinery to be set up. The fact that we work here with an operator L in Hörmander form will allow us to set up a higher order expansion setting. We will use this for the description of the space in which to make sense of the two singular PDEs we want to analyse. However, this a priori useful setting is in direct conflict with one of the main technical tools introduced by Gubinelli, Imkeller and Perkowski in their seminal work [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF].

The case is easier to explain on the example of the (PAM) equation. A solution to that equation is formally given as a fixed point of the map Φ : u Þ Ñ e ´¨L u 0 `Rpuζq, for which we shall need u to be a priori controlled by Z :" Rpζq, to make sense of the product uζ -more will actually be required, but let us stick to this simplified picture here; so the map Φ will eventually be defined on a space of distributions controlled by Z, such as defined in Section 2, where it will be shown to be a contraction. At a heuristic level, for a distribution pu, u 1 q controlled by Z, the product uζ will be given by a formula of the form uζ " Π u pζq `p¨¨¨q.

To analyse the term Rpuζq, and recalling that Z :" Rpζq, it is thus very tempting to write R `Πu pζq ˘" Π u `Z˘`" R, Π u ‰ pζq `p¨¨¨q and work with the commutator " R, Π u ‰ . This is what was done in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] to study the 2-dimensional (PAM) equation on the torus and more general settings; and it somehow leads to a non-natural choice of function space for the remainder f 7 of a paracontrolled distribution in a space-time setting. Unfortunately, we have little information on this commutator, except from the fact that it is a regularizing operator with a quantifiable regularizing effect -it was first proved in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] in their Fourier setting. This sole information happens to be insufficient to push the analysis of the (PAM) or Burgers equations far enough in a 3-dimensional setting. As a way out of this problem, we introduce another paraproduct r Π v p¨q, tailor-made to deal with that problem, and intertwined to Π v p¨q via R, that is R ˝Πv " r Π v ˝R; so r Π is formally the Π operator seen in a different basis r Π " R ˝Π ˝L.

We show in Section 4.1 that Π and r Π have the same analytic properties. In particular, if f P L 8 T C α x with ´2 ă α ă 0, the Schauder estimate proved in proposition 10 shows that r Π v `Rf ˘is an element of the parabolic Hölder space C α`2 . In the end, we shall be working with an ansatz for the solution space of the 3-dimensional (PAM) equation given by distributions/functions of the form u " r Π u 1 pZq `p¨¨¨q.

The introduction of semigroup methods for the definition and study of paraproducts is relatively new; we refer the reader to different recent works where such paraproducts have been used and studied [5, 7, 1, 6, 4].

Intertwined paraproducts

We introduce in this section a pair of intertwined paraproducts that will be used to analyze the a priori ill-posed terms in the right hand side of the parabolic Anderson model equation and multiplicative Burgers system in the next section. We follow here for that purpose the semigroup approach developed in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF], based on the pointwise Calderón reproducing formula f "

ż 1 0 Q pbq t f dt t `Ppbq 1 f,
where f is a bounded and continuous function. This formula says nothing else than the fact that lim tÓ0 P pbq t " Id.

(This is a direct consequence of the fact that the operator ϕ ‹ t tends to the identity operator, since ϕ has unit integral.) We can thus write formally for two continuous and bounded functions f, g

where ∆ ´1pf, gq :"

P pbq 1 ´Ppbq 1 f ¨Ppbq 1 g
stands for the "low-frequency part" of the product of f and g. This decomposition corresponds to an extension of Bony's well-known paraproduct decomposition [START_REF] Bony | Calcul symbolique et propagation des singulariés pour les équations aux dérivées partielles non linéaires[END_REF] to our setting given by a semigroup.

The integral exponent b has not been chosen so far. Choose it here even and no smaller than 6. Using iteratively the Leibniz rule for the differentiation operators V i or B τ , generically denoted D, Dpφ 1 qφ 2 " Dpφ 1 ¨φ2 q ´φ1 ¨Dpφ 2 q, we see that P pbq t ´Qpbq t f ¨Ppbq t g ¯can be decomposed as a finite sum of terms taking the form A I,J k, pf, gq :" Denote by I b the set of all such pI, J, k, q. We then have the identity

P pbq t ´t |I| 2 `kV I B k τ ¯´S pb{2q t f ¨`t |J| 2 ` V J B τ
ż 1 0 P pbq t ´Qpbq t f ¨Ppbq t g ¯dt t " ÿ I b a I,J k, ż 1 0 A I,J k, pf, gq dt t ,
for some coefficients a I,J k, . Similarly, we have

ż 1 0 Q pbq t ´Ppbq t f ¨Ppbq t g ¯dt t " ÿ I b b I,J k, ż 1 0 B I,J k, pf, gq dt t ,
with B I,J k, pf, gq of the form

B I,J k, pf, gq :" S `b 2 t ´!`t |I| 2 `kV I B k τ ˘Ppbq t f ) ¨!`t |J| 2 ` V J B τ ˘Ppbq t g )¯,
for some coefficients b I,J k, . So we have at the end the decomposition

f g " ÿ I b a I,J k, ż 1 0 ´AI,J k, pf, gq `AI,J k, pg, f q ¯dt t `ÿ I b b I,J k, ż 1 0 B I,J k, pf, gq dt t ,
which leads us to the following definition.

Definition. Given f P Ť sPp0,1q C s and g P L 8 pMq, we define the paraproduct Π pbq g pf q by the formula

Π pbq g pf q :" ż 1 0 # ÿ I b ; |I| 2 `ką b 4 a I,J k, A I,J k, pf, gq `ÿ I b ; |I| 2 `ką b 4 b I,J k, B I,J k, pf, gq + dt t ,
and the resonant term Π pbq pf, gq by the formula

ż 1 0 # ÿ I b ; |I| 2 `kď b 4 a I,J k, ´AI,J k, pf, gq `AI,J k, pg, f q ¯`ÿ I b ; |I| 2 `k" |J| 2 ` " b 4 b I,J k, B I,J k, pf, gq + dt t .
With these notations, Calderón's formula becomes f g " Π pbq g pf q `Πpbq f pgq `Πpbq pf, gq `∆´1 pf, gq with the "low-frequency part" ∆ ´1pf, gq :"

P pbq 1 ´Ppbq 1 f ¨Ppbq 1 g ¯.
If b is chosen large enough, then all of the operators involved in paraproducts and resonant term have a kernel pointwisely bounded by a kernel G t at the right scaling. Moreover, (a) the paraproduct term Π pbq g pf q is a finite linear combination of operators of the form

ż 1 0 Q 1' t ´Q2 t f ¨P1 t g ¯dt t with Q 1 , Q 2 P StGC b 4
, and P 1 P StGC r0,2bs . (b) the resonant term Π pbq pf, gq is a finite linear combination of operators of the form ż 1 0

P 1 t ´Q1 t f ¨Q2 t g ¯dt t with Q 1 , Q 2 P StGC b 4 and P 1 P StGC r0,2bs .
Note that since the operators Q ' and P 1 t are of the type Q pcq t , P pcq t or a P pcq t V I , they can easily be composed on the left with another operator Q pdq r ; this will simplify the analysis of the paraproduct and resonant terms in the parabolic Hölder spaces. Note also that Π pbq f p1q " Π pbq pf, 1q " 0, and that we have the identity Π pbq 1 pf q " f ´Ppbq 1 P pbq 1 f, as a consequence of our choice of the normalizing constant. Therefore the paraproduct with the constant function 1 is equal to the identity operator, up to the strongly regularizing operator P One can prove the following continuity estimates in exactly the same way as in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]. Note first that if ω 1 , ω 2 are two space-time weights, then ω :" ω 1 ω 2 is also a space-time weight. Proposition 13. Let ω 1 , ω 2 be two space-time weights, and set ω :" ω 1 ω 2 .

(a) For every α, β P R and every positive regularity exponent γ, we have

› › ∆ ´1pf, gq › › C γ ω À }f } C α ω 1 }g} C β ω 2
for every f P C α ω 1 and g P C β ω 2 . (b) For every α P p´3, 3q and f P C α ω 1 , we have

› › ›Π pbq g pf q › › › C α ω À › › ω ´1 2 g › › 8 }f } C α ω 1
for every g P L 8 pω ´1 2 q, and › › ›Π pbq g pf q

› › › C α`β ω À }g} C β ω 2 }f } C α ω 1
for every g P C β ω 2 with β ă 0 and α `β P p´3, 3q. (c) For every α, β P p´8, 3q with α `β ą 0, we have the continuity estimate

› › ›Π pbq pf, gq › › › C α`β ω À }f } C α ω 1 }g} C β ω 2
for every f P C α ω 1 and g P C β ω 2 . The range p´3, 3q for α (or α`β) is due to the fact that all the operators involving a cancellation used in this estimate satisfy a cancellation of order at least ν `10 ą 3. We simply write 3 in the above statement, which will be sufficient for our purpose. We proved similar regularity estimates for the paraproduct introduced in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF], with a range for α limited to p´2, 1q. This difference reflects the fact that the class of operators L considered in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF], characterized by the first order carré du champ operators, is more general than the class of Hörmander form operators considered in the present work, and allows only for a first order calculus.

These regularity estimates can be refined if one uses the specific weights and p a introduced in Subsection 3.4. Proposition 14. For every α P p´3, 3q and a, ε P p0, 1q with α ´a ´ε P p´3, 3q and f P C α pa , we have ' for every g P L 8

› › ›Π pbq g pf q › › › C α´a´ε À κ ´ε› › ´1g › › 8 }f } C α pa ;
' for every g P C β with β ă 0 and α `β ´a P p´3, 3q

› › ›Π pbq g pf q › › › C α`β´2pa`εq À κ ´ε}g} C β }f } C α pa .
The proof of this result is done along exactly the same lines as the proof of Proposition 13, using as an additional ingredient the elementary Lemma 11.

We shall use the above paraproduct in our study of the parabolic Anderson model equation, and multiplicative Burgers system, to give sense to the a priori undefined products uζ and M ζ u of a C α function u on M with a C α´2 distribution ζ on M, while 2α ´2 ď 0. Our higher order paracontrolled setting is developed for that purpose. As said above, and roughly speaking, we shall solve the Anderson equation pB τ `Lqu " uζ by finding a fixed point to the map Φpuq " e ´¨L u 0 `Rpuζq. We would like to set for that purpose a setting where the product uζ can be decomposed as a sum of the form uζ "

3 ÿ i"1 Π pbq u i pY i q `p¨¨¨q,
for some remainder term p¨¨¨q. We would then have Rpuζq "

3 ÿ i"1 R ´Πpbq u i pY i q ¯`p¨¨¨q,
which we would like to write in the form Rpuζq "

3 ÿ i"1 Π pbq u i `RpY i q ˘`p¨¨¨q,
commuting the resolution operator R with the paraproduct. The commutation is not perfect though and only holds up to a correction term involving the regularizing commutator operator " R, Π g p¨q ‰ , whose regularizing effect happens to be too limited for our purposes. This motivates us to introduce the following operator.

Definition. We define a modified paraproduct r Π pbq setting r Π pbq g pf q :" R ´Πpbq g `Lf ˘¯.

The next proposition shows that if one chooses the parameters 1 that appears in the reference kernels G t , and the exponent b that appears in the definition of the paraproduct, both large enough, then the modified paraproduct r Π pbq g p¨q has the same algebraic/analytic properties as Π pbq g p¨q. Proposition 15. If the ambient space M is bounded, then for a large enough choice of constants 1 and b, the modified paraproduct r Π g pf q is a finite linear combination of operators of the form

ż 1 0 Q 1' t ´Q2 t f ¨P1 t g ¯dt t with Q 1 P GC b 8 ´2, Q 2 P StGC b 4
and P 1 P StGC. If the space M is unbounded, then the result still holds on the parabolic space r0, T sM for every T ą 0, with implicit constants depending on T .

The operators Q 1 t that appears in the decomposition of Π g pf q are elements of StGC r0,2bs , while the operators Q 1 t that appear in the decomposition of r Π g pf q are mere elements of GC b 8 ´2.

Proof -Given the structure of Π pbq g p¨q as a sum of terms of the form

ż 1 0 Q 1' t ´Q2
t p¨q.P which is exactly the decay required in the definition of the class G.

Step 2. Assume now that k ě b 8 . We work with the above formula for the kernel K and use the cancellation effect in the time variable by integrating by parts in λ for transporting the cancellation from time to space variable. So starting from formula (4.2), the "boundary term" in the integration by parts K t |I| 2 V I P pbq t e ´pλ´σqL px, yqptB τ q k´1 ϕ t pτ ´λq is vanishing for λ Ñ 8, and equal to

K t |I| 2 V I P pbq t
px, yqptB τ q k´1 ϕ t pτ ´σq for λ " σ. The latter term satisfies estimate (4.3). So up to a term denoted by p q, bounded as desired, we see that K `px, τ q, py, σq ˘is equal to

p q `ż 8 σ K t |I| 2 `1V I P pbq t Le ´pλ´σqL px, yqp´tB λ q k´1 ϕ t pτ ´λq dλ t 2 ,
where we used that by analyticity of L in L 1 pM q B λ e ´pλ´σqL " ´Le ´pλ´σqL .

Doing k integration by parts provides an identity of the form K `px, τ q, py, σq ˘" p q `ż 8

σ K t |I| 2 `k V I P pbq t L k e ´pλ´σqL px, yqϕ t pτ ´λq dλ t 2
, where p q stands for a term with (4.3) as an upper bound. This procedure leaves us with a kernel which has an order of cancellation at least b 8 in space; we can then repeat the analysis of Step 1 to conclude.

Step 3. The proof that r Q 1 actually belongs to GC b 8 ´2 is very similar, with details largely left to the reader. The above two steps make it clear that the study of r Q 1 reduces to the study of operators with a form similar to that of the elements of StGC r0,2bs . We have provided all the details in Proposition 3 of how one can estimate the composition between such operators and obtain an extra factor encoding the cancellation property. The cancellation result on r Q 1 comes by combining the arguments of Proposition 3 with the two last steps. Let us give some details for the particular case where the family Q belongs to StGC a for some a ě b 8 ´1 and commutes with R; this covers in particular the case where Q is built in space only with the operator L with no extra V i involved. Let us then take s, t P p0, 1q and consider the kernel of the operator

r Q 1 t Q ' s . Note first that r Q 1 t Q ' s " ´Q1 t Q ' s ¯˝`t ´1R " t `s t ´Q1 t Q ' s ¯pt `sq ´1R. Since Q 1 P O b 4 , we know that Q 1 t Q '
s is an operator with a kernel with decay at scale pt `sq 1 2 with an extra factor ´st pt`sq 2 ¯b 8 . We may also consider that

Q 1 t Q ' s " ˆst pt `sq 2 ˙b 16 Q 2 t`s pt `sq ´1R
for some operator Q 2 t`s having b 8 -order of cancellation and a kernel with decay at scale ? s `t. So by what we did in the two first steps we also obtain that Q 2 t`s pt `sq ´1R has a kernel with decay at scale pt `sq 1 2 , for a large enough choice of b. (Indeed, note that Q 2 is very similar to the operators studied in the two first steps: easily analyzed as a function of the space-variable, while, as far as the time-variable is concerned, the composition of convolution preserves the main properties needed on the functions -vanishing moments.) At the end, we conclude that

r Q 1 t Q ' s " ˆst pt `sq 2 ˙b 16 ´1 Q 2 t`s
with Q 2 t`s having fast decreasing kernel at scale ps `tq 1 2 . That concludes the fact that r

Q 1 P GC b 8 ´2.
The following continuity estimate is then a direct consequence of Proposition 15, since the latter implies that we can reproduce the same argument as for the standard paraproduct in Proposition 14.

Proposition 16. For every α P p´3, 3q and a, ε P p0, 1q with α ´a ´ε P p´3, 3q and f P C α pa , we have

› › › r Π pbq g pf q › › › C α´a´ε À κ ´ε› › ´1g › › 8 }f } C α pa , for every g P L 8 .
Last, note the normalization identity

r Π 1 pf q " f ´RP pbq 1 P pbq 1 pLf q for every distribution f P S 1 o ; it reduces to r Π 1 pf q " f ´Ppbq 1 P pbq 1 pf q if f
ˇˇτ"0 " 0. (Use here the support condition on ϕ in the definition of P.) Let us also point out here the strongly regularizing effect of the two operators P We shall fix from now on the parameters b and 1 , large enough for the above result to hold true. Remark 17. The previous Proposition is very interesting because of the following observation: the time-space paraproducts r Π are defined in terms of parabolic cancellations and so do not differentiate the space and the time. Consequently, it is not clear if the time-space paraproducts r Π may be bounded on L 8 T C β for some α ă 0 (with or without weights), Such property would be very useful since the paracontrolled calculus (as shown later in the study of (PAM) for instance) needs to estimate the composition of R (the resolution of heat equation) with the paraproduct. However, following the definition of the paraproduct we have for f P L 8 T C β and g P C β RΠ pbq g pf q " r Π pbq g pRf q. So if f P L 8 T C β for some α P p´2, 0q then Schauder estimates imply that Rf P C α`2 and we may then use the boundedness on Hölder spaces of the modified paraproduct r Π pbq .

In conclusion, these new space-time paraproducts seem to be very natural for the paracontrolled calculus. They allow us to get around a commutation between the initial paraproduct and the resolution operator R (which could be a limitation for a higher order paracontrolled calculus) and fits exactly in what paracontrolled calculus requires to solve singular PDEs, modelled on the heat equation.

Commutators and correctors

We state and prove in this section two continuity estimates that will be useful in our study of the 3-dimensional parabolic Anderson model equation and Burgers system in Section 5.

Definition 18. Let us introduce the following a priori unbounded trilinear operators on S 1 o . Set Rpf, g, uq :" Π pbq u ´Πpbq g pf q ¯´Π pbq ug pf q, and define the corrector Cpf, g, uq :" Π pbq ´r Π pbq g pf q, u ¯´g Π pbq pf, uq.

This corrector was introduced by Gubinelli, Imkeller and Perkowski in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] under the name of commutator. We prove in the remainder of this section that these operators have good continuity properties in some weighted parabolic Hölder spaces.

Proposition 19. Given some space-time weights ω 1 , ω 2 , ω 3 , set ω :" ω 1 ω 2 ω 3 . Let α, β, γ be Hölder regularity exponents with α P p´3, 3q, β P p0, 1q and γ P p´3, 0s. Then if δ :" α `β `γ P p´3, 3q with α `β ă 3, we have

› › Rpf, g, uq › › C δ ω À }f } C α ω 1 }g} C β ω 2 }u} C γ ω 3 , (4.4)
for every f P C α ω 1 , g P C β ω 2 and u P C γ ω 3 ; so the modified commutator defines a trilinear continuous map from

C α ω 1 ˆCβ ω 2 ˆCγ ω 3 to C δ ω .
Proof -Recall that Π pbq g is given by a finite sum of operators of the form A 1 g p¨q :"

ż 1 0 Q 1' t ´Q2 t p¨q P 1 t pgq ¯dt t ,
where Q 1 , Q 2 belong at least to StGC 3 . We describe similarly Π pbq u as a finite sum of operators of the form

A 2 u p¨q :" ż 1 0 Q 3' t ´Q4 t p¨qP 2 t puq ¯dt t .
Thus, we need to study a generic modified commutator

A 2 u `A1
g pf q ˘´A 2 ug pf q, and introduce for that purpose the intermediate quantity Epf, g, uq :"

ż 1 0 Q 3' s ´Q4 s pf q ¨P1 s pgq ¨P2 s puq ¯ds s .
Note here that due to the normalization Π 1 » Id, up to some strongly regularizing operator, there is no loss of generality in assuming that

ż 1 0 Q 1' t Q 2 t dt t " ż 1 0 Q 3' t Q 4 t dt t " Id. (4.5)
Step 1. Study of A 2 u `A1 g pf q ˘´Epf, g, uq. We shall use a family Q in StGC a , for some a ą |δ|, to control the Hölder norm of that quantity. By definition, and using the normalization (4.5), the quantity Q r ´A2 u `A1

g pf q ˘´Epf, g, uq ¯is, for every r P p0, 1q, equal to

ż 1 0 ż 1 0 Q r Q 3' s ! Q 4 s Q 1' t ´Q2 t pf qP 1 t pgq ¯¨P 2 s puq ) ds dt st ´ż 1 0 Q r Q 3' s ´Q4 s pf q ¨P1 s pgq ¨P2 s puq ¯ds s " ż 1 0 ż 1 0 Q r Q 3' s ! Q 4 s Q 1' t ´Q2 t pf q `P1 t pgq ´P1 s pgq ˘¯¨P 2 s puq ) dsdt st ,
where in the last line the variable of P 1 s pgq is the one of Q 3' s , and so it is frozen through the action of Q 4 s Q 1' t . Then using that g P C β with β P p0, 1q, we know by Proposition 7 that we have, for τ ě σ, ω 2 px, τ q ´1ˇ`P 1 s g ˘px, τ q ´`P 1 t g ˘py, σq ˇˇÀ ´s `t `ρ`p x, τ q, py, σq

˘2¯β 2 e cdpx,yq }g} C β ω 2
.

Note that it follows from equation (3.4) that the kernel of Q 4 s Q 1' t is pointwise bounded by G t`s , and allowing different constants in the definition of the Gaussian kernel G, we have G t`s `px, τ q, py, σq ˘`s `t `dpx, yq 2 ˘β 2 e cdpx,yq À ps `tq β 2 G t`s `px, τ q, py, σq ˘. (4.6) So using Lemma 4 and the cancellation property of the operators Q at an order no less than a (resp. 3) for Q (resp. the other collections Q i ), we deduce that

› › ›ω ´1Q r ´A2 u `A1 g pf q ˘´Epf, g, uq ¯› › › 8 À }f } C α ω 1 }g} C β ω 2 }u} C γ ω 3 ż 1 0 ż 1 0 ˆsr ps `rq 2 ˙a 2 ˆst ps `tq 2 ˙3 2 t α 2 ps `tq β 2 s γ 2 ds dt st ,
where we used that γ is negative to control P 2 s puq. The integral over t P p0, 1q can be computed since α ą ´3 and α `β ă 3, and we have

› › ›ω ´1Q r ´A2 u `A1
g pf q ˘´Epf, g, uq

¯› › › 8 À }f } C α ω 1 }g} C β ω 2 }u} C γ ω 3 ż 1 0 ż 1 0 ˆsr ps `rq 2 ˙a 2 s δ 2 ds s À }f } C α ω 1 }g} C β ω 2 }u} C γ ω 3 r δ 2 ,
uniformly in r P p0, 1q because |a| ą δ. That concludes the estimate for the high frequency part. We repeat the same reasoning for the low-frequency part by replacing Q r with Q 1 and conclude that

› › ›A 2 u `A1 g pf q ˘´Epf, g, uq › › › C δ ω À }f } C α ω 1 }g} C β ω 2 }u} C γ ω 3 .
Step 2. Study of A 2 ug ´Epf, g, uq. This term is simpler than that of Step 1 and can be treated similarly. Note that Q r ´A1 g `A2

u pf q ˘´Epf, g, uq ¯is equal, for every r P p0, 1q, to

ż 1 0 Q r Q 3' s ´Q4 s pf qP 2 s pugq ¯ds s ´ż 1 0 Q r Q 3' s ´Q4 s pf q ¨P1 s pgq ¨P2 s puq ¯ds s " ż 1 0 Q r Q 3' s ´Q4 s pf q `P2 s pugq ´P1 s pgq ¨P2 s puq ˘¯ds s .
Now note that since g P C β with β P p0, 1q, we know by Proposition 7, for τ ě σ, ω 2 px, τ q ´1ˇg px, τ q ´`P 1 s g ˘py, σq ˇÀ ω 2 px, τ q ´1ˇg px, τ q ´gpy, σq ˇˇ`ω 2 px, τ q ´1ˇg py, σq ´`P 1 t g ˘py, σq ˇÀ ´s `t `ρ`p x, τ q, py, σq

˘2¯β 2 e cdpx,yq }g} C β ω 2
.

Then the same proof as in Step 1 can be repeated.

As far as the continuity properties of the corrector Cpf, g, uq " Π pbq ´r Π pbq g pf q, u ¯´gΠ pbq pf, uq are concerned, the next result was proved in an unweighted setting in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]Proposition 3.6] for a space version of the paraproduct Π; elementary changes in the proof give the following space-time weighted counterpart.

Proposition 20. Given space-time weights ω 1 , ω 2 , ω 3 , set ω :" ω 1 ω 2 ω 3 . Let α, β, γ be Hölder regularity exponents with α P p´3, 3q, β P p0, 1q and γ P p´8, 3s. Set δ :" pα `βq ^3 `γ. If

0 ă α `β `γ ă 1 and α `γ ă 0 then the corrector C is a continuous trilinear map from C α ω 1 ˆCβ ω 2 ˆCγ ω 3 to C δ ω .
5

Anderson and Burgers equations in a 3-dimensional background

We are now ready to start our study of the parabolic Anderson model equation pB t `Lqu " uζ and the multiplicative Burgers system pB t `Lqu `pu ¨V qu " M ζ u in a 3-dimensional manifold, using the above tools. Here for Burgers system, we consider a collection of three operators V :" pV 1 , V 2 , V 3 q ( 0 " 3). We shall study the (PAM) equation in a possibly unbounded manifold, using weighted Hölder spaces, while we shall be working in a bounded setting for the Burgers equation, as its quadratic term obviously does not preserve any reasonable weighted space.

Getting solutions for the (PAM) equation

Let us take the freedom to assume for the moment that the noise ζ in the above equations is not necessarily as irregular as white noise. We shall fix from now on a finite positive time horizon T . Recall the elementary result on paracontrolled distributions u with derivative u stated in section 2; such distributions are of the form u " e ´Z v 1 , for some more regular factor v 1 . This is indeed what happens formally for any solution fo the (PAM) equation, since uζ " Π u pζq, up to some smoother term, and R ´Πu pζq ¯" Π u pRζq, up to some more regular remainder. Elaborating formally on this remark leads to the introduction of the following distributions, and the choice of representation for a solution of the (PAM) equation adopted below in proposition 21.

Due to the definition of Y i 's, we have some telescoping property:

L r Z ´ 0 ÿ i"1 V i pZq 2 " LRpY 2 `Y3 q ´ 0 ÿ i"1 ´Vi pZ 1 q `Vi pZ 2 q `Vi pZ 3 q ¯2 " Y 2 `Y3 ´ 0 ÿ i"1 3 ÿ j,k"1 V i pZ j qV i pZ k q " W 1 `W2 ´ 0 ÿ i"1 3 ÿ j,k"2 j`kě5 V i pZ j qV i pZ k q.
Since we assume that Z j P C jα pa , it follows that V i pZ j q P L 8 T C jα´1 pa and V i pZ k q P L 8

T C kα´1 pa . Given that j `k ě 5 and α P p1{3, 1{2q, at least one of the two numbers pjα ´1q and pkα ´1q is positive and the other not smaller than 2α ´1. So

U :" L r Z ´ 0 ÿ i"1 V i pZq 2 P W 1 `W2 `L8 T C 2α´1 , (5.3)
and the result follows.

Instead of solving directly (PAM) through paracontrolled calculus, we are going to solve (5.2). So the space of (PAM)-enhanced distributions p ζ for the (PAM) equation is here simply the product space

C α´2 pa ˆ3 ź k"2 L 8 T C α´p5´kq{2 pa ˆ`L 8 T C 2α´1 pa ˘b 0 `2 .

The paracontrolled approach

The study of singular PDEs, such as the Anderson and Burgers equations or (5.2), from a paracontrolled point of view is a four step process. Let us sketch it for equation (5.2) as an example.

(a) Set yourself an ansatz for the solution space, in the form of a Banach space of paracontrolled distributions/functions.

Given 1 3 ă β ă α ă 1 2 , we choose here to work with functions v paracontrolled by the collection

! R `Vi pZ 1 q ˘¯ 0 i"1
, that is with v of the form v "

0 ÿ i"1 r Π pbq v i ´RpV i Z 1 q ¯`v 7 (5.4)
for a remainder v 7 P C 1`α`β p ´a and v i P C β . We refer the reader to Subsection 3.4 for the introduction of weights p a and . Note that we use the r Π paraproduct and not the Π paraproduct. We turn the solution space S α,β `p ζ ˘:" ! pv; v 1 , . . . , v 0 ; v 7 q satisfying the above relations ) into a Banach space by defining its norm as This is where we use the continuity properties of the corrector and different paraproducts. In the specific situations of equation (5.2), given pv; v 1 , . . . , v 0 ; v 7 q in the solution space S α,β `p ζ ˘, one sets

› › pv; v 1 , . . . , v 0 ; v 7 q › › α,β :" › › v 7 › › C 1`α`β p ´a ` 0 ÿ i"1 › › v i › › C β . ( 5
y " L ´´U v `2 0 ÿ i"1 V i pZqV i pvq ānd
shows that it has a decomposition py; y 1 , . . . , y 0 ; y 7 q of the form (5.4). This is where we need all th extra information contained in p ζ. Then, given an initial data v 0 P C 1`α`β p ´a , the application γ : pτ, xq Þ Ñ e ´τ L pv 0 qpxq, belongs to C 1`α`β p ´a and satisfies

Lγ " 0, γ τ "0 " v 0 .

We define a continuous map Φ from the solution space S α,β `p ζ ˘to itself setting Φ :" pv; v 1 , . . . , v 0 ; v 7 q Þ Ñ py `γ; y 1 , . . . , y 0 ; y 7 `γq.

(c) Prove that Φ is a contraction of the solution space.

Recall a parameter κ ą 1 appears in the definition of the special weight . We shall see below that the function y 7 satisfies the estimate

› › y 7 › › C 1`α`β p ´a ď κ ´ε› › pv; v 1 , . . . , v 0 ; v 7 q › › α,β
, for some ε ą 0, and that py 1 , . . . , y 0 q depends only on v and not on v 1 , . . . , v 0 and v 7 . These facts provide a quick proof that Φ ˝Φ is a contraction of the solution space S α,β `p ζ ˘. Indeed, given pv; v 1 , . . . , v 0 ; v 7 q in S α,β `p ζ ˘, set `z `γ; z 1 , . . . , z 0 ; z 7 `γ˘: " Φ ˝2pv; v 1 , . . . , v 0 ; v 7 q P S α,β `p ζ ˘.

We know that

› › z 7 › › C 1`α`β p ´a ď κ ´ε› › py `γ; y 1 , . . . , y 0 ; y 7 `γq › › α,β À κ ´ε› › pv; v 1 , . . . , v 0 ; v 7 q › › α,β
. The paracontrolled structure (5.4) of y and Schauder estimates also give

}y} C 1`β p ´a À › › y 7 › › C 1`β p ´a ` 0 ÿ j"1 κ ´ε}y i } C 1`α À κ ´ε› › Φpv, v 1 , .., v 0 , v 7 q › › α,β À κ ´ε› › pv, v 1 , .., v 0 , v 7 q › › α,β .
So we conclude that y is controlled with a small bound. Since pz 1 , .., z 0 q will be given by y, we then obtain that pz; z 1 , . . . , z 0 ; z 7 q will be controlled in S α,β `p ζ ˘with small norms (relatively to the initial pv; v 1 , . . . , v 0 ; v 7 q) and so that Φ ˝Φ is indeed a small perturbation of the map pv; v 1 , . . . , v 0 ; v 7 q Þ Ñ pγ; 0, . . . , 0; γq. Then it is standard that if κ ´ε is small enough, that is κ is large enough, then we can apply the fixed point theorem to Φ ˝2 and conclude that it has a unique fixed point in the solution space S α,β `p ζ ˘; the same concolusion for Φ follows as a consequence.

(d) Renormalisation step.

The defining relations for Z i found in step (b) actually involve some terms that cannot be defined by purely analytical means when ζ is a white noise, but which make perfect sense for a regularized version ζ ε of ζ. Their proper definition requires a renormalisation procedure that consists in defining them as limits in probability, in some parabolic Hölder spaces, of suitably modified versions of their regularized versions (with ζ ε in place of ζ), which essentially amounts in the present setting to adding to them some deterministic functions or constants. (This may be trickier in other situations as the theory of regularity structures makes it clear.) Given the inductive construction of the Z i , this renormalisation step also needs to be done inductively. At ε fixed, this addition of deterministic quantities in the defining relations for Z i defines another map Φ ε from the solution space to itself that can eventually be equivalent to consider a renormalised equation with noise ζ ε , with ε-dependent terms added in the equation, when compared to the initial equation. Write u ε for its solution. In the end, we get, from the continuity of fixed points of parameter-dependent uniformly contracting maps, a statement of the form: Let Φ stand for the map constructed by taking as reference distributions/functions Z i the limits, in probability, of their renormalised versions. Then the functions u ε converge in probability to the solution u of the fixed point problem of the map Φ.

We shall do here the first three steps of the analysis for both the Anderson and Burgers equations, leaving the probabilistic work needed to complete the renormalisation step to another work -the present article is already big enough not to overload it with 20 or 30 pages more, but will instead give in section 6 some hints as to what is going on.

The deterministic PAM equation

Given what was said in the preceding section, the main work for solving the (PAM) equation consists in proving the following result. Given an enhanced distribution p ζ, one can extend the product operation

v P C 8 c pM q Þ Ñ ´U v `2 0 ÿ i"1 V i pZqV i pvq to the space S α,β `p ζ ˘into an operation p v Þ Ñ ´p U p v `2 ř 0 i"1 V i p p
ZqV i pp vq, so that setting

y :" R " ´p U p v `2 0 ÿ i"1 V i p p ZqV i pp vq ı ,
and y i :" 2vV i pZ 3 q `2V i pvq, there exists y 7 P C 1`α`β p ´a such `y; y 1 , . . . , y 0 ; y 7 ˘is an element of the solution space S α,β `p ζ ˘, and › › › `y; y 1 , . . . , y 0 ;

y 7 ˘› › › α,β À › › › `v; v 1 , . . . , v 0 ; v 7 ˘› › › α,β › › y 7 › › C 1`α`β p ´a À κ ´ε › › › `v; v 1 , . . . , v 0 ; v 7 ˘› › › α,β .
(5.6)

Proof -First, we note that since v satisfies the ansatz (5.4) and 2a ă α ´β, we know from Schauder estimates that

v P C 1`α pa X C 1`β p ´a .
Step 1. We first consider the part U v where we recall that U " W 1 `W2 `W3 for some

W 3 P L 8 T C 2α´1 pa
. Using the paraproduct algorithm, one gets

W 3 v " Π pbq W 3
pvq `Πpbq v pW 3 q `Πpbq pv, W 3 q. By the boundedness of paraproducts, Proposition 13, and Schauder estimates, Proposition 12, we get

Π pbq W 3 pvq P C 2α`β so RΠ pbq W 3 pvq P C 2`3β p ´a Ă C 1`α`β p ´a with › › ›RΠ pbq W 3 pvq › › › C 1`α`β p ´a À κ ´ε}v} C 1`β p ´a
since 2ε `2a ă α ´β and α ă 1. For the resonant part, a similar reasoning with Proposition 13 yields Π pbq pv, W 3 q P C 2α`β so RΠ pbq pv, W q P C 2`3β

p ´a with › › ›RΠ pbq pv, W q › › › C 1`α`β
p ´a À κ ´ε}v} C 1`β p ´a . For the second paraproduct, we use the modified paraproduct and its boundedness, Proposition 16, to have RΠ pbq v pW 3 q " r Π pbq v pRW 3 q, hence since

RW 3 P C 1`2α pa we have RΠ pbq v pW 3 q P C 1`α`β p ´a with › › ›RΠ pbq v pW 3 q › › › C 1`α`β p ´a À κ ´ε}v} C 1`α pa ,
since 4pa `εq ď α ´β. So we have R `W3 v ˘P C 1`α`β p ´a , with an acceptable bound. The term W 2 is an element of L 8 T C 2α´1 pa , so using the same reasoning yields that R `W2 v ˘P C 1`α`β p ´a with an acceptable bound. The term W 1 is an element of L 8 T C α´1 pa , so it is really more singular than the two previous terms. Recall its definition

W 1 " ´2 0 ÿ i"1 V i pZ 1 qV i pZ 3 q with V i pZ 3 q in C α pa , since Z 3 is an element of C 1`α pa . So W 1 is in C α´1 pa ,
› › › C 1`α`β `› › ›RΠ pbq pW 1 , vq › › › C 1`α`β À κ ´ε}v} C 1`α pa .
It remains us to study the paraproduct term Π pbq v pW 1 q " I `II `III, with

I :" ´2 0 ÿ i"1 Π pbq v ´Πpbq V i pZ 3 q `Vi pZ 1 q ˘ĪI :" ´2 0 ÿ i"1 Π pbq v ´Πpbq `Vi pZ 3 q, V i pZ 1 q ˘ĪII :" ´2 0 ÿ i"1 Π pbq v ´Πpbq V i pZ 1 q `Vi pZ 3 q ˘¯.
By easy considerations on paraproducts, the third term III belongs to C 2α´1 pa and RpIIIq P C 1`α`β p ´a , with acceptable bounds, because Z 3 is an element of C 1`α pa . Moreover, since we assume that W 1 " ř 0 i"1 Π pbq `Vi pZ 3 q, V i pZ 1 q ˘is an element of

L 8 T C 2α´1 pa
, the second term II also satisfies RpIIq P C 1`α`β p ´a . Using the regularity of v P C 1`α pa Ă L 8 pa and Proposition 19 for the commutation property, we deduce that

I P ´2 0 ÿ i"1 Π pbq vV i pZ 3 q rV i pZ 1 qs `C4α´2 p 3a
and consequently

RpIq P ´2 0 ÿ i"1 r Π pbq vV i pZ 3 q rRV i pZ 1 qs `C1`α`β p ´a ,
with an acceptable bound for the remainder since 8pa `εq `1 ă 3α ´β.

At the end, we have obtained that

RpU vq P ! 2 0 ÿ i"1 r Π pbq vV i pZ 3 q rRV i pZ 1 qs `C1`α`β p ´a ) ,
which proves that RpU vq is paracontrolled by the collection `RV i pZ 1 q ˘i and the remainder has a bound controlled by κ ´ε.

Step 2. Let now focus on the term ř 0 i"1 V i pZqV i pvq. Fix an index i and write

V i pZqV i pvq " Π pbq V i pvq `Vi pZq ˘`Π pbq V i pZq `Vi pvq ˘`Π pbq `Vi pZq, V i pvq ˘.
The second term is of regularity 2α ´1 and using the modified paraproduct, Schauder estimate and the fact that we have v P C 1`α pa , we see that

R " Π pbq V i pZq pV i pvqq ı " r Π pbq V i pZq pRV i pvqq P C 1`α`β p ´a .
We proceed as follows to study the resonant part. First, since α ą 1{3, we have

Π pbq pV i pZq, V i pvqq P ! 0 ÿ j"1 Π pbq ´Vi pZ 1 q, V i r Π pbq v j " RpV j Z 1 q ‰ ¯`C 3α´1 p 2a
) .

Theorem 24. Let 1 3 ă α ă 1 2 be given. Choose β ă α, the positive parameter a in the weight p a , and ε ą 0, such that 2α `β ą 1 and 8pa `εq ď α ´β.

Then, one can choose a positive parameter κ, in the definition of the special weight , large enough to have the following conclusion. Given v 0 P C 1`α`β 0 p ´a , the map Φ has a unique fixed point pv, v 1 , .., v 0 , v 7 q in S v 0 α,β `p ζ ˘; it depends continuously on the enhanced distribution p ζ, and satisfies the identity v i " 2vV i pZ 3 q `2V i pvq for i " 1, .., 0 . This distribution is a solution of

Lpvq " ´U v `2 0 ÿ i"1 V i p p
ZqV i pp vq with v |τ "0 " v 0 . So that, as we have observed u " e Z v is a solution of (PAM) with initial data v 0 .

If the ambient space M is bounded, then we do not have to take care of the infinity in the space variable, and one can prove a global (in time) result by considering the weight px, τ q " e κτ with a large enough parameter κ.

The stochastic PAM equation

Recall the time-independent white noise over the measure space pM, µq is the centered Gaussian process ξ indexed by L 2 pM q, with covariance

E " ξpf q 2 ‰ " ż f 2 pxq µpdxq.
It can be proved [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] to have a modification with values in the spatial Hölder space C ´ν 2 ´ε pa , for all positive constants ε and a, where ν is the Ahlfors dimension of pM, d, µq -its dimension in our Riemannian setting. We take ν " 3 here. We still denote this modification by the same letter ξ. As made clear in the introduction to this section, the study of the stochastic singular PDE of Anderson pB τ `Lqu " uξ can be done in the present setting. This requires a renormalisation step needed to show that the quantities Ξ " Y j , W j , ... can be defined as elements of suitable functional spaces, as limits in probability of distributions of the form Ξ ε ´λε , where Ξ ε is given by formula (5.1) with ζ " ξ ε :" e ´εL ξ, the regularized version of the noise via the semigroup, and λ ε are some deterministic functions. This renormalisation step is not done here not to overload the present article. Section 6 gives a flavour of what is involved in this process in the present setting. Note that the two dimensional setting was studied in depth in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] -spatial paraproducts were used there instead of space-time paraproducts. Here is the statement which can be proved in the present 3-dimensional setting.

Theorem 25 (Renormalization). Consider ξ a white noise on M and for ε ą 0, denote by ξ ε :" e ´εL ξ its regularized version. Denote by Ξ ε the distributions corresponding to Ξ " Y j , W j , ... that one obtains by replacing ζ by ξ ε . Then for any α ă 1{2

' the distributions Y ε 1 , Y ε 3 , W j,ε 2

The multiplicative Burgers equation

We study in this last section the multiplicative Burgers system pB t `Lqu `pu ¨V q u " M ζ u in the same 3-dimensional setting as before with three operators V :" pV 1 , V 2 , V 3 q forming an elliptic system. Here the solution u " pu 1 , u 2 , u 3 q is a function with R 3 -values and pu ¨V q u has also 3 coordinates with by definition rpu ¨V q us j :"

3 ÿ i"1 u i V i pu j q.
To study this equation, we have to make the extra assumption that the ambient space M is bounded. Indeed the boundeness of the ambient space is crucial here, as using weighted Hölder spaces, it would not be clear how to preserve the growth at infinity dictated by the weight when dealing with the quadratic nonlinearity. In such a bounded framework, we do not need to use spatial weights and consider instead the unweighted Hölder spaces C γ -or rather we work for convenience with a weight in time px, τ q :" e κτ .

(5.9) We stick to the notations of the previous section. The study of Burgers' system requires a larger space of enhanced distributions than the study of the 3-dimensional (PAM) equation; the additional components include those quantities that need to be renormalised to make sense of the term pu ¨V q u, when ζ is an element of C α´2 , such as space white noise.

We first rewrite Burgers system in a more convenient form, as we did for the (PAM) equation. For each cooordinate exponent j " 1, 2, 3, we define Z j α , W j β from ζ j as above. Then consider a function u : M Þ Ñ R 3 defined by u j " e Z j v j with v : M Þ Ñ R 3 . Then observe that u is formally a solution of 3-dimensional Burgers system on M if and only if v is the solution of the system

Lv j " ´U j v `2 3 ÿ i"1 V i pZ j qV i pv j q ´3 ÿ i"1 v i e Z i `Vi v j `vj V i Z j ˘. (5.10)
To treat the nonlinearity, we need to introduce another a priori given element in the enhancement of the noise ζ. Define a 3 ˆ3 matrix Θ setting formally

Θ ij " Π pbq `Zi 1 , V i Z j 1 ˘. Definition 29. Given 1 3 ă β ă α ă 1 2 and a time-independent distribution ζ P C α´2 , a (3d Burgers)-enhancement of ζ is a tuple p ζ :" ´ζ, Y 2 , Y 3 , Ă W 1 , W 2 , pW j 2 q j , Θ ¯, with Y k P L 8 T C α´p5´kq{2 , Ă W 1 , W 2 , W j k P L 8 T C 2α´1 and Θ P `L8 T C 2α´1 ˘.
So the space of enhanced distributions p ζ for the multiplicative Burgers system is the product space

C α´2 pa ˆ3 ź k"2 L 8 T C α´p5´kq{2 pa ˆ`L 8 T C 2α´1 pa ˘5 ˆL8 T C 2α´1 ;
with slightly abuse notations here as the first factors in the above product refer to R 3 -valued distributions/functions, while thelast factr has its values in R 9 . Given such an enhanced distribution p ζ, we define the Banach solution space S α,β p p ζq as in section 5.1.1, replacing the weight p a by the constant 1. Recall the constant κ ą 1 appearsin the time weight (5.9). Theorem 30. Let 1 3 ă α ă 1 2 be given. Choose β ă α, the positive parameter a in the weight p a , and ε ą 0, such that 2α `β ą 1 and 6ε ď α ´β.

Given an enhanced distribution p ζ and p v P S α,β `p ζ ˘, the nonlinear term rN pvqs j :"

3 ÿ i"1 v i e Z i `Bi v j `vj V i Z j ȋs
well-defined and there exists some

z 7 P C 1`α`β with `RrN pvqs, . . . ; z 7 ˘P S α,β `p ζ ȃnd › › › `RrN pvqs, . . . ; z 7 ˘› › › α,β À κ ´ε › › › `v, v 1 , .., v 3 , v 7 ˘› › › α,β .
(5.11)

Proof -We fix a coordinate j " 1, 2, 3 and have to study rN pvqs j "

3 ÿ i"1 v i e Z i `Vi v j `vj V i Z j ˘.
The first quantity is sufficiently regular by itself, and we have Z i P C α , v P C 1`α so for every i " 1, 2, 3 then

v i e Z i V i v j P C α hence R " v i e Z i V i v j ‰ P C 1`α`
β with an acceptable norm (controlled by κ ´ε).

Let us now focus on the second part v i e Z i v j V i Z j . Since v P C 1`α , it is very regular and the problem only relies on defining the product e Z i V i Z j . We first decompose using paraproducts

e Z i V i Z j " Π pbq e Z i pV i Z j q `Πpbq V i Z j pe Z i q `Πpbq pe Z i , V i Z j q.
The second term B ij is bounded in C 2α´1 . The last resonant part is studied through a paralinearization formula (see [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] and references there for example)

e Z i " Π pbq e Z i pZ i q ` 2α which implies with α ą 1{3 A ij :" Π pbq pe Z i , V i Z j q " Π pbq ´Πpbq e Z i pZ i q, V i Z j ¯` 3α ´1 " e Z i Π pbq pZ i , V i Z j ¯` 3α ´1 " e Z i Π pbq pZ i 1 , V i Z j 1 ¯` 3α ´1 " e Z i Θ ij ` 3α ´1 ,
where we have used the commutator estimates. Since we assume that Θ is supposed to be well-defined L 8 T C 2α´1 , we conclude to

A ij P L 8 T C 2α´1 . So we observe that Π pbq A ij pv i v j q `Πpbq pA ij , v i v j q
`vε , u ε , u ε 1 , u ε 2 ˘still satisfies the ansatz. We then complete the proof of Theorem 2, as done for Theorem 1.

6

A glimpse at renormalisation matters We provide in this section a sketch of the proof of the renormalisation step described in Theorem 25, needed to give a complete proof of the well-posedness theorem, Theorem 26; full details will be given in a forthcoming work. Hairer uncovered in [START_REF] Hairer | A theory of regularity structures[END_REF] the rich algebraic setting in which renormalisation takes place within his theory of regularity structures. It provides in particular a clear understanding of which counterterms need to/can be added in the dynamics driven by a regularized noise to get a converging limit when the regularizing parameter tends to 0. Basic renormalisation consists in removing from diverging random terms their expectation. While this operation is sufficient in a number of cases, such as the 2 and 3-dimensional (PAM) equations, or the 1-dimensional stochastic heat equation [START_REF] Hairer | A theory of regularity structures[END_REF][START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF], more elaborate renormalisation procedures are needed in other examples, such as the (KPZ) or Φ 4 3 equations. Hopefully, the kind of renormalisation needed here for the study of the 3-dimensional (PAM) and Burgers equations, is basic, in accordance with the work of Hairer and Labbé [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] on the (PAM) equation in R 3 . We describe it below in elementary terms and refer the reader to a forthcoming work for full details.

Three kinds of terms Ξ 2 , Ξ 3 , Ξ 4 need to be renormalised, with Ξ i formally i-linear in the noise ξ. The terms Y 2 and W j 2 are bilinear, the term Y 3 is 3-linear and the terms W 1 , W 2 are 4-linear. The term Y 2 is relatively easy to analyse, in the line of what we did in our previous work [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]; some more details on the terms W j 2 will be given, and we shall study a toy model for the higher order terms Ξ 3 and Ξ 4 , for which complete computations can be easily performed. We write here ζ for a 'generic' noise that plays the role of a regularized version of the Gaussian space noise ξ; this is a time-independent function/distribution; recall Z 1 " Rpζq. We do not pay attention here to the weights that should be added below as this is mainly a technical issue.

In addition to the Conditions on the operator L that we spelled out in section 3.1, we need to assume here that the following two mild conditions hold.

' The 'first-order' order operators V i (for i " 1, .., 0 ) are anti-self-adjoint with respect to the measure µ, which in particular implies that L is self-adjoint. Note that in a Riemannian setting, where V i is a differentiation operators along a vector field, such an assumption is equivalent to the fact that the corresponding vector fields are divergence-free.

' The operators t p|I|`|J|`1q{2 V I rV i , V j sV J e ´tL have kernels K t satisfying the Gaussian bounds, where r , s denotes the usual commutator between two linear operators (3.

2

)
The examples of section 3.1 satisfy these additional conditions. Let point out here that while the basic renormalisation operation done here consists in substracting from some diverging quantities their expectation, the latter quantities are a priori time and space-dependent functions. Following remark 11 in [START_REF] Gubinelli | Lectures on singular stochastic PDEs[END_REF], it is actually possible to set up a framework where we only need to renormalise some quantities by time-independent functions. Aiming that, the operation of subtracting the expectation seems to not be sufficient. Indeed, as explained in [START_REF] Gubinelli | Lectures on singular stochastic PDEs[END_REF]Remark 11], this technical difficulty can be avoid by modifying the time initial data (for τ " 0) in the operator R of resolution for the heat equation. By such a modification, the suitable expectation becomes a timeindependent function (and so is a good quantity for renormalizing) but we need to develop the whole paracontrolled calculus with distributions singular at time τ " 0. This is fully explained in details in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]Chapter 5], and there is no difficulty to adapt their result to the present setting, to which we stick in this section, to emphasize the basic problems at hand.

Renormalising the quadratic terms

In order to explain and to be more convincing, we present the main arguments for the renormalisation in the case of a doubling space pM, d, µq with general Ahlfors regularity of dimension ν " d (not necessarily ν " 3q. We recall that this means that µpBpx, rqq » r d , uniformly in x P M and r P p0, 1s.

The core idea of the renormalization procedure for Y 2 is best understood on the model term Ξ 2 " Π pbq pζ, Z 1 q. It is given by a linear combination of terms of the form ż 1 0

P t ´Q1 t ζ ¨Q2 t Z 1 ¯dt t " ż 1 0 P t ´Q1 t ζ ¨Q2 t Rζ ¯dt t ,
where the operators Q i are in the class StGC b{4 and P is an element of StGC r0,2bs . We have actually proved along the proof of Proposition 15 that the operator t ´1Q 2 t R is also an operator with cancellation, precisely an element of GC b 8 ´2. So the core of the renormalisation procedure for Ξ 2 happens to be the renormalisation of terms of the form

I 2 :" ż 1 0 P t ´Q1 t ζ ¨Q2 t ζ ¯dt
Since ζ is time-independent, we only have to consider in the previous term the case where the operators Q 1 , Q 2 have cancellation in space. We estimate the size of Q r pI 2 q in terms of r, to see whether or not it belongs to some Hölder space. For ζ white noise, the expectation E " ˇˇQ r pIqpeq ˇˇ2 ı is given by the integral on M 2 ˆr0, 1s 2 of K QrPt 1 pe, e 1 qK QrPt 2 pe, e 2 q E " Q 1 t 1 ξpe 1 qQ 2 t 1 ξpe 1 qQ 1 t 2 ξpe 2 qQ 2 t 2 ξpe 2 q ı (6.1) against the measure νpde 1 qνpde 2 qdt 1 dt 2 . The expectation in (6.1) is estimated with Wick's formula by

E " Q 1 t 1 ξpe 1 qQ 2 t 1 ξpe 1 q ‰ E " Q 1 t 2 ξpe 2 qQ 2 t 2 ξpe 2 q ‰ `E" Q 1 t 1 ξpe 1 qQ 1 t 2 ξpe 2 q ‰ E " Q 2 t 2 ξpe 2 qQ 2 t 1 ξpe 1 q ‰ `E" Q 1 t 1 ξpe 1 qQ 2 t 2 ξpe 2 q ‰ E " Q 1 t 2 ξpe 2 qQ 2 t 1 ξpe 1 q ‰ À pt 1 t 2 q ´d{2 `Gt 1 `t2 pe 1 , e 2 q 2 ,
where d is the homogeneous dimension of the ambiant space M . We recall that stochastic cancellations yields that for shows that the associated distribution is in C 2´d . That computation was essentially already done in Section 5.3 of our previous work [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF]. For the second integral, with dm :" νpde 1 qνpde 2 qdt 1 dt 2 , we have an estimate ż K QrPt 1 pe, e 1 qK QrPt 2 pe, e 2 qG t 1 `t2 pe 1 , e 2 q 2 dm À ż ˆr r `t1 ˙a ˆr r `t2 ˙a pt 1 `t2 q ´ν 2 ż G r`t 1 px, yqG r`t 2 px, zqG t 1 `t2 py, zq dm À ż ˆr r `t1 ˙N ˆr r `t2 ˙a pt 1 `t2 q ´d 2 pr `t1 `t2 q ´d{2 dt 1 dt 2 À r 2´d

for d ă 4, where we have used (3.4). By combining with Kolmogorov's continuity criterion, that also shows that the associated distribution is (almost surely) in C p2´dq

´.

So we see here that the basic renormalisation procedure allows to take into account in (6.1) only the terms where the stochastic cancellations mixes the parameters t 1 and t 2 as well as the variables e 1 and e 2 , which happens to be crucial. This brings us indeed to integrate pt 1 `t2 q ´d{2 pr `t1 `t2 q ´d{2 rather than pt 1 t 2 q ´d{2 . Roughly speaking, each term Q t ζ in the formula

I 2 " ż 1 0 P t `Q1 t ζ ¨Q2 t ζ ˘dt
is of order t ´d{4 , which gives a converging integral only if d ă 2; renormalisation has the effect to turn the integral into a convergent integral for d ă 4: in some sense this operation (with a time-independent noise) allows us to 'compensate' a lack of regularity of order pd{2q ´. One can treat the bilinear term Y 2 in the very same way as done in this paragraph.

Such a basic renormalisation procedure cannot work in all examples, as the Φ 4 3 equation makes it clear. In that 3-dimensional example, the space-time noise ζ has regularity ´5{2, so Z 1 has regularity ´1{2, which makes the expression Z 3 1 undefined. In terms of the operators Q i t ζ, the problem comes from the fact that Q 1 t pZ 1 q ¨Q2 t pZ 1 q ¨Q3 t pZ 1 q is of size t ´p3{2q ´. Even, if the simple renormalisation could compensate a lack of regularity of order p3{2q ´by above mechanism, it would fall short of making it integrable for the Lebesgue measure on r0, 1s. A trickier renormalisation procedure is thus needed, whose roots are well-explained in Hairer's work [START_REF] Hairer | A theory of regularity structures[END_REF] -see also the paracontrolled approach of Catellier and Chouk [START_REF] Catellier | Paracontrolled distributions and the 3-dimensional stochastic quantization equation[END_REF]. up to a satisfying remainder term controlled in terms of t 2α , by

P t " Q 1 t V i Rξ ε ¨Vj Q 2 t RpV i Rξ ε q ‰ peq `Pt " Q 2 t V i Rξ ε ¨Vj Q 1 t RpV i Rξ ε q ‰ peq.
Its expectation can be seen to converge in C 2α´1 to ż K Pt px, yq

" K rV j Q 2 t RpV j Rqs ˚Q1 t V i R py, yq `KrV j Q 1 t RpV j Rqs ˚Q1 t V i R py, yq ı µpdyq,
where ˚denotes the usual adjoint in L 2 pM, dµq (in space) and where the time is fixed in the operator R. By symmetry, it is equal to ż K Pt px, yq " K R ˚Vi R ˚QtViR py, yq ‰ µpdyq where Q t :" Q 2,t V j Q 1 t `Q1,t V j Q 2 t is antisymmetric. Since at time fixed, the spatial operator R ˚is self-adjoint, we deduce that R ˚Vi R ˚Qt V i R is antisymmetric in space and so its kernel is vanishing on the diagonal. This shows as a consequence that

E " w j,ε 2 ı
is bounded in the Hölder space C 2α´1 .

6.2. Toy models for the higher order terms On can get a feeling of what happens for the higher order terms Ξ 3 and Ξ 4 by looking at model quantities whose the structure is the same, or not far from, the different terms that appear in the definition of these terms. To study the trilinear expressions of ζ that appear in Ξ 3 , look at the model quantity

I 3 :" ż 1 0 P t `Q1 t ζ ¨Q2 t ζ ¨tQ 3 t ζ ˘dt.
This quantity is of a slightly different nature than I 2 since its expectation is null as ζ appears an odd number of times. On the other hand, the computation of We can see on such terms that a basic renormalisation procedure suffices to get objects of regularity 0 ´, in dimension 3, such as expected. Indeed, since we work in dimension 3 and so Z 1 has a positive regularity, we see that Ξ 3 and Ξ 4 have a

. 10 )

 10 Here are a few examples. Consider a smooth function m with compact support in r2 ´1, 2s, an integer c ě 1, and a tuple I of indices.

˘Ppbq t g where S pb{2q P StGC b 2

 2 and the tuples I, J and integers k, satisfy the constraint |I|

1 L 1 ,

 11 , denoted by A below, that satisfy the continuity estimate }A} C α ω ÑC β ω À for any α, β P p´3, 3q and any space-time weight ω.

Definition 22. Given 1 3 ă β ă α ă 1 2

 2 and a time-independent distribution ζ P C α´2 pa , a (PAM)-enhancement of ζ is a tuple p ζ :" ´ζ, Y 2 , Y 3 , W 1 , W 2 , pW j 2 q j ¯, with Y k P L 8 T C α´p5´kq{2 pa and W 1 , W 2 , W j k P L 8 T C 2α´1 pa .

. 5 )

 5 (b) Recast the equation as a fixed point problem for a map Φ from the solution space to itself.

Theorem 23. Let 1 3 ă α ă 1 2

 2 be given. Choose β ă α, the positive parameter a in the weight p a , and ε ą 0, such that 2α `β ą 1 and 8pa `εq ď α ´β.

  , are ε-uniformly bounded and converging in C T C α´2 pa , respectively C T C α´1 pa and C T C 2α´1 pa , for every a P p0, 1q;

  yq stands for a geodesic joining x to z and of length dpx, zq, and pX j q stands for a local frame field near px, yq, and it acts here as a first order differential on the first component of K. As a matter of fact, it suffices for the present work to assume that the semigroup has regularity estimates of large enough order. One can keep in mind the following two examples.

	where px,
	the semigroup has regularity estimates at any order, by which we mean that
	for every tuple I, the operators	´t |I| 2 V I	¯e´tL and e ´tL ´t |I| 2 V I ¯have kernels
	K t px, yq satisfying the Gaussian estimate
	ˇˇK t px, yq ˇˇÀ	1 µ `Bpx,	?	tq	˘e´c dpx,yq 2 t	(3.2)
	and the following regularity estimate. For dpx, zq ď	? t
	ˇˇK t px, yq ´Kt pz, yq ˇˇÀ	dpy, zq ? t	1 µ `Bpx,	?	tq	˘e´c dpx,yq 2 t	,	(3.3)
	for some constants which may depend on |I|.
	Let us point out that the regularity property (3.3) for |I| " k can be obtained from
	(3.2) with k `1 writing the "finite-increments" formula
	ˇˇK t px, yq ´Kt pz, yq ˇˇÀ dpx, zq sup	sup	ˇˇX j K t pw, yq	ˇˇ
							j		wPpx,zq

  is an operator of the form p b ptLqe ´tL , for some polynomial p b of degree pb ´1q, with value 1 in 0. Under the above Conditions assumptions, the operators P

		´tL	and	´tB t P	pbq t	:" Q pbq t ,
	with P 0 " Id, so P pbq t pbq
	pbq t	and Q pbq t both satisfy the Gaussian regularity estimates (3.2) at
	any order
	ˇˇˇK

t |I| 2

  Denote by M τ the multiplication operator in R by τ . An easy computation yields that Q where ψpσq :" ϕpσq`σϕ 1 pσq. (For an extension of the present theory to the setting of Sobolev spaces, such as done in the appendix B of

	P t :" P pbq	pbq t b ϕ ‹ t	and	Q pbq t :" ´tB t P t , pbq
		pbq t " Q pbq t b ϕ ‹ t	`P pbq t b ψ t

  We have P 1 P StGC r0,2bs , and it is easy to check that `Q2t ptLq ˘0ătď1 also belongs to StGC |I|{2 V I P pbq t e ´pλ´σqL px, yqptB τ q k ϕ t pτ ´λqIf dpx, yq ď 1, this is exactly the desired estimate. If dpx, yq ě 1 and one works on a finite time interval r0, T s then we keep the information that |λ ´σ| ď T and so the exponentially decreasing term in the Gaussian kernel on the spatial variable allows us to keep in all the previous computations an extra coefficient of the form

				b 4 `2 Ă StGC	b 4 . Insofar as
											RQ 1' t " `Q1 t R ˘',
	we are left with proving that the family Ă Q 1 :" `Q1 t t ´1R ˘0ătď1 belongs to GC	b 8 ´2,
	with Q 1 essentially given here by
											Q 1 t "	´t |I| 2 `kV I B k τ	¯Ppbq t
	with |I| 2 `k ą b 4 . Note in particular that we have either |I| ě b 4 or k ě b 8 . We
	check in the first two steps of the proof that r Q P G in both cases provided b is
	chosen big enough. The third step is dedicated to proving that r Q 1 P GC	b 4 ´1.
	Step 1. Assume here that |I| ě b 4 . The kernel K of Q 1 t ˝pt ´1Rq is given by
			K `px, τ q, py, σq	˘" ż 8	dλ t 2 .	(4.2)
	So by the Gaussian estimates of the operator t q 1{2 , and since |I| ě b 4 , we deduce that	|I| 2 V I P	pbq t e ´pλ´σqL at scale maxpt, λσ
	ˇˇˇK
		|I|
	t	2 V I P
											À	ˆt t `λ	´σ ˙b 8 ´ν 2	µpBpx,	? tqq ´1 ˆ1	`dpx, yq 2 t `λ ´σ ˙´ 1
											À	ˆt t `λ	´σ ˙b 8 ´ν 2 ´ 1	µpBpx,	? tqq	´1 ˆ1	t `dpx, yq 2	˙´ 1
	if b is chosen large enough for b 8 ´ν 2 ´ 1 to be non-negative. Using the smoothness of ϕ we then deduce that ˇˇK `px, τ q, py, σq ˘ˇˇi s bounded above by
	µpBpx,	? tqq ´1 ˆ1	`dpx, yq 2 t	˙´ 1 ż 8 σ ˆt t `λ	´σ ˙b 8 ´ν 2 ´ 1 ˆ1	`τ	´λ t	˙´ 1 dλ t 2
	À	1 t µ `Bpx,	?	tq	˘ˆ1	`dpx, yq 2 t	˙´ 1 ˆ1	`|τ ´σ| t	˙´ 1	.
	So we get the upper bound
	ˇˇK `px, τ q, py, σq	˘ˇˇÀ ν `BM `px, τ q,	?	t	˘˘´1	ˆ1	`dpx, yq 2 `|τ ´σ| t	˙´ 1	.	(4.3)
	¯dt t 4 , it suffices to look at 1 t g b µpB M px, 1qq ´1e with P 1 P StGC and Q 1 , Q 2 P StGC ´c dpx,yq 2 1`T
											ż 1 0	pt ´1RqQ 1' t	´Q2 t ptLqp¨q.P 1 t g	¯dt t	.

σ K t pbq t e ´pλ´σqL px, yq ˇˇˇÀ ˆt t `λ ´σ ˙|I| 2 G t`λ´σ px, yq

  T, U two operators on M and for every points y, z P M can be bounded above by the sum of two integrals, with the first one no greater than ż ż K QrPt 1 pe, e 1 qK QrPt 2 pe, e 2 qpt 1 t 2 q ´ν{2 dνpde 1 qνpde 2 qdt 1 dt 2 À ż ˆr r `t1 ˙a ˆr r `t2 ˙a pt 1 t 2 q ´d 2 dt 1 dt 2 for some positive exponent a, with a relatively sharp upper bound, which happens to be infinite in dimension 2 or larger. Considering I 2 ´E" I 2 ‰ removes precisely in Wick's formula the diverging part, and satisfies the estimate E

	So E	" ˇˇQ r `I2 ˘ˇ2	ı
						" `I2 ´ErI 2 s ˘2ı 1 2 À r 1´d 2 , that
						ż
		E "	T ξpyqU ξpzq	‰	"	K (6.2)

T px, yqK U px, zqdµpxq " K U ˚T pz, yq.

  formula shows that we always work with expectations of a product involving t 1 and t 2 , meaning that we have an estimate of the form QrPt 1 pe, e 1 qK QrPt 2 pe, e 2 qpt 1 t 2 q ´d{2 G t 1 `t2 pe 1 , e 2 qt 1 t 2 dm, with dm " νpde 1 qνpde 2 qdt 1 dt 2 , as above. For d ă 4, this gives the estimateE " |Q r `I3 ˘peq| 2 ‰ À ż ż ˆr r `t1 ˙a ˆr r `t2 ˙a pt 1 t 2 q ´d{2 pr `t1 `t2 q ´d{2 t 1 t 2 dt 1 dt 2 À r ´3d{2`4 ,on which one reads that I 3 has almost surely regularity p´3d{2 `4q ´, and there is no required renormalisation. Indeed, it is coherent with[START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] -equation(5.3), where it is shown also, through the regularity structures theory, that the trilinear (in terms of noise) quantities do not need to be renormalised.

	E	"	ˇˇQ r `I3 ˘peq ˇˇ2		involves a product of six Gaussian random variables. Applying
	ż K The model quantities corresponding to Ξ 4 are of the type Wick's E " |Q r `I3 ˘peq| 2 ‰ À
					ż 1
					I 4 :"	P t	`Q1 t ζ ¨Q2 t ζ ¨tQ 3 t ζ ¨tQ 4 t ζ ˘dt.
					0

I.Bailleul thanks the U.B.O. for their hospitality, part of this work was written there.

F. Bernicot's research is partly supported by the ANR projects AFoMEN no. 2011-JS01-001-01

Acknowledgements. We thank the two referees for their constructive critical remarks on previous versions of this work.

and HAB no. ANR-12-BS01-0013 and the ERC Project FAnFArE no. 637510.

3 D. Frey's research is supported by the ANR project HAB no. ANR-12-BS01-0013.

For a continuous function ζ in C 0 pa , and 1 ď i ď 3, define recursively the following reference distributions/functions Z i :" RpY i q, with Y 1 :" ζ, Y 2 :"

and define p‹q :" ´2

as well as for j P t1, .., 0 u W j 2 :"

Indeed in the p‹q term, only the resonant parts in the products have to be defined, since the parapoducts always have a sense, and so we will focus on the resonant part of p‹q

Defining the Y i 's as elements of

, the distributions W k as element of L 8 T C kα´1 pa , and Ă W 1 , W j 2 as an element of L 8 T C 2α´1 pa , for some 1{3 ă α ă 1{2 and a ą 0, when ζ is a space white noise, is the object of the renormalisation step, which shall be done elsewhere. These conditions ensure, by Schauder estimates, Proposition 9, that Z i is in the parabolic Hölder space C iα pa . Note that assuming W 1 is an element of L 8 T C 2α´1 pa ensures that p‹q is an element of L 8 T C α´1 pa . We assume throughout this section that this data set is given; set Z :" Z 1 `Z2 `Z3 ": Z 1 `r Z.

Proposition 21. The function u is a formal solution of the (PAM) equation if and only if the function v :" e ´Z u is a solution of the equation

with the same initial condition as u at time 0. The letter U stands here for W 1 `W2 `W3 for an explicit distribution

and using the Leibniz rule on V i 's

Consider the modified resonant part Π pbq i pf, gq :" Π pbq pf, V i gq and the corresponding corrector C i pf, g, hq :" Π pbq i `r Π pbq g pf q, h ˘´gΠ pbq i pf, hq. Then since in the study of the resonant part and the commutator, we can change the localization operators, so we can integrate an extra V i operator, we get boundedness of Π pbq i from C α ˆCβ to C α`β´1 as soon as α `β ´1 ą 0, and boundedness of the corrector C i from C α ˆCβ ˆCγ into C α`β`γ´1 as soon as α `β `γ ´1 ą 0, proceeding exactly in the same way as above for Π pbq and C. Using this commutator, we see that ř 0 i"1 Π pbq pV i pZq, V i pvqq is an element of the space

, that is an element of

pa and 2α `β ą 1. In the end, we conclude that

as expected. Observe that V i pZ 3 q is of parabolic regularity p3α ´1q, so vV i pZ 3 q and V i pvq belong to C β .

We can then apply the contraction principle, such as explaned above in Step (c) in section 5.1.1.

and is the solution of the equation

and y 7 given by the previous theorem. Note that the map Φ depends continuously on the enhanced distribution p ζ; so the next well-posedness result is then a direct consequence of Theorem 23.

' there exists deterministic functions

then the tuple 7 `γȃ lso satisfies the ansatz.

Theorem 26. Let 1 3 ă α ă 1 2 be given. Choose β ă α, the positive parameter a in the weight p a , and ε ą 0, such that 2α `β ą 1 and 8pa `εq ď α ´β.

One can choose a large enough parameter κ in the definition of the special weight for the following to hold. There exists a sequence of deterministic functions `λε j ˘0ăεď1 such that if v ε stands for the solution of the renormalized equation

with initial condition v 0 P C 1`α`β 0 p ´a , then v ε converges in probability to a solution v P C 1`α pa .

By reproducing calculations of subsection 5.1, we observe that v ε is solution of equation (5.7) if and only if u ε :" e Z ε v ε is solution of the equation

˘uε , u ε p0q " v 0 .

(5.8)

Theorem 27. Let 1 3 ă α ă 1 2 be given. Choose β ă α, the positive parameter a in the weight p a , and ε ą 0, such that 2α `β ą 1 and 8pa `εq ď α ´β.

One can choose a large enough parameter κ in the definition of the special weight for the following to hold. There exists a sequence of deterministic functions `λε j ˘0ăεď1 such that if u ε stands for the solution of the renormalized equation (5.8) with initial condition v 0 P C 1`α`β 0 p ´a , then u ε converges in probability to a distribution u P C α p 2a .

Remark 28. This result is also coherent with the one of [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF] by Hairer and Labbé. Indeed, in [25, Equation (5.

3)], where the quantities of order 'odd' (in terms of the white noise) has no renormalization correction terms (as for us) as well as for W j 2 (which is of order 'even' but involving an extra derivative 'V j '). This latter term will be more explained at the end of Section 6, and we will see why this extra derivative with symmetrical properties implies that the correction term (for the renormalization) is null, as in [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF].

is well-defined in C 3α whose evaluation through R is then bounded in C 1`α`β with acceptable bounds. And since

q this is also controlled in C 1`α`β by Schauder estimates and we conclude to

It remains the quantity with B ij (instead of A ij ). Here we only know that B ij belongs to C 2α´1 (and not L 8 T C 2α´1 as for A ij ) but we can take advantage of the fact that B ij is a paraproduct. Indeed as before we have

is well-controlled in C 1`α`β due to Schauder estimates (Proposition 10). In conclusion, we have obtained that

which exactly shows that RrN pvqs j is paracontrolled by the collection RV i Z j q i .

Corollary 31. Under the assumptions of Theorem 30 on the positive parameters α, β, a, ε, and given p u P S α,β `p ζ ˘with u P C α´2a´2ε , set v :" R `p u p ζ ´pu ¨V qu ˘. Then the tuple `v, u, u 1 , u 2 ˘satisfies the structure equation (5.4), with

where κ is the constant appearing in the definition (5.9) of the weight .

Proof of Theorem 2 -Well-posedness of Burgers system follows as a direct consequence. Theorem 2 on the convergence of the solutions to a renormalised ε-dependent equation to the solution of the Burgers equation is thus obtained as a direct consequence of this well-posedness result together with an additional renormalisation step that will be done in a forthcoming work. The 3 ˆ3 matrixvalued functions d ε is the one renormalizing the quantities pΘ i,j q ε 1ďi,jď3 . By tracking the changes (in the proof of Theorem 30), induced by a renormalisation of Θ ε into Θ ε ´dε in L 8

T C 2α´1 pa , we see that if `uε , u ε 1 , u ε 2 , u ε 3 ˘satisfies Ansatz (5.4) with Z ε i , and setting v ε :" R `pu ε ¨V qu ε ´dε pu ε 1 , u ε 1 q ˘, the tuple

The study of the term W j,ε 2 :"

done differently from the study of Y 2 , as can be expected from comparing our setting with the setting of regularity structures for the 3-dimensional setting, investigated in Hairer and Labbé's work [START_REF] Hairer | Multiplicative stochastic heat equations on the whole space[END_REF]. While the same reasoning shows that recentering W j 2 around its expectation makes it converge in the right space, there is actually no need to renormalize this term.

To see this, we replace in a first step the study of the above quantity by a similar one where the spacetime paraproduct Π pbq and resonent term Π pbq p¨, ¨q is replaced by a space paraproduct π pbq and resonent operator π pbq p¨, ¨q introduced and studied in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF] -they are defined in the exact same way as Π b , but without the time convolution operation. Continuity properties were proved for such spatial paraproduct in [START_REF] Bailleul | Heat semigroup and singular PDEs (with an appendix[END_REF], and we shall use in addition an elementary comparison result between this spatial paraproduct and our space-time paraproduct proved by Gubinelli, Imkeller and Perkowski in their setting [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]Lemma 5.1]. A similar statement and proof holds with the two paraproducts Π pbq and π pbq ; we state it here for convenience.

Lemma 32. Let ω 1 , ω 2 be two space-time weights. If u P C α ω 1 for α P p0, 1q and v P L 8 T C β ω 2 for some β P p´3, 3q then

Setting w j,ε 2 :"

‰ ānd using the comparison lemma and then the continuity estimates of each paraproduct, we see that W j,ε 2 ´wj,ε 2 is equal to

so it is an element of C 2α´1 . So in order to estimate W j,ε 2 is the suitable Hölder space we only need to study its "spatial" counterpart w j,ε 2 . This can be done as follows.

As W j,ε 2 , the quantity w j,ε 2 is quadratic as a function of the noise, however we are going to see that its expectation is already bounded in C 2α´1 , as a consequence of some symmetry properties -this explains why w j,ε 2 is directly converging in C 2α´1 , with no renormalization needed along the way. The term w j,ε 2 can indeed be written as a finite sum of integrals in time of terms of the form

where the localizing operators P t and Q t are only in space. Using the above additional geometric assumptions on the operator, the previous integral can be estimated, smaller lack of regularity than Ξ 2 . So it is 'easier' to renormalize these higher order terms than the first one.