Marco Furlan
email: furlan@ceremade.dauphine.fr

Massimiliano Gubinelli
email: gubinelli@iam.uni-bonn.de

Paracontrolled

Paracontrolled quasilinear SPDEs

come

Introduction

We show how to renormalise a class of general quasilinear equations of which one of the simplest examples is the following parabolic SPDE: @ t u(t; x) ¡ a(u(t; x))u(t; x) = (x); u(0; x) = u 0 (x); x 2 T 2 ; t > 0;

with a: R ! [; 1] for > 0 a uniformly bounded C 3 diusion coecient, and ka (k) k L 1 6 1 for k = 0; :::; 3. We assume that 2 C ¡2 (T 2 ) with 2/3 < < 1 where C (T 2 ) is the Besov space B 1;1 (T 2 ). This would apply to the space white noise on T 2 , for example. In this case we only expect that u(t; ) 2 C (T 2 ) and the term a(u(t; ))u(t; ) is not well dened when 2 ¡ 2 < 0. Eq. ( 1) is a quasilinear generalisation of the twodimensional periodic parabolic Anderson model (PAM).

Let us remark from the start that the framework we will consider below allows to deal with a class of equations of the form a 1 (u(t; x))@ t u(t; x) ¡ a 2 (u(t; x))u(t; x) = (a 3 (u(t; x)); t; x); x 2 T 2 ; t > 0;

(

where a 1 ; a 2 are suciently smooth non-degenerate coecients and (z; t; x) is a Gaussian process with covariance E[(z; t; x)(z 0 ; t 0 ; x 0 )] = F (z; z 0 )Q(t ¡ t 0 ; x ¡ x 0 ); x; x 0 2 T 2 ; t; t 0 ; z; z 0 2 R;

with F a smooth function and Q a distribution of parabolic regularity > ¡4 / 3. This includes the space white noise discussed before or a time white noise with a regular dependence on the space variable or some noise mildly irregular in space and time. Also the scalar character of the equation or of the non-linear diusion coecient will not play any specic role and we could consider vectorvalued equations with general diusion coecients provided the template problem [START_REF] Bailleul | Quasilinear generalized parabolic anderson model equation[END_REF] below remains uniformly parabolic.

For the sake of clarity and simplicity we will discuss mainly the basic example (1) since this contains already most of the technical diculties. The fact that one can handle models as general as (2) can be considered a direct byproduct of the techniques we will introduce below.

Recently Otto and Weber [START_REF] Otto | Quasilinear SPDEs via rough paths[END_REF] and Bailleul, Debussche and Hofmanova [START_REF] Bailleul | Quasilinear generalized parabolic anderson model equation[END_REF] investigated quasilinear SPDEs in the context of pathwise methods and in a range of regularities compatible with the ones we will consider in this paper.

In [START_REF] Otto | Quasilinear SPDEs via rough paths[END_REF] the authors obtained a priori estimates for equations of the form @ t u(t; x) ¡ a(t; x)@ x 2 u(t; x) = f (u(t; x))(t; x);

(t; x) 2 T 2
where both space and time variables take values in a one dimensional periodic domain and their noise can be white in time but colored in space, essentially behaving like a distribution of parabolic regularity in (¡4 / 3; 1). In order to do so they introduce a specic notion of modelled function and related estimates.

Bailleul, Debussche and Hofmanová in [START_REF] Bailleul | Quasilinear generalized parabolic anderson model equation[END_REF] obtain local well-posedness for the generalised parabolic Anderson model equation @ t u(t; x) ¡ a(u(t; x))u(t; x) = g(u(t; x))(x) t > 0; x 2 T 2 :

(3)

The authors obtain the same result as the one presented in Section 6 of our work, without the machinery of nonlinear paraproducts introduced here, but using only the basic tools of paracontrolled analysis and some clever transformations.

On the other hand, we remark that the apparently innocuous vectorial formulation of (3) @ t u(t; x) ¡ a ij (u(t; x)) @ 2 @x i @x j u(t; x) = g(u(t; x)) t > 0; x 2 T 2 is out of reach of the techniques used in [START_REF] Bailleul | Quasilinear generalized parabolic anderson model equation[END_REF], while can be treated awlessly in our framework.

Let us state one simple result that can be obtained via the theory developed in this paper: Theorem 1. Fix 2/3 < < 1. Let 2 C ¡2 (T 2 ) be a space white noise with zero average on the torus, u 0 2 C an initial condition and a: R ! [; 1] for some > 0, a 2 C 3 (R) and ka (k) k L 1 6 1 8k 2 0; :::; 3. Let ( " ; u 0;" ) ">0 be a family of smooth approximations to ; u 0 obtained by convolution with a rescaled smoothing kernel and u " the classical solution to the Cauchy problem @ t u " ¡ a(u " )u " = " + " a 0 (u " ) a(u " ) 2 ; u(0) = u 0;" : (4)

Then we can choose the constants ( " ) ">0 and a random time T > 0 in such a way that the family of r.v. (u " ) ">0 L T (T 2 ) almost surely converge as " ! 0 to a random element u 2 L T (T 2 ), where L T is the parabolic space C([0; T ]; C (T 2 )) \ C /2 ([0; T ]; C 0 (T 2 )). This element can be characterised as the solution to a paracontrolled singular SPDE (see below for more details).

In order to devise a suitable formulation of eq. ( 1) and obtain a theory with u 2 C we decompose the non-linear diusion term in the l.h.s. with the help of Bony's paraproduct [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF] and write @ t u ¡ a(u) u = + (u) [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF] with

(u) := a(u) u + a(u) u (6)
where ; are standard paraproducts and denotes the resonant product (see below for precise denitions). Now the l.h.s. is always well dened irrespective of the regularity of the function u and the problem becomes that of controlling the resonant product a(u) u appearing in the r.h.s. . A key point of the analysis put forward below is that this term can be expected to be of regularity 2 ¡ 2 > ¡ 2 so better than the leading term .

Our approach can be described as follows. For an equation of the form

@ t u ¡ a 1 (u)u = a 2 (u);
we consider at rst a parametric template problem with constant coecients @ t #(; t; x) ¡ 1 #(; t; x) = 2 (t; x); [START_REF] Bailleul | Quasilinear generalized parabolic anderson model equation[END_REF] where now = ( 1 ; 2 ) are xed numbers. A nonlinear paraproduct will allow us to modulate the parametric solution # with the coecient a(u) = (a 1 (u); a 2 (u)) as to capture the most irregular part of the solution u itself. As a consequence, the paracontrolled Ansatz

u = (a(u); #) + u ]
will dene a regular remainder term u ] which solves a standard PDE. With this decomposition the resonant products appearing in the equation can be estimated along the lines of the standard paracontrolled arguments introduced in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] and all the arguments introduced there can be extended in a straightforward manner to the quasilinear setting. This approach has been inspired by the parametric controlled Ansatz of Otto and Weber [START_REF] Otto | Quasilinear SPDEs via rough paths[END_REF]. At variance with their approach we use the parametric Ansatz in the context of the paradierential calculus and consider more general noise terms.

Usefulness of paraproducts in the analysis of non-linear PDEs is by now well established: see for example the seminal paper of Meyer [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF], the early review of Bony [START_REF] Bony | Microlocal Analysis and Applications Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held[END_REF], the recent books of Alinhac and Gérard [START_REF] Alinhac | Opérateurs pseudo-diérentiels et théorème de Nash-Moser[END_REF] and Bahouri, Chemin, and Danchin [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]. Let us mention also the interesting paper of Hörmander [START_REF] Hörmander | The Nash-Moser theorem and paradierential operators[END_REF] where paradierential operators allows to bypass the NashMoser xpoint theorem in some applications where the loss of regularity prevents straightforward use of standard Banach xpoint theorem. The main observation in that paper is that, with the aid of paradierential operators, it is possible to identify a corrected problem for which standard Banach xpoint applies.

Paracontrolled calculus for singular SPDEs has beed introduced by Gubinelli, Imkeller and Perkowski [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] (see also the lecture notes [START_REF] Gubinelli | Lectures on singular stochastic PDEs[END_REF]) and used to study various equations like the KPZ equation [START_REF] Gubinelli | KPZ reloaded[END_REF], the dynamic 3 4 model [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF] in d = 3 and its global wellposedness [START_REF] Mourrat | Global well-posedness of the dynamic $\Phi4_3$ model on the torus[END_REF], the spectrum of the continuous Anderson Hamiltonian in d = 2 [START_REF] Allez | The continuous Anderson hamiltonian in dimension two[END_REF]. By using heatsemigroup techniques paracontrolled calculus has been extended to the manifold context by Bailleul and Bernicot [START_REF] Bailleul | Heat semigroup and singular PDEs[END_REF].

Nonlinear generalisation of the classic bilinear paraproducts already appeared in the notion of paracomposition introduced by Alinhac [START_REF] Bony | Microlocal Analysis and Applications Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held[END_REF]. Nonlinear versions of rough paths have been considered by one of the authors in order to study the Kortewegde Vries equation [START_REF] Gubinelli | Rough solutions for the periodic Kortewegde Vries equation[END_REF]. Nonlinear Young integrals were used by one of the authors in joint work with Catellier [START_REF] Catellier | Averaging along irregular curves and regularisation of ODEs[END_REF] to study the the regularising properties of sample paths of stochastic processes processes. See also the related work of Hu and Le [START_REF] Hu | Nonlinear Young integrals and differential systems in H\"older media[END_REF] on dierential systems in Hölder media. Relevant to this discussion of non-linear variants of rough paths is the work of Bailleul on rough ows [START_REF] Bailleul | Flows driven by rough paths[END_REF] and their application to homogeneisation [START_REF] Bailleul | Rough ows and homogenization in stochastic turbulence[END_REF]. By looking at the composition f (g(x)) as the action of the distribution g(x) on the function f , nonlinear constructions can be linearised at the price of working in innitedimensional spaces: this is the approach chosen by Kelly and Melbourne to avoid nonlinear generalisations of rough path theory in their study of homogeneisation of fastslow system with random initial conditions [START_REF] Kelly | Deterministic homogenization for fast-slow systems with chaotic noise[END_REF]. It is worth mentioning also Kunita's theory of semimartingale vector elds [START_REF] Kunita | Stochastic dierential equations and stochastic ows of dieomorphisms[END_REF] which occupy a place in stochastic analysis quite similar to that which these recent developments occupy in the landscape of rough paths/paracontrolled distributions theories.

Paracontrolled calculus is currently limited to rst order computations. This limitation is also ubiquitous in the present work. Even if, in practice, this is not a big issue, and the calculus is still able to deal with a large class of problems, it makes the paracontrolled approach less appealing for a general theory of singular SPDEs. Let us remark that recently Bailleul and Bernicot [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF] developed an higher order version of the paracontrolled calculus. However, apart from these recent development, whose impact is still to be assessed, the most general theory for singular SPDEs has been developed by Hairer [19,[START_REF] Hairer | Singular stochastic PDEs[END_REF][START_REF] Friz | A Course on Rough Paths: With an Introduction to Regularity Structures[END_REF] under the name of regularity structures theory. Regularity structures are a vast generalisation of Lyons' rough paths [START_REF] Lyons | Dierential equations driven by rough signals[END_REF][START_REF] Lyons | System Control and Rough Paths[END_REF][START_REF] Lyons | Dierential Equations Driven by Rough Paths: Ecole d'Eté de Probabilités de Saint-Flour XXXIV-2004[END_REF] which give eective tools to describe non-linear operations acting on certain spaces of distributions, their renormalization by subtraction of local singularities and their use to solve singular SPDE. Regularity structures have been successfully applied to all the models mentioned so far [START_REF] Hairer | A theory of regularity structures[END_REF][START_REF] Hairer | Solving the KPZ equation[END_REF], to other models like the SineGordon model [START_REF] Hairer | The dynamical sine-Gordon model[END_REF] (which however can also be handled via paracontrolled techniques) and to study weak universality conjectures [START_REF] Hairer | A class of growth models rescaling to KPZ[END_REF][START_REF] Hairer | Large scale behaviour of 3d phase coexistence models[END_REF]. In their current instantiation it does not seem possible to solve quasilinear SPDEs via regularity structures. The results of the present paper hint to the fact that a non-linear version of regularity structures is conceivable, at least in principle. Indeed one can imagine models depending on additional parameters and modelled distributions acting as evaluations of the parametric models at certain spacetime dependent values of the parameters. It would be interesting to pursue further this intuition.

The structure of the paper is the following. In Section 2 we introduce our basic tools: the non-linear paraproduct decomposition and some related commutation lemmas. In Section 3 we introduce the paracontrolled Ansatz which allows to transform the singular problem (1) into a wellbehaved PDE. In Section 4 we discuss the apriori estimates, the uniqueness of the solution of the transformed PDE and its continuity w.r.t. the random data and the initial condition, we introduce also the algebraic structure which allows to renormalise the model. Section 5 deals with the renormalization of the stochastic data and the construction of the enhanced noise associated to white noise. Section 6 deals with the extension of the results to more general equations, in particular with equation (3) or with noise whose law depends on the solution itself. Finally Appendix A reviews some reference material on Besov spaces and proves some technical lemmas.

Notations. We will denote C := B 1;1 (T 2 ) the Zygmund space of regularity 2 R on the torus T 2 . See Appendix A for the denition of the general Besov spaces B p;q , the LittlewoordPaley operators ( i ) i>¡1 and the basic properties thereof needed in this paper. If V is a Banach space and T > 0, we denote C T V the space of Hölder functions in

C T V := C([0; T ]; V ). We introduce parabolic spaces L T := C T /2 C 0 \ C T C with norm kf k L T = kf k C T /2 C 0 + kf k C T C : (8)
Moreover for convenience we denote C T := C T C . We will avoid to note explicitly the time span T whenever this does not cause ambiguities. We will need also spaces for functions of (; t; x) where is an additional parameter in [; 1] for 2 (0; 1) which we denote

C k V with norm kF k C k V = sup 2[;1] sup n=0;:::;k k@ n F (; )k V ; ( 9
)
where V is a Banach space of functions on [0; T ] T 2 , in our case V = C T or V = L T .

We will denote by K i;x (y) = 2 2i K(2 i (x ¡ y)) the kernel of the LittlewoodPaley operator i and

Q i;t (s) = Q i (t ¡ s) = 2 2i Q(2 2i (t ¡ s))
a smoothing kernel at scale 2 2i in the time direction where Q is a smooth, positive function with compact support in R + and mass 1. We introduce also the shortcut P i;x = K <i¡1;x = P j <i¡1 K i;x . Another notation shortcut widely used in this article is to write R x;y for integrals on T 2 or R with respect to the measures dx and dy without specifying the integration bounds, whenever this does not create ambiguity. Finally, we will note f sy tx = f (t; x) ¡ f (s; y) and f sy tx = f (s; y) + (f (t; x) ¡ f (s; y)) for 2 [0; 1].

Nonlinear paraproducts

In this section we introduce the nonlinear paraproduct and related results that will be used in Section 3 to analyse equation [START_REF] Alinhac | Opérateurs pseudo-diérentiels et théorème de Nash-Moser[END_REF].

Let g: [0; T ] T 2 ! R, and h: R [0; T ] T 2 ! R be smooth functions. We can decompose the composition h(g(); ) via nonlinear paraproducts as follows. Dene (g; h)(t; x) := X q Z y;z P q;x (y)K q;x (z)h(g(t; y); t; z) (10)

(g; h)(t; x) := X kq Z y;z K k;x (y)K q;x (z)h(g(t; y); t; z) ( 11
)
(g; h)(t; x) := X k Z y;z K k;x (z)P k;x (y)h(g(t; z); t; y) : (12)
This gives a map (g; h) 7 ! } (g; h) := (g; h) + (g; h) + (g; h) = h(g(); ) [START_REF] Friz | A Course on Rough Paths: With an Introduction to Regularity Structures[END_REF] that can be uniquely extended to

} : C T C 2 C T ! C T ^ 2 (0;
k (g; h)k C T . khk C C T ; k (g; h)k C T ^(+ ) . kg k C T khk C 1 C T ; and k (g 1 ; h) ¡ (g 2 ; h)k C T . kg 1 ¡ g 2 k C T L 1kh k C 1 C T ; k (g 1 ; h) ¡ (g 2 ; h)k C T ^(+ ) . kg 1 ¡ g 2 k C T L 1(kg 1 k C T + kg 2 k C T )khk C 2 C T +kg 1 ¡ g 2 k C T khk C 1 C T : Moreover if + > 0 we have also k (g; h)k C T + . khk C 1 C T kgk C T ; k (g 1 ; h) ¡ (g 2 ; h)k C T + . kg 1 ¡ g 2 k C T L 1(k g 1 k C T + kg 2 k C T )khk C 2 C T +kg 1 ¡ g 2 k C T khk C 1 C T : In particular if + > 0 the composition } (g; h) = h(g();
) is linear in h and locally Lipshitz in g:

k } (g; h)k C T . khk C 1 C T kg k C T ; k } (g 1 ; h) ¡ } (g 2 ; h)k C T . kg 1 ¡ g 2 k C T (1 + kg 1 k C T + kg 2 k C T )khk C 2 C :
Proof. Using the fact that

k k h(g(t; y); t; )k L 1 . 2 ¡k khk C C T ; k k h(g(t; y); t; ) ¡ k h(g(t; y 0 ); t; )k L 1 . 2 ¡k khk C 1 C T kg k C T C jy ¡ y 0 j ; and k k h(g 1 (t; y); t; ) ¡ k h(g 2 (t; y); t; )k L 1 . 2 ¡k khk C 1 C T kg 1 ¡ g 2 k C T L 1;
we obtain the bounds on (g; h), (g; h), (g; h) and (g 1 ; h) ¡ (g 2 ; h). We proceed to estimate the term (g 1 ; h) ¡ (g 2 ; h). We will use the following notation for brevity:

g 2z 1 y : =g 1 (t; y) ¡ g 2 (t; z) and g 2z 1 y : =g 2 (t; z) + (g 1 (t; y) ¡ g 2 (t; z)): Then Z y;z K k;x (z)P k;x (y)[h(g 1 (t; z); t; y) ¡ h(g 2 (t; z); t; y)] = Z y;z; 2[0;1] K k;x (z)P k;x (y)[@ h( g 2z 1z ; t; y)g 2z 1z ¡ @ h( g 2x 1x ; t; y)g 2x 1x ]
.

Z y;z;t2[0;1];2[0;1] K k;x (z)P k;x (y)@ 2 h( ( g 2 1 ) x z ; t; y)(g 1x 1z ¡ g 2x 2z )g 2z 1z + Z y;z; 2[0;1] K k;x (z)P k;x (y)@ h( g 2x 1x ; t; y)(g 2z 1z ¡ g 2x 1x )
.

kg 1 ¡ g 2 k C T L 1(kg 1 k C T C + kg 2 k C T C )khk C 2 C T C 2 ¡ k X q<k ¡1 2 ¡q +kg 1 ¡ g 2 k C T C khk C 1 C T C 2 ¡ k X q<k¡1 2 ¡q :
With the same reasoning we can bound the norm of (g 1 ; h) ¡ (g 2 ; h).

We will need the following time-smoothed nonlinear paraproduct

(g; h)(t; x) := X i Z y;s Q i;t (s)P i;x (y)( i h(g(s; y); t; ))(x); ( 14
) with Q 2 C c 1 (R) with total mass 1, and
Q i;t (s) := 2 2i Q(2 i (t ¡ s))
as specied in the introduction. In [START_REF] Gubinelli | Rough solutions for the periodic Kortewegde Vries equation[END_REF] we use the convention that a continuous function t 7 ! g(t) on R + is extended to R by dening g(t) = g(0) for t 6 0. This preserves the Hölder norms of index in [0; 1]. The modied nonlinear paraproduct enjoys similar bounds to the regular one.

Lemma 3. Let g 2 C T L 1 , g 2 [; 1] and h 2 C 1 L T for 2 (0; 2). Then k (g; h)k C T . khk C C T and k (g; h)k L T . khk C L T :
Moreover, (g; h) is linear in h and:

k (g 1 ; h) ¡ (g 2 ; h)k C T . kg 1 ¡ g 2 k C T L 1kh k C 1 C T ; k (g 1 ; h) ¡ (g 2 ; h)k L T . kg 1 ¡ g 2 k C T L 1kh k C 1 L T :
Proof. The norm k (g; h)k C T C can be treated in the same way as in Lemma 2. We estimate k (g; h)k C T /2 C 0 as follows:

k j (g; h)(t 1 ) ¡ j (g; h)(t 2 )k L 1 . sup x Z z K j;x (z) X i j Z y;s Q i;t 1 (s)P i;z (y)[ i h(g(s; y); t 1 ; z) ¡ i h(g(s; y); t 2 ; z)] +sup x Z z K j ;x (z) X ij Z y;s [Q i;t 1 (s) ¡ Q i;t 2 (s)]P i;z (y) i h(g(s; y); t 2 ; z) . kh(; t 1 ; ) ¡ h(; t 2 ; )k C C 0 + jt 1 ¡ t 2 j /2 khk C C T :
The second inequality can be obtained easily with the same techniques used so far.

Remark 4. Using the Fourier support properties of the kernel P q;x it is easy to see that it has mass 1. Therefore, the paraproducts ( 10) and ( 14) when g(t; x) = g is a constant smooth function become 8(t; x) 2 R T 2 :

(g ; h)(t; x) = h(g ; t; x) = } (g ; h); (g ; h)(t; x) = h(g ; t; x):

Nonlinear commutator.

The next technical ingredient is a commutator lemma between the non-linear paraproduct of ( 14) and the standard resonant product. It will be needed below to analyse a term of the form (g; h) (g; h), then we will specialise our discussion to this specic structure. Notice that in the following the various spacetime operators act pointwise in the parameter , in the sense that, for example:

(h h)(; t; x) = (h(; t; ) h(; t; ))(x): Lemma 5. We dene the map : C 1 ([0; T ]; T 2 ) C 2 C 1 ([0; T ]; T 2 ) ! C 1 ([0; T ]; T 2 ) by (g; h) := [ (g; h) (g; h)] ¡ } (g; h h):
Then for all 2 (0; 1), < R, " > 0 such that 2 ¡ 2 + ¡ " > 0 and g 2 [; 1], we have

k(g; h)k C T 2 ¡2+ ¡" . kgk L T khk C 1 C T 2 and k(g 1 ; h) ¡ (g 2 ; h)k C T 2 ¡2+ ¡" . kg 1 ¡ g 2 k C T L 1(kg 1 k L T + kg 2 k L T )khk C 2 C T 2 +kg 1 ¡ g 2 k L T khk C 1 C T 2 :
As a consequence can be uniquely extended to a locally Lipshitz function

: L T C 2 C T ! C T 2 ¡2+¡" : Proof. Let A(t; z) := (g; h) (g; h)(t; z)
. We can approximate A(t; z) with its value for a xed g = g(t; z), to obtain

q A(t; x) = Z z K q;x (z)( (g; h) (g; h))(t; z) = Z z K q;x (z)( (g(t; z); h) (g(t; z); h))(t; z) + Z z K q;x (z)(( (g; h) ¡ (g(t; z); h)) (g(t; z); h))(t; z) (15) + Z z K q;x (z)( (g; h) ( (g; h) ¡ (g(t; z); h)))(t; z): ( 16
)
with the remainders ( 15) and ( 16). To estimate the rst term notice that j [ (g(t; z); h)](t; z) = j (g(t; z); h)(t; z) and that, by Remark 4

i (g(t; z); h)(t; z) = Z z 0 K i;z (z 0 ) (g(t; z); h)(t; z 0 ) = Z z 0 K i;z (z 0 )h(g(t; z); t; z 0 ) = i h(g(t; z); t; )(z):
This yields

( (g(t; z); h) (g(t; z); h))(t; z) = X i j i (g(t; z); h) = X i j i h(g(t; z); t; )(z) j [h(g(t; z); t; )](z) = } (g; h h)(t; z):
We proceed then to estimate ( 15) and ( 16). We obtain

Z z K q;x (z)[( (g; h) ¡ (g(t; z); h)) (g(t; z); h)](t; z) = Z z K q;x (z) X i j&q ( i (g; h)(t; z) ¡ i (g(t; z); h)(t; z)) j (g(t; z); h)(t; z):
Using Lemma 3 we have j j (g(t; z); h)(t; z)j . 2 (2¡)j khk C C T :

Lemma 6 gives

j i ( (g; h) ¡ (g(t; z); h))(t; z)j . 2 ¡(+¡")i kgk L T khk C 1 C T ;
and thus ( 15) is bounded by

2 ¡(2 +¡2¡")q kg k L T khk C T 2 .
We can easily bound [START_REF] Gubinelli | KPZ reloaded[END_REF] in the same way, and this proves the rst inequality. For the second inequality, Lemma 3 yields

j j (g 1 ; h)(t; z) ¡ j (g 2 ; h)(t; z)j . 2 (2¡)j kg 1 ¡ g 2 k C T L 1kh k C 1 C T ;
and using the second inequality of Lemma 6 we obtain the desired bound. The extension of to L T C 2 L T is standard (see e.g. the proof of the commutator lemma in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], Lemma 2.4).

Lemma 6.

With the same assumptions of Lemma 5 we have

j i (g; h)(t; z) ¡ i (g(t; z); h)(t; z)j . 2 ("¡¡)i kgk L T khk C 1 C T and j i (g 1 ; h)(t; z) ¡ i (g 1 (t; z); h)(t; z) ¡ i (g 2 ; h)(t; z) + i (g 2 (t; z); h)(t; z)j .2 ("¡¡)i kg 1 ¡ g 2 k L T khk C 1 C T + kg 1 ¡ g 2 k C T L 1(kg 1 k L T + kg 2 k L T )khk C 2 C T Proof. [ i (g; h) ¡ i (g(t; z); h)](t; z) = X ki Z x; y;s; K i;z (x)Q k;t (s)P k ¡1;x (y)@ k h(
jK i;z (x)Q k;t (s)P k¡1;x (y)jk@ k hk C T L 1jt ¡ sj (¡")/2 kg k C T /2¡"/2 L 1 + X ki Z
x; y;s;

jK i;z (x)Q k;t (s)P k¡1;x (y)jk@ k hk C T L 1jy ¡ z jkg k C T . 2 ¡(¡")i 2 ¡i k@ hk C T kgk C T C 0 + kgk C T
where we used the notation g tz sy = g(s; y) ¡ g(t; z), g tz sy = g(t; z) + (g(s; y) ¡ g(t; z)) and Lemma 22. This proves the rst bound.

The second inequality can be obtained in the same way with the techniques already used here and in Lemma 2.

Approximate paradierential problem.

In this section we construct an approximate solution to the equation

(@ t ¡ g )u = f ; u(0; ) = 0; (17)
with data f 2 C ¡2 and g 2 L T , for some xed ; 2 (0; 1). The idea is to obtain it via a certain class of paradierential operators. We introduce the operator L acting on functions of (; t; x) by (L U )(; t; x) := @ t U (; t; x) ¡ U (; t; x): [START_REF] Hairer | Solving the KPZ equation[END_REF] We will also use the notation L 1 := @ t ¡ 1 with 1 2 R.

Observe that if u does not depend on we can dene

(g; L )u := (g; L u) (19)
and from denition [START_REF] Bony | Microlocal Analysis and Applications Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held[END_REF] with h = L u we obtain (g; L )u = @ t u ¡ g u.

We can describe the commutation between the dierential operator L and the para- product (g; ) via the following estimate:

Lemma 7. Let 2 (0; 1), 2 R. Let U 2 C 2 C T and g 2 L T such that g 2 [; 1]. Dene (g; U ) := R 1 + R 2
with R 1 and R 2 as in ( 22), [START_REF] Hairer | Large scale behaviour of 3d phase coexistence models[END_REF]. Then for every " > 0

k (g; U )k C T + ¡2¡" . (1 + kg k C T L 1)kg k L T kU k C 1 C T : (20) Moreover, (g; U ) is linear in U and k (g 1 ; U ) ¡ (g 2 ; U )k C T + ¡2¡" . kg 1 ¡ g 2 k L T (1 + kg 1 k L T + kg 2 k L T )kU k C 2 C T :
In particular, we have

(g; U ) = (g; L U ) ¡ (g; L ) (g; U ) 2 C T C + ¡2¡" ( 21
)
whenever this expression makes sense.

Proof. We start considering g 2 C 1 ([0; T ]; T 2 ) and U 2 C 2 C 1 ([0; T ]; T 2
), and prove (21) in this setting. Notice that (g(t; y); L g(t;y) U ) = L U (g(t; y)). As a consequence, we can estimate

(g; L U )(t; x) ¡ (g; L (g; U ))(t; x) = (g; L U )(t; x) ¡ X k Z y P k;x (y)(L g(t;y) k (g; U ))(t; x) = (g; L U )(t; x) ¡ X k Z y P k;x (y)(@ t k (g; U ))(t; x) + X k Z y P k;x (y)g(t; y)( k (g; U ))(t; x) = (g; L U ¡ @ t U )(t; x) + X k Z y P k;x (y)g(t; y)( k (g; U ))(t; x) + X k Z P k;x (y)g(t; y)( k [; (g; )]U )(t; x) ¡ X k Z P k;x (y)( k [@ t ; (g; )]U )(t; x)
with the commutators

[; (g; )]U := (g; U ) ¡ (g; U ); [@ t ; (g; )]U := @ t k (g; U ) ¡ k (g; @ t U ):
We have

(g; L U ¡ @ t U )(t; x) + X k Z y P k;x (y)g(t; y)( k (g; U ))(t; x) = R 1 (t; x)
with the denition R 1 (t; x) := X k;i Z y;z y 0 ;s P k;x (y)K k;x (z)P i;z (y 0 )Q i;t (s)[g(t; y) ¡ g(s; y 0 )] i U (g(s; y 0 ); t; z) [START_REF] Hairer | The dynamical sine-Gordon model[END_REF] and

X k Z y P k;x (y)[g(t; y)( k [; (g; )]U )(t; x) ¡ ( k [@ t ; (g; )]U )(t; x)] = R 2 (t; x)
with the denition

R 2 (t; x) := X k;i Z y;y 0 ;s P k;x (y)K k;x (z)Q i;t (s)g(t; y)P i;z (y 0 ) i U (g(s; y 0 ); t; z) +2 X k;i Z y;y 0 ;s P k;x (y)K k;x (z)Q i;t (s)g(t; y)rP i;z (y 0 )r i U (g(s; y 0 ); t; z) ¡ X k;i Z y;y 0 ;s P k;x (y)K k;x (z)@ t Q i;t (s)P i;z (y 0 ) i U (g(s; y 0 ); t; z): (23)
Indeed:

([@ t ; (g; )]U )(t; x) = X i Z y;s (@ t Q i;t )(
j q R 1 (t; x)j . X k q 2 ¡(¡")k kg k C T /2 C 0 + 2 ¡k kg k C T 2 (2¡)k kU k C C T :
By the spectral support properties of the commutators we have that

k[; (g; )]U k C T + ¡2 . kg k C T kU k C 1 C T ; k q [@ t ; (g; )]U k C T L 1 . 2 (2+"¡¡ )q kg k C T /2 C 0 + 2 (2¡¡ )q kg k C T kU k C 1 C T :
This yields

kR 2 k C T + ¡2¡" . (1 + kgk C T L 1)kg k L T kU k C 1 C T :
We have so far proved [START_REF] Hairer | Singular stochastic PDEs[END_REF] and then [START_REF] Hairer | A class of growth models rescaling to KPZ[END_REF] follows by continuity. The local Lipshitz dependence on g can be obtained via similar computations.

Remark 8. If f does not depend on we consider the parametric problem

(@ t ¡ )U f (; t) = f ; U f (; 0) = 0; 2 [; 1]; (25)
which is solved by

U f (; t) = Z 0 t e (t¡s) f ds: Remark that @ U f (; t) = Z 0 t e (t¡s) (t ¡ s)f ds and @ 2 U f (; t) = Z 0 t e (t¡s) (t ¡ s) 2 2 f ds:
We have, thanks to the well-known Schauder estimates of Lemma 21 (since > ):

kU f k C 2 L T := sup n=0;1;2 sup 2[;1] k@ n U f ()k L T . kf k C T ¡2 ( 26
)
We dene then

u(t; x) := (g; U f )(t; x) (27)
and observe that u(t; x) is an approximate solution of equation [START_REF] Gubinelli | Lectures on singular stochastic PDEs[END_REF], indeed

(@ t ¡ g )u = (g; L (g; U f )) = (g; L U f ) ¡ (g; U f ) = f ¡ (g; U f )
and the estimation in Lemma 7 together with the bound (26) yield immediately the following inequality:

k (g; U f )k C T + ¡2¡" . kg k L T (1 + kg k C T L 1)kf k C T ¡2 : (28)

Paracontrolled Ansatz

In order to give a meaning to the PDE in [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF] with initial condition u 0 2 C , our initial goal will be to get informations on solutions = (g) of the equation

@ t ¡ g = ; for a xed g 2 C T , 2/3 < < 1, g 2 [; 1]
. Using the results of Section 2.2, we consider to this eect the parametric problem

(@ t ¡ )#(; t) = ;
for 2 [; 1]. We will consider the stationary solution of this problem which has the form

#(; x) = Z 0 1 e s ds = (¡) ¡1 (29)
and in order for [START_REF] Lyons | Dierential Equations Driven by Rough Paths: Ecole d'Eté de Probabilités de Saint-Flour XXXIV-2004[END_REF] to be well dened we impose that the noise has zero mean on T 2 (this is a simplifying assumption which can be easily removed, e.g. at the price of adding a linear term to the equation). We can control (29) by bounding its Littlewood-Paley blocks with a Bernstein lemma for distributions with compactly supported Fourier transform ([3], Lemma 2.1) to obtain:

k#k C 2 L T = k#k C 2 C T . k k C ¡2 : ( 30
)
We dene now for every t 2 [0; T ] (t; x) := (a(u); #):

Thanks to Lemma 3 we have the bound kk L T . k#k C L T . k k C ¡2 .
We observe that this denition together with Lemma 7 gives

@ t ¡ a(u) = ¡ (a(u); #) with k (a(u); #)k C T 2¡2¡" . ka(u)k L T 2 k k C T ¡2
. We expect then (a(u); #) to be bounded in C T 2¡2¡" for any " > 0. At this point let us introduce the Ansatz

u = + u ] : (31)
Remark 9. Notice that we are not making any assumption on the existence of such u, which is the subject of Section 4. Our aim here is to nd the equation that a couple (u; u ] ) 2 C T C T 2 verifying (31) must solve, in order for u to solve [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF].

Observe that @ t u ¡ a(u) u = (@ t ¡ a(u) ) + (@ t ¡ a(u) )u ] = + (@ t ¡ a(u) )u ] ¡ (a(u); #):

It follows that u ] must solve ( (@ t ¡ a(u) )u ] = (u) + (a(u); #) u ] (t = 0) = u 0 ] := u 0 ¡ (a(u 0 ); #)(t = 0) 2 C ( 32
)
with (u) = a(u) u + a(u) u, and if we can make sense of the resonant term a(u) u, it is reasonable to expect u ] (t; ) 2 C 2 8t 2 (0; T ]. Indeed, take U ] := U Q to be the solution of

L U ] () := (@ t ¡ )U ] () = Q U ] (; t = 0) = 0 (33)
for some Q = Q(u ] ) to be determined and 2 [; 1]. Using again Lemma 7 as shown in Remark 8 we have

(@ t ¡ a(u) ) (a(u); U ] ) = Q(u ] ) ¡ (a(u); U ] ):
For 2 [; 1] we dene P t u 0 ] () := e t u 0 ] so that L (P t u 0 ] ) = 0, with L as in (18). We set u ] := (a(u); U ] ) + (a(u); P u 0 ] ) : (34)

Taking

Q(u ] ): =(u) + (a(u); #) + (a(u); U ] ) + (a(u); P u 0 ] ) ;
we obtain that U ] solves equation [START_REF] Otto | Quasilinear SPDEs via rough paths[END_REF] if and only if u ] solves equation [START_REF] Mourrat | Global well-posedness of the dynamic $\Phi4_3$ model on the torus[END_REF]. As we will see, Q(u ] )(t) belongs to C 2¡2 8t 2 (0; T ] but not uniformly as t ! 0. However it belongs to C ¡2 uniformly as t ! 0. It remains to control the resonant term a(u) u appearing in (u). We have

a(u) u = a(u) + a(u) u ] : By paralinearization (see Theorem 25) a(u) = a 0 (u) u + R a (u) with kR a (u)k C T 2 . 1 + kuk C T 2
, and then

a(u) = (a 0 (u) u) + R a (u) :
In order to use the commutator lemma (Lemma 26) we can estimate a 0 (u), recalling that 2 (0; 1), as

ka 0 (u)k C T . ka 00 k L 1ku k C T
and write

a(u) = a 0 (u)(u ) + C(a 0 (u); u; ) + R a (u) :
Then, Ansatz [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF] gives

a(u) = a 0 (u)( ) + a 0 (u)(u ] ) + C(a 0 (u); u; ) + R a (u) :
Summarizing, we have:

(u) = a 0 (u)( ) + a(u) u + a 0 (u)(u ] ) +C(a 0 (u); u; ) + R a (u) + a(u) u ]
Thanks to the nonlinear commutator (Lemma 5), we can decompose the resonant term to obtain

(u) = a(u) u + a 0 (u)(u ] ) + C(a 0 (u); u; ) + R a (u) + a(u) u ] +a 0 (u)(a(u); #) + a 0 (u) } (a(u); 2 )
and (a(u); #) 2 C T 3¡2¡" if u 2 L T . Here we dened

2 (; x) := (# #)(; x) = X ij i #(; )(x) j [#(; )](x) (35)
Finally, recalling the decomposition of u ] in two terms (34) we obtain

(u) = a 0 (u) } (a(u); 2 ) + 1 (u) + 2 (u)
where

1 (u) := a(u) u + C(a 0 (u); u; ) + R a (u) + a 0 (u)(a(u); #) +a 0 (u)( (a(u); U ] ) ) + a(u) (a(u); U ] ) ; 2 (u) := a 0 (u)( (a(u); P u 0 ] ) ) + a(u) (a(u); P u 0 ] ):
Thanks to Lemma 2 the terms a 0 (u) } (a(u); 2 ) and 1 (u) can be estimated in Section 5). On the other hand the term 2 (u)(t) can be estimated in C 2¡2 only for strictly positive times t > 0 due to the lack of regularity of the initial condition u 0 ] which a priori lives only in C . Note moreover that the specic form of allows to deduce that if we replace 2 by ~2 = 2 ¡ H with H 2 C 2 C T 2¡2 then this is equivalent to consider an equation for u of the form @ t u(t; x) ¡ a(u(t; x))u(t; x) = (x) ¡ a 0 (u(t; x))H(a(u(t; x)); t; x): Let us resume this long discussion in the following theorem:

C T 2¡2 , provided 2 2 C 2 C T 2¡2 (see
Theorem 10. Assume that 2 C 0 ; u 0 2 C 2 ; H 2 C 2 C T 0 . u 2 C T 1 C 2 is the classical solution to the equation @ t u(t; x) ¡ a(u(t; x))u(t; x) = (x) ¡ a 0 (u(t; x))H(a(u(t; x)); t; x); u(0) = u 0 ; (36) up to time T > 0 if u = (a(u); # + U ] + P u 0 ] );
where # is the solution to eq. ( 29) and U ] is the solution to the PDE

(@ t ¡ )U ] () = F (u; U ] ; u 0 ] ) U ] (; 0) = 0 2 [; 1] (37)
with

F (u; U ] ; u 0 ) = a 0 (u) } (a(u); 2 ) + 1 (u) + 2 (u) + (a(u); #) + (a(u); U ] ) + (a(u); P u 0 ]
)

and 2 = # # ¡ H. Denition 11. For any 2 R we dene X C 2 C C 2 C 2¡2 the closure of the image of the map (; H) 2 C 2 C 2 C 2 C 0 7 ! J (; H) = (; ¡ H) 2 C 2 C 2 C 2 C 0 (in the topology of C 2 C C 2 C 2¡2 ).
We call the elements in X enhanced noises. In the next section we will exploit the space X for 2/3 < < 1 to solve equations (37) and (31).

Local wellposedness

The main result of this section is the local wellposedness for equations [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF] and (37) when (#; 2 ) 2 X and u 0 2 C for 2/3 < < 1. This yields a unique solution to (36), thanks to Theorem 10. Theorem 12. Let > 2 /3: Then for any (#; 2 ) 2 X and u 0 2 C there exists a time T > 0 depending only on k(#; 2 )k X and ku 0 k up to which the system of equations ( 31) and ( 37) has a unique solution

(u; U ] ) 2 L T C 2 L T 2
for all < such that 2 + > 2. For any xed > 0 there exist a ball B C X such that the solution map

: (u 0 ; #; 2 ) 2 B 7 ! (u; U ] ) 2 L C 2 L 2
is well dened and Lipshitz continuous in the data.

Remark 13. The proof is based on a Picard xed point argument. In order to obtain a small-time contraction we carry on our estimates of U ] in the space

C 2 L T 2 C 2 L T 2 .

Proof. (Theorem 12)

Let G T = L T C 2 L T 2 .
We introduce the map

¡: (u; U ] ) 2 G T 7 ! (¡ u (u; U ] ); ¡ U ](u; U ] )) 2 G T by ¡ u (u; U ] ) := (a(u); #) + (a(u); ¡ U ](u; U ] )) + (a(u); P u 0 ] )
and

(@ t ¡ )¡ U ](u; U ] )() = F (u; U ] ; u 0 ] ); ¡ U ](u; U ] )()(0) = 0; 2 [; 1];
We will establish that this map is a contraction in the space G T : First, we have to show that there exists a ball B G T such that ¡(B) B. We have the bound

kP u 0 ] k C 1 L T . ku 0 ] k C T .
It is easy to obtain, using the estimates of Section 2 and Lemma 21:

Z 0 T e ¡(t¡s) [ 1 (u) + (a(u); #) + (a(u); U ] ) + (a(u); P u 0 ] )] s ds C 2 L T 2 . T (1 + kuk L T ) 4 (1 + k k C ¡2) 2 ku 0 ] k C ¡ 1 + kU ] k C 2 L T 2
for some > 0.

By the assumption that (#; 2 ) 2 X we deduce that there exists M > 0 such that

k 2 k C 2 C T 2¡2 6 M . We have Z 0 T e ¡(t¡s) [a 0 (u) } (a(u); 2 )] s ds C 2 L T 2 . T ¡ (1 + kuk C T ) 2 k 2 k C 2 C T 2¡2 :
To bound the term 2 (u) we observe that

kP t u 0 ] k C 2 C 2 . t ¡ 2 ku 0 ] k C thanks to Lemma 21. This gives Z 0 T e ¡(t¡s) 2 (u) s ds C 2 L T 2 . T ¡ (1 + kuk C T )(1 + k k C ¡2)ku 0 ] k C and then ¡ U ](u; U ] ) is bounded in C 2 L T 2
for T small enough. We have also

k¡ u (u; U ] )k L T . k k C ¡2 + ku 0 ] k C + k¡ U ](u; U ] )k C L T . k k C ¡2 + ku 0 ] k C + T 2 ¡ 2 k¡ U ](u; U ] )k C 2 L T 2
and these bounds show that ¡(B) B. The contractivity of ¡ U ](u; U ] ) can be obtained in the same way. Now consider ¡ u (u; U ] ): we have

k (a(u 1 ); U 1 ] ) ¡ (a(u 2 ); U 2 ] )k L T . T 2 ¡ 2 ¡ kU 1 ] ¡ U 2 ] k C L 2 + ku 1 ¡ u 2 k C T L 1kU 2 ] k C 1 L 2 while for the other terms in ¡ u (u 1 ; U 1 ] ) ¡ ¡ u (u 2 ; U 2 ] ) we remark that sup s2[0;t] ku 1;s ¡ u 0 ¡ u 2;s + u 0 k L 1 . t "/2 ku 1 ¡ u 2 k C [0;t] "/2 L 1 :
Then 80 < " < , using Lemma 3 and Lemma 22:

k (a(u 1 ); #) ¡ (a(u 2 ); #)k L T . ka(u 1 ) ¡ a(u 2 )k C T L 1k# k C 1 L T . ku 1 ¡ u 2 k C T L 1k k C ¡2 . T "/2 ku 1 ¡ u 2 k C T "/2 L 1 k k C ¡2 . T "/2 ku 1 ¡ u 2 k L T k k C ¡2 :
With the same reasoning we estimate k (a(u 1 );

P u 0 ] ) ¡ (a(u 2 ); P u 0 ] )k L T . T "/2 ku 1 ¡ u 2 k C T "/2 L 1 kP u 0 ] k C 1 L T . T "/2 ku 1 ¡ u 2 k L T ku 0 ] k C T
and then ¡ is a contraction for small times. The uniqueness of the solution

(u; U ] ) 2 L T C 2 L T 2
and the Lipshitz continuity of the localized solution map can be proved along the same lines via standard arguments.

Renormalization

At this point we want to construct an enhanced noise associated to the white noise . Already in the standard setting of the generalised PAM model with constant diusion matrix, the construction of the enhancement requires a renormalization since the resonant product # # is not well dened.

Let 2 S(T 2 ) be a cuto function and let " (x) = " ¡2 (x / "). Then dene a regularised noise by " = " and let # " = (¡) ¡1 " . Notice that

H " () := E[# " (; x) # " (; x)] = E[# " (; x)# " (; x)] = ¡ X k2Z 2 nf0g ^"(k) 2 2 jkj 2 = ¡ " 2
where

" := X k2Z 2 nf0g ^"(k) 2 jkj 2 ' jlog "j
as " ! 0. Subtracting the diverging quantity H " to # " # " and then taking the limit as " ! 0 delivers a nite result.

Theorem 14. Take < 1 and let " = ( " ; 2;" ): =( " ; # " # " ¡ H " ). Then the family ( " ) " X converges a.s. and in L p to a random element = (; 2 ) 2 X .

Proof. The proof is a mild modication of the proof for PAM [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. In order to establish the required C 2 C T 2¡2 regularity for 2 we follow the computations for the case where the diusion coecient is constant. We have only to discuss the additional regularity in the parameter. In order to do so observe that 2;" () = X i j

J i # " () j # " ()K
where JK denotes the Wick product with respect to the Gaussian structure of . Then we have

@ 2;" () = X ij J i @ # " () j # " ()K + X ij J i # " () j @ # " ()K; and @ 2 2;" () = X i j J i @ 2 # " () j # " ()K + X ij J i # " () j @ 2 # " ()K + X ij 2J i @ # " () j @ # " ()K:
Now the computations relative to the regularities of these additional stochastic objects are equivalent to those for the term 2;" where one or two instances of # " () are replaced by Gaussian elds of similar regularities of the form @ # " () and @ 2 # " (), a direct inspection of the proof allows us to deduce that we have almost sure C 2¡2 regularity for these terms and also for random elds @ n 2;" for any nite n. This allows also to deduce that the random eld is a.s. smooth in the parameter . Similar computations allow to prove continuity in " for " > 0. The rest of the proof is standard.

In conclusion we see that in order to be able to use this convergence result we need to modify our approximate PDE and consider instead

@ t u " ¡ a(u " )u " = " ¡ a 0 (u " )H " (a(u " ))
which gives the renormalised equation (4).

Our wellposedness results for the paracontrolled formulation of this equation together with the convergence result in Theorem 14 allow to deduce that u " ! u in C T for any 2 / 3 < < < 1 and that the limiting process u satises a modied version of eq. ( 1), namely

@ t u ¡ a(u)u = ; u(0) = u 0 ;
where a(u)u denotes a renormalized diusion term given by a(u)u := a(u) u + a 0 (u) } (a(u); 2 ) + 1 (u) + 2 (u):

(38)

Nonlinear source terms

Let us start by discussing the presence of a u dependent r.h.s. in eq. ( 1). We want to solve

@ t u ¡ a 1 (u)u = a 2 (u)
where a 1 is a non-linear diusion coecient as before and a 2 : R ! R is another bounded function with suciently many bounded derivatives. We rewrite this equation as

(a(u); L )u = a 2 (u) + a 1 (u) u + a 2 (u) + a 1 (u) u + a 2 (u)
where now a(u) = (a 1 (u); a 2 (u)) is a vector valued non-linearity. Since we don't need u to depend on any parameter = ( 1 ; 2 ), we have dened L as L () := @ t ¡ 1 and used the identity (a(u); L )u = (@ t ¡ a 1 (u) )u, similarly to what we have done in [START_REF] Hairer | A theory of regularity structures[END_REF].

Notice that the nonlinear paraproduct can be extended trivially to the vector valued case in such a way that, for example,

((g 1 ; g 2 ); h)(t; x) = X i Z y;s
Q i;t (s)P i;x (y)( i h((g 1 (s; y); g 2 (s; y)); t; ))(x):

As before we make the Ansatz

u = (a(u); #) + u ]
where now # solves L ()#() = (@ t ¡ 1 )#() = 2 ;

for = ( 1 ; 2 ) 2 [; 1] [¡L; L]
where L is a large but xed constant. The bounded domain is important to be able to have uniform estimates and reuse the estimates proved above in the simple situation of 2 = 1. The solution of this equation is

#(; ) = 2 Z 0 1 e 1 s ds = ¡ 2 1 ¡1 :
Observe that

(a(u); L )u = (a(u); L ) (a(u); #) + (a(u); L )u ]
and recall that (Lemma 7) (a(u); L ) (a(u); #) = (a(u); L #) + (a(u); #). Now

(L #)() = (@ t ¡ 1 )#(; t; x) = (); = ( 1 ; 2 ) 2 [; 1] [¡L; L]
with ()(t; x) = 2 (x) and then (a(u); L #) = (a(u); ) = a 2 (u) : In conclusion (a(u); L ) (a(u); #) = a 2 (u) + (a(u); #) and the equation for u ] reads

(a(u); L u ] ) = a 1 (u) u + a 2 (u) + [a 2 (u) ¡ a 2 (u) ] +a 1 (u) u + a 2 (u) ¡ (a(u); #)
where now all the terms on the r.h.s. can be treated as remainder terms. Let us just remark that the commutation term a 2 (u) ¡ a 2 (u) has a standard treatment via Lemma 27. Of course, the rst two terms require to be treated as resonant terms. Note that, modulo terms of order C T 3¡2 (or E /2 C 2¡2 as dened in Lemma 21) the terms a 1 (u) u + a 2 (u) are equivalent to

a 1 0 (u) } (a(u); # #) + a 2 0 (u)( (a(u); #) )
and that by computations similar to those of the previous sections one can prove that

( (a(u); #) ) = } (a(u); # ) + C T 3¡2
so the resonant terms are comparable to the sum of the two terms

a 1 0 (u) } (a(u); # #) + a 2 0 (u) } (a(u); # )
which require renormalization of the form

a 1 0 (u)a 2 (u) 2 a 1 (u) 2 " ¡ a 2 0 (u)a 2 (u) a 1 (u) " ( 39
)
and the convergence follows with the same arguments of Section 5.

We remark that the structure of the second renormalisation term, which is due to the r.h.s. in the equation, is the same of that found by Bailleul, Debussche and Hofmanova in [START_REF] Bailleul | Quasilinear generalized parabolic anderson model equation[END_REF].

Remark 15. Our approach works straightforwardly for the equation

@ t u(t; x) ¡ a ij (u(t; x))@ ij 2 u(t; x) = g(u(t; x))
with a: R ! M 2 (R) such that

P i;j a(u) ij x i x j > C jxj 2 8x 2 R 2 for C > 0 and @ ij 2 := @ 2 @ x i @x j .
To see that, let a(u) := (a ij (u); g(u)) 2 R 5 and = ( i;j ; g ) 2 R 5 . Let L () := @ t ¡ ij @ ij 2 and () := g with the uniform ellipticity condition

P i;j ij x i x j > C jxj 2 8x 2 R 2 .
It is easy to verify that Lemma 7 and Lemma 5 hold within this setting, just considering nonlinear paraproducts for functions depending on 5 parameters. We have then:

u = (a(u); # + U ] + P u 0 ] )
with #() stationary solution of L #() = (), P t u 0 ] := e i j @ ij 2 t u 0 ] and U ] () which solves

L U ] () = } ((a(u); a 0 (u)); 1 ) + } ((a(u); a 0 (u)); 2 ) + Q(u; U ] ) with Q(u; U ] ) 2 C 2¡2¡" , 1 2 C 2 C 0 2 C 2¡2 = #() ij 0 @ ij 2 #(), 2 (; 0 ) = #() g 0 and U ] (t = 0) = 0. Note that we can write # as #() = g Z 0 1 e t i j @ i j 2 dt # ^(k) = g ^(k) ij k i k j ; k 2 Z 2 nf0g:
From the uniform ellipticity condition we have that k#k C k C . k k C , and Schauder estimates analogous to those of Lemma 21 hold as well.

Now consider the renormalization. We have

H 1 " (; 0 ) := E( 1 (; 0 )) = ¡ g 2 X k2Z 2 nf0g ^"(k) 2 P i;j ij 0 k i k j ( P i;j ij k i k j ) 2 ; H 2 " (; 0 ) := E( 2 (; 0 )) = g g 0 X k2Z 2 nf0g ^"(k) 2 P i;j ij k i k j : The convergence of 1 " ¡ H 1 " , 2 " ¡ H 2 " in C (; 0 ) k C 2¡2 (T 2
) can be obtained with the techniques used in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], Section 5.2.

Full generality

Within the framework of the present work we are actually able to treat equations of the form

@ t u(t; x) ¡ a 1 (u(t; x))u(t; x) = (a 2 (u(t; x)); x) (40)
where ( 2 ; x) is a Gaussian process with covariance

E[( 2 ; x)( ~2; x ~)] = F ( 2 ; ~2)(x ¡ x ~)
where F is a smooth covariance function. Let as before 2/3 < < 1. In this case we can take as a parametric equation L ()# := @ t #(; t; x) ¡ 1 #(; t; x) = ( 2 ; x) whose solution # is a Gaussian process, smooth with respect to the variable = ( 1 ; 2 ) which we assume taking value in a compact subset of R 2 for which 1 > > 0 with xed . Letting a(u) = (a 1 (u); a 2 (u)) we can rewrite the l.h.s. of eq. ( 40) in the form @ t u ¡ a 1 (u)u = } (a(u); L u) and the r.h.s. as (a 2 (u(t; x)); x) = } (a(u); )

where (; x) = ( 2 ; x). Now we perform the paraproduct decomposition to get (a(u); L u) ¡ (a(u); ) = (a(u); ) + (a(u); Du) + (a(u); ) + (a(u); Du):

with D() := 1 . Let P t () := e t 1 as before, and invoke the paracontrolled Ansatz in the usual form u = (a(u);

# + U ] + P u 0 ] ):
Using that (a(u); L (a(u);

# + U ] + P u 0 ] )) = (a(u); L (# + U ] + P u 0 ] )) + (a(u); # + U ] + P u 0 ] )
and observing that we can take L # = and that L P u 0 ] = 0 to get

(a(u); L U ] ) = F (u; U ] )
where F (u; U ] ) = (a(u); ) + (a(u); Du) + (a(u); ) + (a(u); Du) +[ (a(u); ) ¡ (a(u); )] ¡ (a(u);

# + U ] + P u 0 ] ) which is solved by U ] satisfying L U ] = F (u; U ] ):
Indeed (a(u); F (u; U ] )) = F (u; U ] ), since F (u; U ] ) does not depend on the additional parameter. Remark that the term (a(u); ) ¡ (a(u); ), which does not appear in the simpler case, can be treated with Lemma 28.

It remains now to discuss the handling of the resonant products under the paracontrolled assumption, namely (a(u); ) and (a(u); Du). Next lemma is a paralinearization result adapted to our non-linear context. and observe that the rst term is equal to u ((a(u); a 0 (u)); DZ) while the second term can be easily estimated in C T +2 .

Using this result and Lemma 28 we can expand (a(u); ) = u ((a(u); a 0 (u)); D)

+ C T 3¡2 = (a(u); #) ((a(u); a 0 (u)); D) + C T 3¡2 = } ((a(u); a 0 (u)); # D) + C T 3¡2
and similarly, noting that ((a(u); a 0 (u)); (DD)u) = ((a(u); a 0 (u)); (DD) (a(u); #)) + C T L U ] = } ((a(u); a 0 (u)); # D + # (DD)#) + C T 3¡2 :

This equation can be solved essentially as we did in the simpler context. We see that the general enhancement has the form

(; # D + # (DD)#)
which of course will require renormalization like we did before. In particular

(# D + # (DD)#)(; 0 ) = #() 2 0 @ 2 ( 2 ; ) + #() 1 0 #() = ¡ 2 0 1 ( ¡1 ( 2 ; )) @ 2 ( 2 ; ) + 1 0 1 2 ( ¡1 ( 2 ; )) ( 2 ; )
where we used that 1 #() = ¡( 2 ; ). Now observe that E[( ¡1 " ( 2 ; )) " ( 2 ; )] = ¡F ( 2 ; 2 ) " and that E[( ¡1 " ( 2 ; )) @ 2 " ( 2 ; )] = ¡(@ 1 F )( 2 ; 2 ) " with @ 1 F denoting the derivative with respect to the rst entry.

In the end the renormalized enhanced noise is obtained as the limit in X of ( " ; 2;" ) where 2;" (;

0 ) = ¡ 2 0 1 ( ¡1 " ( 2 ; )) @ 2 " ( 2 ; ) + 1 0 1 2 ( ¡1 " ( 2 ; )) " ( 2 ; ) ¡ H " (; 0 ) with H " (; 0 ) = 2 0 1 (@ 1 F )( 2 ; 2 ) " ¡ 1 0 1 2 F ( 2 ; 2 ) " :
We remark that if we take F ( 2 ; ~2) = 2 ~2 we reobtain the situation treated in Section 6, indeed in this case } ((a(u); a 0 (u)); H " ) = a 2 0 (u)a 2 (u)

a 1 (u) " ¡ a 1 0 (u)a 2 (u) 2 a 1 (u)
; x 0 )] = F (; 0 )Q(t ¡ t 0 ; x ¡ x 0 )
with F a smooth function and Q a distribution of parabolic regularity > ¡4/ 3. First note that the coecient a 1 (u) 2 [; 1] in front of the time derivative can be eliminated trivially by dividing. In order to handle the time dependence of the noise, the framework of this paper will still apply, provided we consider spacetime paraproducts instead of paraproducts which act only on the space variable. This can be done exactly following the lines of the paper [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] where time paraproducts were considered in the paracontrolled approach to solutions to SDE driven by gaussian signals.

The constraint of regularity > ¡4/3 does allow to treat a noise which is white in time and smooth in space, but not a spacetime white noise. It is well known that the rst order paracontrolled approach on which the present paper is based does not allow to treat this kind of irregular signals in full generality.

We also use the notation

S j f = X i< j ¡1 i f
and notice that j f (x) = Z K j ;x (y)f (y)dy; S j f (x) = Z P j ;x (y)f (y)dy with K j ;x (y) = 2 dj K(2 j (x ¡ y)), P j ;x (y) = P i<j ¡1 K i;x (y), K radial with zero mean. Every dyadic block has a compactly supported Fourier transform and it belongs therefore to C 1 . It is easy to see that f = P j >¡1 j f = lim j!1 S j f for all f 2 D 0 . For 2 R, the Hölder-Besov space C is given by C = B 1;1 (T d ; R), where for p; q 2 [1; 1] we dene

B p;q = B p;q (T d ; R) = 8 < : f 2 D 0 : kf k B p; q = X j>¡1 (2 j k j f k L p ) q ! 1/q < 1 9 = ; ;
with the usual interpretation as `1 norm in case q = 1. Then B p;q is a Banach space and while the norm kk B p; q depends on (; ), the space B p;q does not, and any other dyadic partition of unity corresponds to an equivalent norm. If 2 (0; 1) n N, then C is the space of bc times dierentiable functions whose partial derivatives of order bc are ( ¡ bc)-Hölder continuous (see page 99 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]). Note however, that for k 2 N the space C k is strictly larger than C k , the space of k times continuously dierentiable functions.

The following lemma gives useful characterisation of Besov regularity for functions that can be decomposed into pieces which are localized in Fourier space.

Lemma 19.

1. Let A be an annulus, let 2 ℝ, and let (u j ) be a sequence of smooth functions such that F u j has its support in 2 j A , and such that ku j k L 1 . 2 ¡j for all j. Then u = X j>¡1 u j 2 C and kuk . sup j >¡1 f2 j ku j k L 1g:

2. Let B be a ball, let > 0, and let (u j ) be a sequence of smooth functions such that F u j has its support in 2 j B , and such that ku j k L 1 . 2 ¡j for all j. Then u = X j>¡1 u j 2 C and kuk . sup

j >¡1 f2 j ku j k L 1g:
The Bernstein inequalities of the next lemma are extremely useful when dealing with functions with compactly supported Fourier transform. Lemma 20. Let A be an annulus and let B be a ball. For any k 2 N 0 , > 0, and 1 6 p 6 q 6 1 we have that

1. if u 2 L p is such that supp(F u) B , then max 2N d :jj=k k@ uk L q . k k+d 1 p ¡ 1 q kuk L p ; 2. if u 2 L p is such that supp(F u) A , then k kuk L p . k max 2N d :jj=k k@ uk L p :
We recall the following standard heat kernel estimations (see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF], Chapter 2). t¡s) v s ds and P t u 0 = e t u 0 , with 8), [START_REF] Bony | Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires[END_REF] and introduce the norm

Lemma 21. (Schauder estimates)

Let V t = R 0 t e (
> . We dene L T and C k L T , C k C T for k 2 N as in (
kvk E T C = sup t2[0;T ] t kv(t; )k C :
Then for any 2 [0; 1) and 2 R:

sup t2[0;T ] t kV t k C k C ¡2 . sup t2[0;T ] t + kv t k C ¡2 8 2 [0; 1) kV k C k L T ¡2 . T kvk C T ¡2 8 2 [0; 1) kV k C k L T ¡2 . T ¡ kv k E T C ¡2 8 2 [0; 1); 8 2 [0; ] kV k C k L T . T ¡ 2 +1¡ kv k E T C 8 2 [ ¡ 2; ); 8 2 h 0; ¡ 2 + 1 i kP t u 0 k C k C . t ¡ ¡ 2 ku 0 k C 8 < kP t u 0 ¡ P s u 0 k C k C . s ¡ jt ¡ sj ¡ 2 + ku 0 k C 8t = / s 2 R + ; 6 + 2; 2 0; 1 ¡ ¡ 2
We need the following interpolation lemma:

Lemma 22. Let 2 (0; 2), 0 < " < and u 2 L T . Then kuk C T /2¡"/2 L 1 . kuk L T Proof. sup s= / t ku t ¡ u s k L 1 jt ¡ sj /2¡"/2 6 sup s= / t " X i6n k i u t ¡ i u s k L 1 jt ¡ sj /2¡"/2 + X i>n k i u t ¡ i u s k L 1 jt ¡ sj /2¡"/2
# and choosing 2 ¡n¡1 6 jt ¡ sj 1/2 6 2 ¡n we obtain

X i<n k i u t ¡ i u s k L 1 jt ¡ sj /2¡"/2 . kuk C T /2 C 0 X i6n jt ¡ sj "/2 X i>n k i u t ¡ i u s k L 1 jt ¡ sj /2¡"/2 . kuk C T X i>n 2 ¡i 2 ¡(+")n
and this gives the result.

Terms of the type ka(u(t; x))k C 0 with a: R ! R cannot be estimated directly with their Hölder norm. In the following lemma we note some bounds used in Section 4. The third term can be estimated as the rst one when k > ¡1. Otherwise we just bound it as Z z;w; 2[0;1]

K i;x (z) X j6N
[ j u(t; ) ¡ j u(s; )](z) K ¡1;z (w)a 0 ( u sw tw ) . ku t ¡ u s k C 0ka 0 k L 1:

For the three terms together we have the bound

ka(u t ) ¡ a(u s )k C 0 . ku t ¡ u s k C 0(ka 0 k L 1 + ka 00 k L 1ku k C T )
With the same technique we obtain

ka(u 1 ) ¡ a(u 2 )k C T /2 C 0 . ka 0 k L 1ku 1 ¡ u 2 k C T /2 C 0 +ka 00 k L 1ku 1 ¡ u 2 k C T /2 C 0 ku 1 ¡ u 2 k C T +ka 000 k L 1ku 1 ¡ u 2 k C T /2 C 0 ku 1 ¡ u 2 k C T 2
and this gives the second estimate. The third one can be obtained easily.

A.1. Bony's paraproduct.

In terms of LittlewoodPaley blocks, the product f g can be decomposed as f g = X j>¡1 X i>¡1 i f j g = f g + f g + f g:

Where the paraproducts f g and f g and the resonant product f g are dened as

f g = g f := X j >¡1 X i=¡1 j ¡2
i f j g and f g := X ji¡j j61 i f j g:

We will often use the shortcuts P i j for P ji¡j j61 and P i.j for P i<j ¡1 . Of course, the decomposition depends on the dyadic partition of unity used to dene the blocks j , and also on the particular choice of the pairs (i; j) in the diagonal part. The choice of taking all (i; j) with ji ¡ j j 6 1 into the diagonal part corresponds to property 2 in our denition of dyadic partitions of unity.

Bony's crucial observation [START_REF] Bony | Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires[END_REF][START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF] is that the paraproduct f g (and thus f g) is always a well-dened distribution. Heuristically, f g behaves at large frequencies like g (and thus retains the same regularity), and f provides only a frequency modulation of g. The only diculty in constructing f g for arbitrary distributions lies in handling the resonant product f g. The basic result about these bilinear operations is given by the following estimates. Bony proved also a basic paralinearisation result, soon after improved by Meyer. We give here a particular version suited to our purposes. Theorem 25. Let 2 (0; 1), f 2 (C ) k and F 2 C 3 (R k ; R) then

R F (f ) := F (f ) ¡ F 0 (f ) f 2 C 2 with kR F (f )k C 2 . kF k C 2(1 + kf k C ) 2 :
Moreover the map f 7 ! R F (f ) is locally Lipshitz and

kR F (f ) ¡ R F (f ~)k C 2 . kF k C 3(1 + kf k C + kf ~kC ) 2 kf ~¡ f k C :
The additional key ingredient at the core of the paracontrolled approach is the following commutation result proved in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], Lemma 2.4: Lemma 26. Assume that ; ; 2 R are such that + + > 0 and + < 0. Then for f ; g; h 2 C 1 the trilinear operator C(f ; g; h) := ((f g) h) ¡ f (g h)

3¡2=(

  (a(u); a 0 (u)); (DD)#) + C T 3¡2 where (DD)(; 0 ) = 1 0 , we have (a(u); L u) = u ((a(u); a 0 (u)); (DD)u) + C T 3¡2 = (a(u); #) ((a(u); a 0 (u)); (DD)#) + C T 3¡2 = } ((a(u); a 0 (u)); # (DD)#) + C T 3¡2 Finally the equation for U ] reads

Lemma 23 . 2 C 0

 2320 Let a 2 C b 3 uniformly bounded and u 2 L T = C T \ C T /, then

Theorem 24 .

 24 (Paraproduct estimates) For any 2 R and f ; g 2 D 0 we havekf gk C . kf k L 1kg k C ; (41)and for < 0 furthermorekf g k C + . ; kf k C kgk C : (42)For + > 0 we havekf g k C + . ; kf k C kg k C : (43)

Lemma 2. (Nonlinear paraproduct estimates) Let g

				1); 2 R; + > 0;
	thanks to the following bounds:
	g 2 [; 1], and h 2 C	2 C T	for any 2 R. Then	2 C T	for some 2 (0; 1),

  This shows that (21) holds for smooth functions.With the techniques used in Lemma 6 we can estimate

s)P i;x (y)( i U (g(s; y); t; x));

([; (g; )]U )(t; x) = X i Z y;s Q i;t (

s)P i;x (y)( i U (g(s; y); t; x)) (24) +2 X i Z y;s Q i;t (s)rP i;x (y)(r i U (g(s; y); t; x)):

  ka(u)k L T . kak L 1 + ka 0 k L 1 kuk C T /2 C 0 + kuk C T + ka 00 k L 1ku k C T /2 C 0 kuk C The bound on ka(u)k C T is trivial. We estimate ka(u t ) ¡ a(u s )k C 0 as T . i2 ¡i ka 00 k L 1ku t ¡ u s k C 0ku k C T : sw tw ) . ku t ¡ u s k C 0kP j ;z (w)a 0 ( u sw tw )k L 1 . ku t ¡ u s k C 0ka 0 k L 1

	Z							Z
	z	K i;x (z)[a(u(t; z)) ¡ a(u(s; z))]	=	z; 2[0;1]	K i;x (z)a 0 ( u sz tz )[u(t; z) ¡ u(s; z)]
		6	Z	z;w; 2[0;1]	K i;x (z)	X j.ki	[ j u(t; z) ¡ j u(s; z)] K k;z (w)a 0 ( u sw tw )
			+	Z	z;w; 2[0;1]	K i;x (z)	X k.ji	[ j u(t; z) ¡ j u(s; z)] K k;z (w)a 0 ( u sw tw )
			+	Z	z;w; 2[0;1]	K i;x (z)	X kj &i	[ j u(t; z) ¡ j u(s; z)] K k;z (w)a 0 ( u sw tw )	:
	If k > ¡1 we have Z
	The second term is
				Z			K i;x (z) X
					z;w; 2[0;1]

T

. 1 + kuk L T + kuk L T

2

;

ka(u 1 ) ¡ a(u 2 )k L T . ku 1 ¡ u 2 k L T (1 + ku 1 k C T + ku 2 k C T ) 2 ; ka(u 1 ) ¡ a(u 2 )k C T 0 . ku 1 ¡ u 2 k C T 0 (1 + ku 1 k C T + ku 2 k C T ) : Proof. w K k;z (w)a 0 ( u sw tw ) = Z w K k;z (w)[a 0 ( u sw tw ) ¡ a 0 ( u sz tz )]

and then the rst term above becomes Z z;w; ;

K i;x (z) X j.ki [ j u(t; z) ¡ j u(s; z)] K k;z (w)a 00 ( ( u s t ) z w )[u tz tw ¡ u sz sw ] .ka 00 k L 1ku t ¡ u s k C 0 X j.ki Z w jK k;z (w)jjw ¡ z j kuk C ji

[ j u(t; ) ¡ j u(s; )](z) P j ;z (w)a 0 ( u

Paracontrolled quasilinear SPDEs

Appendix A. Besov spaces

In this Appendix we collect some classical results from harmonic analysis needed in the paper. For a gentle introduction to Littlewood-Paley theory and Besov spaces see the recent monograph [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF], where most of our results are taken from. There the case of tempered distributions on R d is considered. The Schauder estimates for the heat semigroup are classical and can be found in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Gubinelli | Lectures on singular stochastic PDEs[END_REF].

Fix d 2 N and denote by T d = (R/(2Z)) d the d-dimensional torus. We focus here on distributions and SPDEs on the torus, but everything in this Appendix applies mutatis mutandis on the full space R d , see [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. The space of distributions D 0 = D 0 (T d ) is dened as the set of linear maps f from

for all ' 2 C 1 . In particular, the Fourier transform

is dened for all f 2 D 0 , and it satises jF f (k)j 6 jP (k)j for a suitable polynomial P . We will also write f ^(k) = F f (k). Conversely, if (g(k)) k2Z d is at most of polynomial growth, then its inverse Fourier transform

denes a distribution, and we have

To see this, it suf- ces to note that the Fourier transform of ' 2 C 1 decays faster than any rational function (we say that it is of rapid decay). Indeed, for 2 N 0 d we have jk g ^(k)j = jF (@ g)(k)j 6 k@ gk L 1 (T d ) for all k 2 Z d . As a consequence we get the Parseval formula hf ; 'i = (2) ¡d P k f ^(k)' ^(k) for f 2 D 0 and ' 2 C 1 . Linear maps on D 0 can be dened by duality: if A: C 1 ! C 1 is such that for all k 2 N there exists n 2 N and C > 0 with sup jj6k k@ (A')k L 1 6 C sup jj6n k@ 'k, then we set h t Af ; 'i = hf ; A'i. Dierential operators are dened by h@ f ; 'i = (¡1) jj hf ; @ 'i. If ': Z d ! C grows at most polynomially, then it denes a Fourier multiplier

Littlewood-Paley blocks give a decomposition of any distribution on D 0 into an innite series of smooth functions.

Denition 18. A dyadic partition of unity consists of two nonnegative radial functions

, where is supported in a ball B = fjxj 6 cg and is supported in an annulus A = fa 6 jxj 6 bg for suitable a; b; c > 0, such that 1. + P j >0 (2 ¡j ) 1 and 2. (2 ¡j ) 0 for j > 1 and (2 ¡i )(2 ¡j ) 0 for all i; j > 0 with ji ¡ j j > 1.

We will often write ¡1 = and j = (2 ¡j ) for j > 0.

Dyadic partitions of unity exist, see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]. The reason for considering smooth partitions rather than indicator functions is that indicator functions do not have good Fourier properties. We x a dyadic partition of unity (; ) and dene the dyadic blocks

allows for the bound

and can thus be uniquely extended to a bounded trilinear operator

We will need the following two lemmas to compare standard and time-smoothed paraproducts. The rst one has essentially the same proof as [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], Lemma 5.1. Lemma 27. Let 2 (0; 2), 2 R. Then for every " > 0 we have the bound

C 0 khk C T :

The second lemma has a standard proof.