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

M Hoshino
email: hoshino@math.kyushu-u.ac.jp

Paracontrolled calculus and regularity structures (1)

We start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models.

-Introduction

Starting with his groundbreaking work [START_REF] Hairer | A theory of regularity structures[END_REF], M. Hairer has developed with his co-authors [START_REF] Bruned | Algebraic renormalization of regularity structures[END_REF][START_REF] Chandra | An analytic BPHZ theorem for Regularity Structures[END_REF][START_REF] Bruned | Renormalising SPDEs in regularity structures[END_REF] a theory of subcritical singular stochastic partial differential equations (PDEs) that provides now an automated blackbox for the basic understanding of a whole class of stochastic PDEs. Equations of this class all share the common feature of involving ill-defined products of distributions with functions or distributions. The methodology of regularity structures for the study of a given singular stochastic PDE takes its roots in T. Lyons' theory of rough paths, such as reshaped by M. Gubinelli [START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Gubinelli | Ramification of rough paths[END_REF]. It requires first to identify a proper space of enhanced noises. The raw random noise that appears in the equation needs to be lifted into a random noise taking values in that enhanced space. This is typically a probabilistic task, mostly independent of the details of the dynamics under study, once the appropriate space of enhanced noises has been constructed from the equation. (That space happens to be equation-independent in the rough differential equation setting, while it is equation-dependent in a PDE setting.) The lifting task typically involves stochastic or Gaussian calculus in a rough paths setting; it involves the difficult implementation of a renormalisation procedure in the singular stochastic PDE setting. This step somehow takes care of the core problem: defining the product of two random distributions as a random variable rather than taking the product of two realizations of these random variables. These enhanced noises come under the form of a model in regularity structures. This is a deterministic object, and the previous step takes care of constructing a random model. Having a model is somewhat equivalent to having a definition of the product of a number of otherwise possibly ill-defined quantities. A restricted class of spacetime functions or distributions is then described in regularity structures theory under the form of a space-time indexed family of jets describing them locally around each space-time point. Given any choice of model, a consistency relation ensures that coherent jets describe indeed true space-time functions or distributions. This is the role of the reconstruction operator; coherent jets are modelled distributions. It happens then that one can reformulate the formal ill-posed equation into the space of jets as a well-posed, model-dependent, fixed point equation in a well-chosen space of jets. For the random model built from a renormalisation procedure in [START_REF] Chandra | An analytic BPHZ theorem for Regularity Structures[END_REF], the space-time function/distribution associated with the solution of the fixed point equation on the jet space can be shown to be the limit in probability of solutions of a family of well-posed space-time stochastic PDEs driven by regularized noises, as the regularization parameter tends to 0 -this is the content of [START_REF] Bruned | Renormalising SPDEs in regularity structures[END_REF]. The fact that some of the terms in these modified and regularized stochastic PDEs blow up as the regularization parameter goes to 0 is a feature of the singular nature of the initial equation.

Let us emphasize that the multiplication problem is fundamentally dealt with on the ground of the following heuristic argument. If one can make sense of the product of a number of reference quantities, one can make sense of the product of quantities that look like the reference quantities. This is what motivates the introduction of jets on scene.

The choice of a jet space to describe a possible solution to a singular stochastic PDE is not the only possible. As a matter of fact, Gubinelli, Imkeller and Perkowski devised in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] a Fourier-based approach to the study of singular stochastic PDEs whose scope has been extended in [START_REF] Bailleul | Heat semigroup and singular PDEs[END_REF][START_REF] Bailleul | Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations[END_REF][START_REF] Bailleul | Higher order paracontrolled calculus[END_REF]. The heuristic remains the same, but paraproducts are used as a mean of making sense of what it means to look like a reference distribution or function. This choice of representation makes the technical details of paracontrolled calculus rather different from their regularity structures counterparts, and paracontrolled calculus remains to be systematized. Despite that fact, it happens to be possible to make a close comparison between the two settings. We start that comparison in this work by providing an 'explicit' paracontrolled representation of the reconstruction operator. This is the operator that associates to a coherent jet a space-time distribution. All notions and notations in the statement are properly defined below.

Theorem 1. Let a concrete regularity structure T " pT `, T q be given, together with a model M " pg, Πq on it.

(1) One can construct functions rr¨ss M : T Þ Ñ C β 0 pR d q and rr¨ss g : T `Þ Ñ C 0 pR d q, such that rrσss M P C |σ| pR d q, and rrτ ss g P C |τ | pR d q, for every homogeneous σ P T and τ P T `, -all rrσss M , and rrτ ss g are continuous function of the model pg, Πq, and the following holds true. for some rrf τ ss g P C γ´|τ | pR d q. Moreover, the map f Þ Ñ ´rrf ss M , `rrf τ ss g ˘τPB from D γ pT , gq to C γ pR d q ˆśτPB C γ´|τ | pR d q, is continuous. This is the content of Proposition 12 and Theorem 14. Any regularity exponent a P R is allowed in the above statement. The inductive definition of rr¨ss M , Proposition 12, will make it clear that rrσss M can be understood as the 'part' of Πσ of regularity C |σ| pR d q. The quantity rrτ ss g has a similar meaning for the function gpτ q. Theorem 1 provides a much refined version of the paraproduct-based construction of the reconstruction operator from Gubinelli, Imkeller and Perkowski' seminal work [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. Notice that this statement is not related to any problem about singular stochastic PDE. The treatment of such equations involves the additional ingredient of an abstract integration operator and the additional notion of admissible model. We provide an explicit paracontrolled-based parametrization of that set of models under some canonical structure assumptions on the regularity structure. The fact that identity (1.3) holds true with rr¨ss M in place of rr¨ss for any model M " pg, Πq, is part of the proof of item (1) of Theorem 1.

We work throughout with the usual isotropic Hölder spaces. All the results presented here have direct analogues involving anisotropic Hölder spaces, such as required for applications to parabolic singular stochastic PDEs. The proofs of all results are strictly identical. We refrain from putting ourselves in that setting so as not to overload the reader with additional technical details and keep focused on the main novelty. The reader will find relevant technical details in the work [START_REF] Martin | A Littlewood-Paley description of modelled distributions[END_REF] of Martin and Perkowski. No previous knowledge of regularity structures or paracontrolled calculus is needed in this work, that is mostly self-contained, with the exception of elementary facts on paraproducts recalled in Appendix A. We have thus given at few places full proofs of statements that were first proved elsewhere. The material has been organized as follows. Section 2 sets the scene of regularity structures under a convenient form for us: Concrete regularity structures, models and modelled distributions are introduced, together with a number of elementary identities and examples. Theorem 1 is proved in Section 3, while Section 4 takes care of Theorem 2.

Notations -We use exclusively the letters α, β, γ, θ to denote real numbers, and use the letters σ, τ, µ to denote elements of T or T `. We agree to use the shorthand notation s p`q to mean both the statement s and the statement s `.

-Basics on regularity structures

Regularity structures are the backbone of expansion devices for the local description of functions and distributions in R d . The usual notion of local description of a function, near a point x P R d , involves Taylor expansion and amounts to comparing a function to a polynomial centered at x f p¨q »

ÿ k f k pxq p¨´xq k , near x. (2.1)
The sum over k is finite and the approximation quantified. One gets a local description of f near another point x 1 writing

f p¨q » ÿ ďk f k pxq ˆk ˙p¨´x 1 q px 1 ´xq k´ » ÿ ¨ÿ k; ďk f k pxq ˆk ˙px 1 ´xq k´ 'p¨´x 1 q . (2.2)
A more general local description device involves an R d -indexed collection of functions or distributions pΠ x τ qp¨q, with labels in a finite set B " tτ u. Consider the real vector space T spanned freely by B. Functions or distributions are locally described as

f p¨q » ÿ τ f τ pxqpΠ x τ qp¨q, near each x P R d .
This implicitely assumes that the coefficients f τ pxq are function of x. One has tτ u " tku and pΠ x kqp¨q " p¨´xq k , in the polynomial setting. Like in the former setting, in a general local description device the reference objects

pΠ x 1 τ qp¨q " `Πx pΓ xx 1 τ q ˘p¨q (2.3)
at a different base point x 1 are linear combinations of the Π x σ, for a linear map

Γ xx 1 : T Ñ T,
and one can switch back and forth between local descriptions at different points. The linear maps Γ xx 1 are thus invertible and one has a group action of an R d ˆRd -indexed group on the local description structure T . Whereas one uses the same polynomial-type local description for the f k as for f itself in the usual C α setting, there is no reason in a more general local description device to use the same reference objects for f and for its local coefficients, especially if the pΠ x τ qp¨q are meant to describe distributions, among others, while it makes sense to use functions only as reference objects to describe the functions f τ . A simple setting consists in having all the f τ locally described by a possibly different finite collection B `" tµu of labels, in terms of reference functions g yx pµq, with f τ pyq » ÿ µPB f τ µ pxqg yx pµq, near x.

One thus has both

f p¨q » ÿ τ PB f τ pxqpΠ x τ qp¨q » ÿ τ PB, µPB `f τ µ pyqg yx pµqpΠ x τ qp¨q (2.4) and f p¨q » ÿ σPB f σ pyqpΠ y σqp¨q.
Consistency dictates that the two expressions coincide, giving in particular the fact that the coefficients f τ µ pyq are linear combinations of the f σ pyq. Re-indexing identity (2.4) and using the notation σ{τ for the µ corresponding to τ µ » σ, one then has

f p¨q » ÿ σPB,τ PB f σ pyq g yx pσ{τ qpΠ x τ qp¨q. (2.5)
The transition map Γ xy : T Ñ T , from (2.3) is thus given in terms of the splitting map

∆ : T Ñ T b T `, ∆σ " ÿ τ τ b pσ{τ q
that appears in the above decomposition, with

Π y σ " ÿ τ PB g yx pσ{τ qΠ x τ so Γ xy σ " ÿ τ PB g yx pσ{τ qτ.
If one further expands f σ pyq in (2.5) around another reference point z, one gets

f p¨q » ÿ τ,σ,νPB f ν pzq g zy pν{σqg yx pσ{τ qpΠ x τ qp¨q » ÿ νPB f ν pzqpΠ z νqp¨q » ÿ τ,νPB
f ν pzq g zx pν{τ qpΠ x τ qp¨q.

(2.6)

Here again, consistency requires that the two expressions coincide, giving the identity ÿ σPB g zy pν{σqg yx pσ{τ q " g zx pν{τ q in terms of another splitting map If we insist that the family of reference functions g yx pµq, µ P B `, be sufficiently rich to describe locally an algebra of functions, it is convenient to assume that the linear span T `of B `has an algebra structure and the maps g yx on T `are characters of the algebra -multiplicative maps. Building on the example of the polynomials, it is also natural to assume that T `has a grading structure; an elementary fact from algebra then leads directly to the Hopf algebra structure that appears below in the definition of a concrete regularity structure.

We choose to record the essential features of this discussion in the definition of a concrete regularity structure given below; this is a special form of the more general notion of regularity structure from Hairer' seminal work [START_REF] Hairer | A theory of regularity structures[END_REF]. The reader should keep in mind that the entire algebraic setting can be understood at a basic level from the above consistency requirements on a given local description device. We refer the reader to Sweedler's book [START_REF] Sweedler | Hopf algebras[END_REF] for an accessible reference on Hopf algebras. Given two statements s and s `, recall the convention that we agree to write s p`q to mean both the statement s and the statement s `.

Concrete regularity structures

Definition -A concrete regularity structure T " pT `, T q is the pair of graded vector spaces

T `": à αPA `T ὰ ,
T ":

à βPA T β
such that the following holds.

' The index set A `Ă R `contains the point 0, and A ``A `Ă A `; the index set A Ă R is bounded below, and both A `and A have no accumulation points in R. Set

β 0 :" min A.
' The vector spaces T ὰ and T β are finite dimensional.

' The set T `is an algebra with unit 1, with a Hopf structure with coproduct

∆ `: T `Ñ T `b T `,
such that ∆ `1 " 1 b 1, and, for τ P T ὰ ,

∆ `τ P $ & % τ b 1 `1 b τ `ÿ 0ăβăα T β b T ὰ´β , . - , (2.7)
' One has T 0 " x1y, and for any α, β P A `, one has T ὰ T β Ă T ὰ`β .

' One has a splitting map

∆ : T Ñ T b T `,
of the form

∆τ P $ & % τ b 1 `ÿ βăα T β b T ὰ´β , .
-

(2.8)
for each τ P T α , with the right comodule property p∆ b Idq∆ " pId b ∆ `q∆.

(2.9)

Let B ὰ and B β be bases of T ὰ and T β , respectively. We assume B 0 " t1u. Set

B `:" ď αPA `Bὰ , B :" ď βPA B β .
An element τ of T p`q α is said to be homogeneous and is assigned homogeneity |τ | :" α. The homogeneity of a generic element τ P T p`q is defined as |τ | :" maxtαu, such that τ has a non-null component in T p`q α . We sometimes denote by T :" `pT `, ∆ `q, pT, ∆q ȃ concrete regularity structure.

Note that we do not assume any relation between the linear spaces T ὰ and T β at that stage. Note also that the parameter β in (2.8) can be non-positive, unlike in (2.7). For an arbitrary element h in T , set h "

ÿ βď|h| h β P à βď|h| T β .
We use a similar notation for elements of T `. For γ P R, set T ăγ :"

à βăγ T β , T ằγ :" à αăγ T ὰ .
The homogeneous spaces T β and T ὰ being finite dimensional, all norms on them are equivalent; we use a generic notation } ¨}β or } ¨}α for norms on these spaces. For simplicity, we write }h} α :" }h α } α .

(2.10)

To have a picture in mind, think of T and T `as sets of possibly labelled rooted trees, with T `consisting only of trees with positive tree homogeneities -a homogeneity is assigned to each labelled tree. This notion of homogeneity induces the decomposition (2.10) of T into linear spaces spanned by trees with the same homogeneity; a similar decomposition holds for T `. The coproduct ∆ `τ is typically a sum over subtrees σ of τ with the same root as τ , and τ {σ is the quotient tree obtained from τ by identifying σ with the root. One understands the splitting ∆τ of an element τ P T in similar terms. See e.g. Section 2 and Section 3 of [START_REF] Bruned | Algebraic renormalization of regularity structures[END_REF].

Notation. Given σ, τ P B p`q , we use the notation σ ď p`q τ to mean that σ appears as a left hand side of one of the tensor products in the sum defining ∆ p`q τ ; we write τ { p`q σ for the corresponding right hand side, so we have, for τ P B p`q ∆ p`q τ " ÿ σPB p`q σ b pτ { p`q σq. Write σ ă p`q τ to mean further that σ is different from τ . The notations τ { p`q σ and σ ă p`q τ are only used for τ and σ in B p`q .

Decomposing ∆τ in the basis B b B `of T b T `as ∆τ ":

ÿ σPB,θPB
`p∆τ q σθ σ b θ, one has τ {σ " ÿ θPB `p∆τ q σθ θ. We have a similar expression for τ { `σ; for σ, τ P B `, τ { `σ " ÿ θPB `p∆ `τ q σθ θ.

(2.11) With these notations, the right comodule property (2.9) writes for all τ P B ÿ σPB p∆τ q σc p∆σq ab " ÿ θPB `p∆τ q aθ p∆ `θq bc (2.12) for all a P B and b, c P B `. The identity from Lemma 3 is a direct consequence of the co-associativity property

p∆ `b Idq∆ `" pId b ∆ `q∆ `,
of the coproduct ∆ `, and the right comodule identity (2.9).

Lemma 3. For σ ă `τ in B `, we have

∆ `pτ { `σq " ÿ σď `ηď `τ pη{ `σq b pτ { `ηq " pτ { `σq b 1 `1 b pτ { `σq `ÿ σă `ηă `τ pη{ `σq b pτ { `ηq.
(2.13)

For σ ă τ in B, we have

∆ `pτ {σq " ÿ σďηďτ pη{σq b pτ {ηq. ( 2

.14)

A character g on the algebra T `is a linear map g : T `Ñ R such that gpτ 1 τ 2 q " gpτ 1 qgpτ 2 q for any τ 1 , τ 2 P T `. The antipode A of the Hopf algebra structure turns the set of characters of the algebra T `into a group G `for the convolution law ˚defined by

pg 1 ˚g2 qτ :" pg 1 b g 2 q∆ `τ, τ P T `.
The identity of the group is the counit 1 1 , the dual basis vector of the unit 1, and the inverse g ´1 " g ˝A. One associates to a character g of T `the map p g :" pId b gq∆ : T Þ Ñ T, from T to itself. We have { g 1 ˚g2 " p g 1 ˝p g 2 for any g 1 , g 2 P G `, as a consequence of the comodule property (2.9). Also, for any τ P T , ´p gpτ q ´τ ¯P T ă|τ | , as a consequence of the structural identity (2.8). Remark that for any concrete regularity structure T " `pT `, ∆ `q, pT, ∆q ˘, then

T `:" `pT `, ∆ `q, pT `, ∆ `qȋ
s also a concrete regularity structure. For g P G `, set p g `τ :" pId b gq∆ `; (2.15) this map sends T `into itself.

Remark. For g P G `, the map p g is denoted by Γ g in Hairer's work [START_REF] Hairer | A theory of regularity structures[END_REF]; we prefer the former Fourier-like notation.

We now come to the definition of the reference objects Π g

x τ and g yx pσq used to give local descriptions of distributions and functions in a regularity structure setting, as in the introduction to this section. They come under the form of a model.

Models

Recall β 0 " min A P R. Given a function ϕ on R d , and

x P R d , 0 ă λ ď 1, set ϕ λ x p¨q :" λ ´dϕ `λ´1 p¨´xq ˘.
Definition -A model over a regularity structure T is a pair pg, Πq of maps

g : R d Ñ G `, Π : T Ñ C β 0 pR d q
with the following properties.

' Set g yx :" g y ˚g´1

x for each x, y P R d . One has

}g} :" sup τ PB `sup xPR d |g x pτ q| `sup τ PB `sup x,yPR d |g yx pτ q| |y ´x| |τ | ă 8. (2.16) ' The map Π is linear. Set Π g x :" pΠ b g ´1 x q∆ for each x P R d . Fix r ą |β 0 ^0|. One has }Π} g :" sup σPB }Πσ} C β 0 `sup σPB sup ϕ,0ăλď1,xPR d ˇˇ@ Π g x σ, ϕ λ x Dˇλ |σ| ă 8, (2.17)
where ϕ runs over all functions ϕ P C r pR d q, with associated norm no greater than 1 and support in the unit ball.

In Hairer's original work [START_REF] Hairer | A theory of regularity structures[END_REF], the notations Π x and Γ yx are used instead of Π g x and x g yx , respectively. In (2.16) and (2.17), we assume global bounds over R d , while Hairer only assumes in [START_REF] Hairer | A theory of regularity structures[END_REF] the previous bounds in any compact subset of R d . In this paper, we work on the globally bounded case for simplicity. Our result may be extended into the locally bounded case using the weighted norms }f } L 8

w " sup xPR d w ´1pxq|f pxq| instead of }f } L 8 . For comparison, and given a ă 0, note that a distribution Θ on R d is an element of C a pR d q iff one has a bound ˇˇ@ Θ, ϕ λ x DˇˇÀ λ a , for any 0 ă λ ď 1, uniformly in x P R d and ϕ P C r pR d q, of unit norm in that space and support in the unit ball, for r " |tau|. We stress that Πτ is only an element of C β 0 pR d q; identity (2.17) conveys the idea that Π g x τ behaves at point x like an element of C |τ | pR d q. Emphasize that g acts on T `, while Π acts on T , and note that g plays on T `the same role as Π on T ; For τ P T `and σ P T , one has

g yx pτ q " `gp¨qpyq b g ´1 x ˘∆`τ , pΠ g x σqpyq " `Πp¨qpyq b g ´1 x ˘∆σ, (2.18)
in a distributional sense for the latter. Note also the fundamental relation

Π g y " Π g x ˝x g xy , (2.19)
for all x, y P R d ; it comes from the comodule property (2.9). The map Π can be recovered from each map Π g x , as we have Π " pΠ g x b g x q∆, (2.20) as a consequence of the comodule property (2.9)

pΠ g x b g x q∆ " `Π b g ´1 x b g x ˘p∆ b Idq∆ " `Π b g ´1 x b g x ˘pId b ∆ `q∆ " pΠ b 1 1 q∆ " Π.
Examples. 1. Bounded polynomials structure. For any smooth function f on R d , and r ą 0, the Taylor expansion property

f pyq ´ÿ |k|ăr B k f pxq k! py ´xq k " Op|y ´x| r q.
is usually lifted to a modelled distribution

f pxq :" ÿ |k|ăr B k f pxq k! X k ,
over the canonical polynomial regularity structure, under the model pΠX k qpxq " x k and g x pX k q " x k . Since they are not bounded functions, we modify this expansion by using smooth and bounded functions behaving like polynomials in local sets. The following elementary claim is proved in Appendix B. where B r pf q :" }f } C r b , if r P N, or B r pf q :" }f } C r , if r P p0, 8qzN.

We lift expansion (2.21) to an appropriate regularity structure as follows. Let X :" tX i e u ePE,1ďiďd be a family of symbols, and let T `pX q be the commutative free algebra with unit 1, generated by these symbols. We define a coproduct ∆ `: T `pX q Ñ T `pX q b T `pX q by

∆ `1 " 1 b 1, ∆ `X i
e " X i e b 1 `1 b X i e , which turns T `pX q into a Hopf algebra. By defining the homogeneity | ¨| by |X i e | " 1, we have the graded Hopf algebra T `pX q. Let T pXq be the subspace spanned by the bounded polynomials tX k e u ePE,kPN d , where

X k e :"

d ź i"1 `Xi e ˘ki , k " pk i q d i"1 P N d .
Denote by ∆ : T pXq Ñ T pXq b T `pX q the restriction of ∆ `to T pXq, which turns T pXq into a right comodule over T `pX q. By definition, we have the concrete regularity structure T pXq :" `T `pX q, T pXq ˘. The canonical model pg, Πq on T pXq is defined by g x pX k e q " pΠX k e qpxq " x k e .

(2.22)

The following elementary result, proved in Appendix B provides the canonical lift of a smooth function to this bounded polynomials regularity structure. See the paragraph on modelled distributions for the definition of D r pT pXq, gq and the associated norm ||| ¨||| D r .

Proposition 5. For any given f P C 8 b pR d q and r ą 0, define the T pXq-valued function

f pxq :" ÿ ePE ÿ |k|ăr B k pφ e f qpxq k! X k e , x P R d .
Then f P D r pT pXq, gq, and |||f ||| D r À B r pf q.

2. Canonical model on T `. As another example of model over some regularity structure, consider the regularity structure T `associated with any regularity structure T , and assume we are given a function g : R d Þ Ñ G `that satisfies estimate (2.16). For τ P T `, set Π g τ pxq :" g x pτ q.

(2.23) Estimate (2.17) holds as a consequence of (2.16), so pg, Π g q is a model on T `" pT `, T `q. This justifies to say simply that g is a model on T `. l Equation (2.20) giving Π in terms of Π g x and g x writes explicitly Πτ "

ÿ σďτ Π g
x pσqg x pτ {σq,

for τ P B, that is Π g x τ " Πτ ´ÿ σăτ g x pτ {σqΠ g x σ. (2.24) Furthermore expanding Π g x σ, one has Π g x τ " Πτ ´ÿ σ 1 ăτ g x pτ {σ 1 qΠσ 1 `ÿ σ 2 ăσ 1 ăτ g x pτ {σ 1 qg x pσ 1 {σ 2 qΠ g x σ 2 .
Iterating this expansion gives a representation of Π g x in terms of g x and Π Π g

x τ " Πτ ´ÿ ně1 p´1q n´1 ÿ σn㨨¨ăσ 1 ăτ g x pτ {σ 1 q ¨¨¨g x pσ n´1 {σ n q Πσ n ;

(2.25) the sum is finite. Similarly, since g y " g yx ˚gx , by definition, Lemma 3 provides for any σ ď p`q τ P B p`q the relation g yx `τ { p`q σ ˘" g y `τ { p`q σ ˘´g x `τ { p`q σ ˘´ÿ σă p`q σ 1 ă p`q τ g x `τ { p`q σ 1 ˘gyx `σ1 { p`q σ ˘.

A repeated expansion then gives a representation of g yx pτ { p`q σq in terms of g y and g x g yx `τ { p`q σ ˘" g y `τ { p`q σ ˘´g x `τ { p`q σ ÿ ně1 p´1q n´1 ÿ σă p`q σnă p`q ¨¨¨ă p`q τ g x `τ { p`q σ 1 ˘¨¨¨g x `σn´1 { p`q σ n ˘´g y `σn { p`q σ ˘´g x `σn { p`q σ ˘¯.

(2.26)

Modelled distributions

Recall notation (2.10) for the notation }h} α for α P A and h P T .

Definition -Let g : R d Ñ G `satisfy (2.16). Fix a regularity exponent γ P R. One defines the space D γ pT , gq of distributions modelled on the regularity structure T , with transition g, as the space of functions f : R d Ñ T ăγ such that

rsf rs D γ :" max βăγ sup xPR d › › f pxq › › β ă 8, }f } D γ :" max βăγ sup x,yPR d › › f pyq ´x g yx f pxq › › β |y ´x| γ´β ă 8. Set |||f ||| D γ :" rsf rs D γ `}f } D γ .
For a basis element σ P B, and an arbitrary element h in T , denote by h σ its component along the σ direction. For a modelled distribution f p¨q " ř σPB f σ p¨q σ in D γ pT , gq, and σ 0 P B, we have ´f pyq ´x g yx f pxq ¯σ0 " f σ 0 pyq ´f σ 0 pxq ´ÿ τ ąσ 0 g yx pτ {σ 0 q f τ pxq.

(2.27)

As an example, given a basis element τ P B, set

h τ pxq :" ÿ σăτ g x pτ {σqσ. ( 2
R : D γ pT , gq Ñ C β 0 pR d q satisfying the property ˇˇ@Rf ´Πg x f pxq, ϕ λ x D ˇˇÀ }Π} g › › f › › D γ λ γ , (2.29)
uniformly in f P D γ pT , gq, ϕ P C r pR d q with unit norm and support in the unit ball, x P R d and 0 ă λ ď 1. Such an operator is unique if the exponent γ is positive.

A distribution satisfying identity (2.29) is called a reconstruction of the modelled distribution f . See Theorem 3.10 in Hairer' seminal work [START_REF] Hairer | A theory of regularity structures[END_REF]. We provide in Theorem 14 below an explicit representation for the reconstruction operator R building on paracontrolled calculus. Notice from the definition of Π g

x that the constraint

ˇˇ@ Π g x τ, ϕ λ x DˇˇÀ λ |τ |
, that needs to be satisfied by a model, is equivalent to the estimate ˇˇ@Πτ

´ÿ σăτ g x pτ {σqΠ g x σ, ϕ λ x D ˇˇÀ λ |τ | .
This means that Πτ , with τ P B, is a reconstruction of the modelled distribution h τ P D |τ | pT , gq defined above in (2.28). Recall that uniqueness in the reconstruction theorem implies that if f takes values in a function-like sector of T , then Rf " 1 1 pf q -see e.g. Proposition 3.28 in Section 3.4 of [START_REF] Hairer | A theory of regularity structures[END_REF].

-Explicit formula for the reconstruction operator

We prove Theorem 1 giving an explicit description of the reconstruction operator in this section.

From Taylor local description to global paracontrolled representation

We describe here some simple properties of a natural two-parameter extension of the elementary paraproduct built from Littlewood-Paley blocks, and refer the reader to Appendix A for background on Littlewood-Paley decomposition. The notations ∆ i and Q i for the i th Littlewood-Paley block and its kernel are recalled in Appendix A. For j ě 1, define the operator S j :" ř iďj´2 ∆ i , and its smooth kernel P j :"

ř iďj´2 Q i .
The Hölder spaces C α pR d q are defined as Besov spaces B α 88 pR d q -see Appendix A. For a two-variable real-valued distribution Λ on R d ˆRd , and j ě 1, set pQ j Λqpxq :" ij P j px ´yqQ j px ´zq Λpy, zqdydz;

we abuse notations using the integral notation. Set PΛ :"

ÿ jě1 Q j Λ.
We often write PΛ " P y,z `Λpy, zq ȋn order to display the integrated variables. With that notation, we have the consistency relation P f g " P y,z `f pyqgpzq ˘, f, g P L 8 , between the paraproduct operator P and its two-parameter extension. For α ą 0, and a measurable real-valued function F on R d ˆRd , set

|||F ||| C α 2 :" sup y,zPR d
|F py, zq| |y ´z| α .

Proposition 8.

(a) Let Λ be a real-valued distribution on R d ˆRd . If }Q j Λ} L 8 À 2 ´jα , for all j ě 1, for some α P R, then PΛ P C α pR d q, and

}PΛ} C α À sup jě1 2 jα }Q j Λ} L 8 .
(b) Let α ą 0, and a real-valued measurable function F on R d ˆRd be given, with

|||F ||| C α 2 ă 8. Then PF P C α pR d q, and }PF } C α À |||F ||| C α 2 .
Proof -(a) Since F P j is supported in

x P R d ; |x| ă 2 j ˆ2 3 ( and
F Q j is supported in x P R d ; 2 j ˆ3 4 ă |x| ă 2 j ˆ8 3 ( , the integral ż Q i px ´wqP j pw ´yqQ j pw ´zqdw
vanishes if |i ´j| ě 5. Hence ∆ i pPΛq " ř |i´j|ď4 ∆ i pQ j Λq and we have

}∆ i pPΛq} L 8 ď ÿ |i´j|ď4 }∆ i pQ j Λq} L 8 À ÿ |i´j|ď4 }Q j Λ} L 8 À ÿ |i´j|ď4 2 ´αj À 2 ´αi .
(b) It is sufficient to show that }Q j F } L 8 À 2 ´αj for all j ě 2. By the scaling properties P j p¨q " 2 pj´2qd P 2 p2 j´2 ¨q and Q j p¨q " 2 pj´2qd Q 2 p2 j´2 ¨q, we have

|Q j F pxq| À ij |P j px ´yqQ j px ´zq||y ´z| α dydz " 2 ´αpj´2q ij |P 2 p2 j´2 x ´yqQ 2 p2 j´2 x ´zq||y ´z| α dydz " 2 ´αpj´2q ij |P 2 pyqQ 2 pzq||y ´z| α dydz À 2 ´αj .
The next proposition is the key step to the representation of the reconstruction operator given in Theorem 6.10 of [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. We state it and prove it here under a slightly more general form. See the proofs of Lemma 6.8, Lemma 6.9 and Theorem 6.10 therein. Proposition 9. Let γ P R and β 0 P R be given together with a family Λ x of distributions on R d , indexed by x P R d . Assume that one has

sup xPR d }Λ x } C β 0 ă 8
and one can decompose pΛ y ´Λx q under the form Here the operator S is defined by

Λ y ´Λx " N ÿ "1 c yx Θ x (3.1) for N finite, R d -indexed distributions Θ x ,
S f :" f ´P1 f " f ´Py,z `1pyqf pzq ˘(3.5)
for any f P S 1 pR d q. This is a smooth function that depends continuously on f ; if f P C α pR d q with α P R, then for any r ą 0,

}S f } C r À }f } C α .
Hence we have for any i ě 1, ˇˇ∆ i `PpΛq ´Λx ˘pxq ˇˇď ÿ |j´i|ď4 ÿ ij ˇˇQ i px ´yqP j py ´uq ˇˇ|u ´x| γ´β 2 ´iβ dudy `op2 ´iγ q.

Then (3.4) follows from elementary estimates and the bounds ż

R d |P j |pxq |x| r dx À 2 ´jr , ż R d |Q j |pxq |x| r dx À 2 ´jr , (3.6)
that holds for any positive exponent r.

(ii) If γ ą 0, estimate (3.4) implies that the sum

f Λ pxq :" ÿ jě´1 ∆ j `PpΛq ´Λx ˘pxq,
defines an element f Λ of C γ pR d q; this is proved in point (iii) below. Then we have, for any

x P R d , ˇˇ@ PpΛq ´fΛ ´Λx , P i px ´¨q Dˇˇ" ˇˇˇˇÿ jďi´2 ∆ j `PpΛq ´Λx ˘´S i pf Λ qpxq ˇˇˇ" ˇˇˇˇf Λ pxq ´ÿ jąi´2 ∆ j `PpΛq ´Λx ˘´S i pf Λ qpxq ˇˇˇÀ ÿ jąi´2 ˇˇ∆ j pf Λ qpxq ˇˇ`ÿ jąi´2 ˇˇ∆ j `PpΛq ´Λx ˘pxq ˇˇÀ 2 ´iγ .
Uniqueness of f Λ follows from the fact that P i converges to a Dirac mass at 0, so if f (iii) We follow the argument in Section 6 of [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. We decompose

f Λ " f ďj`1 Λ `f ąj`1 Λ , where f ďj`1 Λ pxq :" ÿ iďj`1 ∆ i `PpΛq ´Λx ˘pxq.
We consider

∆ j f Λ " ∆ j f ďj`1 Λ `∆j f ąj`1 Λ
. For the second term, by the estimate (3.4) one has

› › ∆ j f ąj`1 Λ › › L 8 À ÿ iąj`1 2 ´iγ À 2 ´jγ .
For the first term, since Q j ˚Qďj`1 " Q j , one has

∆ j f ďj`1 Λ pyq " ż Q j py ´xqQ ďj`1 `PpΛq ´Λx ˘pxqdx " ż Q j py ´xqQ ďj`1 ´PpΛq ´Λy `ÿ c yx Θ x ¯pxqdx " Q j `PpΛq ´Λy ˘pyq `ÿ ż Q j py ´xq c yx `Qďj`1 Θ x ˘pxq dx.
The first term is estimated by (3.4). The second term is bounded by 2 ´jγ by assumption.

In the end, we have

› › ∆ j f ďj`1 Λ › › L 8 À 2 ´jγ .
If Λ x stand for Π g x f pxq, for a modelled distribution f P D γ pT , gq and a model pg, Πq, one has Λ y ´Λx "

ÿ σPB ´f pyq ´x g xy f pyq ¯σ Π g x σ,
and Λ satisfies the assumptions of Proposition 9, from condition (2.17) on models and the definition of a modelled distribution. As in Lemma 6.3 of [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], we can extend the condition (2.17) for any rapidly decreasing smooth functions ϕ. Identities (3.2) and (3.3) are equivalent to saying that PpΛq ´fΛ 1 γą0 is a reconstruction of f -see Lemma 6.6 of [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. This is the content of Theorem 6.10 in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF].

We prove in Theorem 14 below that P y,z `pΠ g y f pyqqpzq ˘has an explicit form, up to some remainder in C γ pR d q. The mechanism at work in the proof of this fact lies in Proposition 10. Following [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF], set R ˝pa, b, cq :" P a pP b cq ´Pab c. It was proved in Appendix C1 of [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF] that the map R ˝is continuous from L 8 pR d q ˆCr 1 pR d q Ĉr 2 pR d q into C r 1 `r2 pR d q, for r 1 P p0, 1q and r 2 any regularity exponent in R. The next proposition provides a refined continuity result for the operator R.

Proposition 10. Pick a positive regularity exponent α. Assume we are given a function f P L 8 pR d q and a finite family pa n , b n q 1ďnďN of elements of L 8 pR d q ˆL8 pR d q such that one has

f pyq ´f pxq " N ÿ n"1 a n pxq `bn pyq ´bn pxq ˘`f 7 yx , (3.7)
for any x, y P R d , for a two-parameter remainder f 7 with finite α-Hölder norm |||f 7 ||| C α 2 ă 8. Then, for any regularity exponent β P R and g P C β pR d q, we have

N ÿ n"1 R ˝pa n , b n , gq P C α`β pR d q.
Proof -Recall from equation (3.5) the definition of the smooth function S g, for any g P C β pR d q, with β P R, and note the identity R ˝pa, b, cq " P ´apxq `Pb´bpxq c ˘pyq ¯´P ab pS cq.

Applying the two-parameter P-operator to identity (3.7), we see that from a (wonderful) telescopic sum simplification. This is where something is happening. Define then rrτ ss g by the formula S gpτ q `Px,y `gyx pτ q 8 ÿ n"1 p´1q n´1 ÿ 1ă `σn`1 ă `¨¨¨ă `σ1 ă `τ R ˝´gpτ { `σ1 q ¨¨¨gpσ n´1 { `σn q, gpσ n { `σn`1 q, rrσ n`1 ss g ¯.

N ÿ n"1 R ˝pa n , b n , gq " R ˝p1, f,
It follows from the estimate |g yx pτ q| À |y ´x| |τ | , and Proposition 8 that P x,y `gyx pτ q ˘P C |τ | pR d q. Corollary 11 takes care of the sum and shows that it defines an element of

C |τ | pR d q.
' The proof of the regularity statement for rrτ ss M , for τ P B, proceeds by induction, similarly as above, using identity (2.25) giving Π g x τ in terms of Π only, as an input. Applying the two-parameter operator P to (2.25), gives (3.8) for a choice of rrτ ss M equal to S Πpτ q `Px,y `Πg x pτ qpyq

8 ÿ n"1 p´1q n´1 ÿ 1ăσ n`1 㨨¨ăσ 1 ăτ
R ˝´gpτ {σ 1 q ¨¨¨gpσ n´1 {σ n q, gpσ n {σ n`1 q, rrσ n`1 ss M ¯.

Since Π g x τ " Π g x 1 y g x 1 x τ " Π g x 1 τ `řσăτ g x 1 x pτ {σqΠ g x 1 σ
, one can use Proposition 9 to conclude that P x,y `Πg x pτ qpyq ˘belongs to C |τ | pR d q.

Recall from Example 2 in Section 2 that given a model pg, Πq on T , the concrete regularity structure T `is endowed with an associated canonical model M g :" pg, Π g q " pg, gq.

Remark that

rr¨ss M g " rr¨ss g , on T `, so the above statement is really about rr¨ss M g and rr¨ss M . The proof makes it clear that the g-brackets rrτ ss g depend only on g. We extend by linearity the operators rr¨ss g , rr¨ss M to T ànd T , respectively.

Remark. One can make the link with the setting introduced in [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF], and give a different representation of the brackets under the assumption that we are given an operator I that acts on smooth functions and an abstract integration operator I : T Þ Ñ T , on the regularity structure T , together with a naive interpretation operator Π, such that Π is multiplicative and ΠpIτ q " IpΠτ q, for all τ P T -see Section 4. Compute rr˝Ip˝q 2 ss M as another example. We have

Π `˝Ip˝q 2 ˘" Z 2 ζ " P Z 2 ζ `2P ζ P Z Z `Pζ `a pZ, Zq ˘`a `2P Z Z `apZ, Zq, ζ ˘.
To make the link with the defining relation (3.8) for rr˝Ip˝q 2 ss M , we use the corrector operator C and the operator S from [START_REF] Bailleul | Higher order paracontrolled calculus[END_REF]. This gives for Π `˝Ip˝q 2 ˘the expression `p∆τ q σθ rrθss g " ÿ θ,ρ,κPB `, 1ă `ρă `θp∆τ q σθ p∆ `θq ρκ P gpκq rrρss g `rrτ {σss g , and on the other hand, and since P f 1 " 0, for any f P S 1 pR d q, ÿ σăηăτ P gpτ {ηq rrη{σss g " ÿ σăηăτ,κ,ρPB `,1ă `ρp∆τ q ηκ p∆ηq σρ P gpκq rrρss g . The statement then follows from the right comodule identity (2.9), such as expressed in coordinates in identity (2.12), and the structural assumption (2.8) on the splitting map ∆. (3.9)

P Z 2 ζ `2P Z `Pζ Z ˘`2Spζ, Z,
Each coefficient f τ also has a representation

f τ " ÿ τ ăµ;|µ|ăγ P f µ rrµ{τ ss g `rrf τ ss g , (3.10)
for some rrf τ ss g P C γ´|τ | pR d q. Moreover, the map

f Þ Ñ ´rrf ss M , `rrf τ ss g ˘τPB from D γ pT , gq to C γ pR d q ˆśτPB C γ´|τ | pR d q, is continuous.
Proof -Recall from Proposition 9 that, there exists a function g P C γ pR d q such that P x,y ´`Π g x f pxq ˘pyq ¯´g1 γą0 (3.11) is the reconstruction Rf . We have from identity (2.25) giving Π g x in terms of g and Π, the finite expansion

`Πg

x f pxq ˘p¨q "

8 ÿ n"0 p´1q n ÿ σn㨨¨ăσ 1 ăσ 0 ´f σ 0 g `σ0 {σ 1 ˘¨¨¨g `σn´1 {σ n ˘¯pxq pΠσ n qp¨q.
So applying the two-parameter paraproduct operator P on both sides, and using the same (fantastic) telescopic sum as in the proof of Proposition 12, we get, with an obvious notation,

P x,y ´`Π g x f pxq ˘pyq ¯" pC 8 q `ÿ σ 0 PB P f σ 0 rrσ 0 ss M `8 ÿ n"1 p´1q n ÿ σ n`1 ăσn㨨¨ăσ 1 ăσ 0 R ˝´f σ 0 g `σ0 {σ 1 ˘¨¨¨g `σn´1 {σ n ˘, gpσ n {σ n`1 q, rrσ n`1 ss M ¯.
From (2.27), for each σ P B we have

f σ pyq ´f σ pxq " `f pyq ´x g yx f pxq ˘σ `ÿ σăσ 0 f σ 0 pxqg yx pσ 0 {σq " `f pyq ´x g yx f pxq ˘σ `ÿ ně0 p´1q n ÿ σăσn㨨¨ăσ 1 ăσ 0
`f σ 0 gpσ 0 {σ 1 q ¨¨¨gpσ n´1 {σ n q ˘pxq `gy pσ n {σq ´gx pσ n {σq ˘.

Proposition 10 applies and tells us that the sum of the R ˝-terms defines an element of Hölder regularity pγ ´|σ n`1 |q `|σ n`1 | " γ. The claim of the theorem on Rf comes from this fact and identity (3.11). To get the paracontrolled representation of f τ , note from Lemma 6 that f {τ " ř µěτ ;|µ|ăγ f µ pµ{τ q P D γ´|τ | pT `, gq, and apply the result just proved to the reconstructions of the modelled distribution. Theorem 14 refines over the paraproduct-based construction of the reconstruction operator given by Gubinelli, Imkeller and Perkowski in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF], where Rf is proved to be of the form P x,y `pΠ g

x f pxqqpyq ˘, up to a C γ pR d q term. See our extension in Proposition 9 above. The point of our refined representation of the family `Rf , f τ ˘as a paracontrolled system lies in Theorem 2, proved in the next section. It parametrizes the class of "admissible" models used for the study of singular stochastic PDEs, in terms of the brackets rrτ ss M , with |τ | ď 0. The forthcoming work [START_REF] Bailleul | Paracontrolled calculus and regularity structures[END_REF] will give a similar description of more general models pg, Πq and modelled distributions in D γ pT , gq, in terms only of the bracket data, extending the main result of [START_REF] Martin | A Littlewood-Paley description of modelled distributions[END_REF] on Besov-type characterizations D γ pT , gq.

One advantage of the explicit construction of the reconstruction operator given by Theorem 14 is that this representation is flexible enough to work in other functional settings than the present B γ 88 -type space D γ pT , gq. The continuity properties of the paraproduct operator on Besov, Triebel-Lizorkin or Sobolev-Slobodeckij spaces are well-known, and allow for a direct approach to reconstruction in these spaces, in the line of the recent works [START_REF] Hairer | The reconstruction theorem in Besov spaces[END_REF][START_REF] Hensel | Modelled distributions of Triebel-Lizorkin type[END_REF][START_REF] Ch | Stochastic Analysis with Modelled Distributions[END_REF][START_REF] Singh | An elementary proof of the reconstruction theorem[END_REF].

Before giving the next statement, note that the restriction to T ď0 of the splitting ∆ turns T ď0 :" ´pT `, ∆ `q, pT ď0 , ∆q īnto a regularity structure. The next statement is essentially contained in Proposition 3.31 from [START_REF] Hairer | A theory of regularity structures[END_REF]; we give the details here to provide a self-contained document. the distribution Πτ appears then as another reconstruction of h τ .

' If one picks now a basis vector µ P B, with |µ| ą 0, then h µ P D |µ| pT , gq has a unique reconstruction, equal to Πµ, that is characterized by the data `Πg x σ, g x pµ{σq; x P R d , σ ă µ ˘, from the defining property (2.29) of a reconstruction. An elementary induction then shows the existence of a unique extension of Π to T that satisfies the property Πτ " Rh τ , for every τ P B with positive homogeneity.

-Parametrization of the set of admissible models 4.1 Usual models

We introduce in this section a notion of usual model on a concrete regularity structure, motivated by some identity satisfied by gpτ {X k q in the usual setting; see Equation (4.4) below. Its introduction is motivated by the fact that usual models pg, Πq are entirely determined by the Π map, under a mild structure assumption on T `and ∆. The definition of a usual model requires that we work with concrete regularity structures where T and T `are related with one another, unlike the results of the previous section.

Let T " `pT `, ∆ `q, pT, ∆q ˘be a concrete regularity structure. If T contains the usual polynomial structure T pXq, one can expand the coproduct ∆τ of any τ P BztX k u k , as

∆τ " ∆τ `ÿ kPN d X k k! b D k τ, ∆τ :" ÿ σ‰X k σ b pτ {σq, where D k τ :" k!pτ {X k q. Applying Π b g ´1
x , we have

Π g x τ " Πg x τ `ÿ kPN d p¨q k k! g ´1 x pD k τ q, Πg x :" pΠ b g ´1 x q ∆.
Setting

f x pD k τ q :" ´ÿ PN d x ! g ´1 x pD k` τ q,
or equivalently,

g ´1 x pD k τ q :" ´ÿ PN d p´xq ! f x pD k` τ q,
gives a Taylor-like expansion formula for Π g x τ , under the form of the identity

Π g x τ " Πg x τ ´ÿ kPN d p¨´xq k k! f x pD k τ q.
Since the derivatives B k y pΠ g x τ qpyq vanishes at y " x for any |k| ă |τ |, one has

f x pD k τ q " 1 |k|ă|τ | B k y p Πg x τ qpyq ˇˇy"x . (4.1)
Given α P R, define a linear projection map P ąα : T Þ Ñ T setting P ąα pτ q :" τ 1 |τ |ąα , for every τ P B.

Lemma 16. For any τ P BztX k u k , one has

g x pD k τ q " ´Bk y p Πg x P ą|k| b g x q∆τ ( pyq ¯ˇˇy "x . (4.2)
Proof -It suffices to show that

g x pD k τ q " ÿ σ‰X f x pD k σqg x pτ {σq; (4.3)
we get (4.2) by inserting (4.1) into (4.3). We start from the formula

∆ `pτ {X k` q " ÿ σ‰X m pσ{X k` q b pτ {σq `ÿ m ˆk ` `m m ˙Xm b pτ {X k` `mq.
Since τ {X k` P T `zx1y, applying g ´1 x b g x to the preceding identity gives

0 " ÿ σ‰X m g ´1 x pD k` σqg x pτ {σq `ÿ m p´xq m m! g x pD k` `mτ q, that is 0 " ´ÿ m p´xq m m! ˜ÿ σ‰X m f x pD k` `mσqg x pτ {σq ´gx pD k` `mτ q ¸.
Identity (4.3) is obtained as a consequence, since

0 " ÿ ,m x ! p´xq m m! ˜ÿ σ‰X m f x pD k` `mσqg x pτ {σq ´gx pD k` `mτ q " ÿ σ‰X m f x pD k σqg x pτ {σq ´gx pD k τ q.
We use in the present work the bounded polynomial structure rather than the usual polynomial structure. We work with concrete regularity structures for which the following assumptions hold true.

Assumption (A)

The bounded polynomial structure T pXq " xX k e y e,k is contained in T , and the polynomial ring T `pX q " xX k 1 e 1 ¨¨¨X kn en y e 1 ,...,en,k 1 ,...,kn is included in T `.

We do not make a difference in the notations between the two copies in T and T `of the bounded polynomial structure. Definition 17. Let T be a concrete regularity structure satisfying Assumption (A1). We say that the model pg, Πq is usual if one has g x pX k e q " pΠX k e qpxq " x k e , and

g x pD k e τ q " ´Bk y ! φ e `Π g x P ą|k| b g x ˘∆τ ) pyq ¯ˇˇy "x . (4.4)
Lemma 18. Let tζ x u xPR d Ă S 1 pR d q be a family of distributions. If there exist α P R and a positive constant C such that one has ˇˇxζ x , ϕ λ x y ˇˇď Cδ λ , uniformly over ϕ P C r pR d q, with unit norm and support in the unit ball, λ P p0, 1s and x P R d , then the sum pI e k pζ x qqpxq :"

A ζ x , B k x `φe pxqKpx, ¨q˘D :" ÿ ně0 A ζ x , B k x `φe pxqK n px, ¨q˘E (4.6)
converges for any |k| ă α `θ, e P E and x P R d .

Proof -Pick x P R d . Let ϕ be any smooth function with support in ty P R d ; |y ´x| ă λu, such that one has sup

| |ďr λ d`| | }B ϕ} L 8 ď 1,
for some λ P p0, 1s. Since ϕ x,λ pyq :" λ d ϕpx `λyq has unit norm in C r pR d q and ϕ " pϕ x,λ q λ x , we have |xζ x , ϕy| ď Cλ α , from the assumption of the lemma. Pick k P N d . Since ϕpyq :" B k

x pφ e pxqK n px, yqq is supported in ty P R d ; |y ´x| ă 2 ´nu and }B ϕ} L 8 À 2 pd`|k|`| |´θqn , we thus have

ˇˇAζ x , B k x `φe pxqK n px, ¨q˘E ˇˇÀ 2 p|k|´α´θqn , (4.7)
and a converging sum in (4.6) if |k| ă α `θ.

Note that we cannot even make sense of ş R d Kpz, yqζ x pyq dy, for z ‰ x. Were we able to define that function as a regular function of z, it would have a regularity structure lift in the canonical polynomial structure. Lemma 18 allows to define an avatar for the lift at point x only of the non-existing function `pφ e Iqζ x ˘p¨q, under the form of the quantity

ÿ ePE,|k|ăα`θ X k e k!
`pφ e Iqζ x ˘pxq.

It follows from Lemma 18 and the assumption β 0 ą ´θ, that one can make sense of IpΠτ qpxq for any x P R d , under the form of the converging sum IpΠτ qpxq :"

ÿ ně0 @
Πτ, K n px, ¨qD .

Regularity structures with an abstract integration operator

In addition to Assumption (A), we make the following set of assumptions on the concrete regularity structure T . (4.10) Remark -Note that our notion of admissible model is more general than the corresponding notion introduced by Bruned, Hairer and Zambotti in [START_REF] Bruned | Algebraic renormalization of regularity structures[END_REF]; Definition 6.9 in that work. Their admissible Π-maps, together with the positive twisted antipode from Proposition 6.2 in [START_REF] Bruned | Algebraic renormalization of regularity structures[END_REF], are used in definition 6.8 therein to build a g-map and models pg, Πq that are admissible model on T in our sense, with all φ e " 1. This is a direct consequence of Lemma 6.10 in [START_REF] Bruned | Algebraic renormalization of regularity structures[END_REF] and the following equalities.

Assumption (B) The sets T `and T are related via the integral operators in the following

Then
Π

Parametrization of the set of admissible models

We prove Theorem 2 in this section, giving a parametrization of the set of admissible models by a product of Hölder spaces of non-positive regularity exponent. A similar parametrization of the space of branched rough paths was achieved in the recent work [START_REF] Tapia | The geometry of the space of branched rough paths[END_REF] of Tapia and Zambotti, with very different tools. The next result applies in particular in the former setting, when formulated in terms of regularity structures. We need the following structural assumptions on T `and T .

Assumption (C)

' The basis B `of T `is a commutative monoid with unit Start by noting that T ăα :" pT pαq , T ăα q is a regularity structure for any α P A. Define inductively on α P A the maps

Π ăα : T ăα Þ Ñ C β 0 pR d q, and g pαq : T pαq Þ Ñ C b pR d q,
with g pαq x pX k e q " x k e , initializing the induction. Write M ăα for the model pg pαq , Π ăα q on T ăα . Set α `:" mintβ ą α; β P Au. Given a basis vector τ P B α , the function h τ :" ř σăτ g pαq pτ {σq σ is an element of D α pT ăα , g pαq q. Define Π ăα `τ as equal to either ÿ σăτ P g pαq pτ {σq rrσss `rrτ ss, if |τ | ď 0, or R Măα ph τ q, if |τ | ą 0, where R Măα stands for the reconstruction operator on D α `Tăα , g pαq ˘associated with the model M ăα . We have in both cases ˇˇ@ pΠ ăα `qg pαq

x τ, ϕ λ x DˇˇÀ λ α , from Corollary 15. Define then an extension g pα `q of g pαq onto T pα `q by requiring that it is multiplicative and by defining g pα `qpI ek τ q from identity (4.11), with Π ăα `in the role of Π. Boundedness of g pα `q is checked by induction. Given Assumption (C) on the regularity structure T , closing the induction step amounts to proving that ˇˇg pα `q yx pI ek τ q ˇˇÀ |y ´x| yx pI ek τ q ˇˇin the left hand side, which closes the induction.

Proof of Lemma 22 -We follow closely the proof of Schauder estimates for modelled distributions -Theorem 5.12 of [START_REF] Hairer | A theory of regularity structures[END_REF]. Note first that by (4.11) one can decompose Υ e α h τ under the form `Υe α h τ ˘pxq " I e0 h τ pxq `J e pxqph τ pxqq `pN e h τ qpxq, with J e pxqh τ pxq :" Remark then, as in Lemma 5.16 of [START_REF] Hairer | A theory of regularity structures[END_REF], that we have for any x, y P R d

x g yx ``I e``J e pxq ˘" `Ie``J e pyq ˘x g yx .

We give a direct proof. By definition, we have x g yx ``I e``J e pxq ˘τ " ∆ i f ∆ j g, apf, gq :"

ÿ σ g yx pτ {σqI e`σ `ÿ k X k e k! g yx pI ek τ q `ÿ k, ;|k` |ă|τ |`θ X k e k!
ÿ i,jě´1 |i´j|ď1
∆ i f ∆ j g, for any f, g P S 1 pR d q, as long as they converge. We then have formally f g " P f g `Pg f `apf, gq.

The basic continuity results for these operators are summarized as follows.

Proposition 23. Let α, β P R.

B -Bounded polynomials

This appendix is a follow-up of example 1 in Section 2 describing bounded polynomials and their associated regularity structure. We give the proofs of Proposition 4 and Proposition 5. Set Λ :" p 1 4 Zq d , and, for any λ " pλ i q i"1..d P Λ, define U λ :" Lemma -One can construct tuples `ϕλ , pψ i λ q i"1..d ˘λPΛ of smooth real-valued functions on R d , with compact support, with the following properties.

ś d i"1 `λi ´3
(a 1 ) One has ϕ λ pxq ě 0, for any x P R d , and ϕ λ pxq " 0 for any x P U c λ , for any λ P Λ. (a 2 ) One has ř λPΛ ϕ λ pxq " 1, for any x P R d . (a 3 ) For any N ą 0, there is a constant C N independent of α such that ˇˇpB ϕ λ qpxq ˇˇď C N dpx, U c λ q N , for any λ P Λ, x P R d , and | | ď N .

(b) The functions ψ i λ are uniformly bounded and ψ i λ pyq ´ψi λ pxq " y i ´xi , for any x, y P U λ .

Proof -We let the reader construct a partition of unity tϕ λ u satisfying assumptions a 1 to a 3 . The third property is ensured if we impose ϕ λ pxq » exp ˜´1 ˇˇx i ´pλ i ˘3 16 q ˇˇw hen x P U λ is near BU λ X x i " λ i ˘3 16 ( . For each λ P Λ, we choose a smooth function ψ i λ such that ψ i λ pxq "

# x i ´λi x P U λ , 0 x R V λ :" ś d i"1 `λi ´1 4 , λ i `1 4 ˘,
and |ψ i λ pxq| ď 1, for any x P R d . Recall we define B r pf q :" }f } C r b if r P N, and B r pf q :" }f } C r if r P p0, 8qzN. Lemma -For any f P C Since they are sums of functions with disjoint supports, we have φ e , x i e P C 8 b pR d q. Proof of Proposition 4 -For x " px i q d i"1 , we define |x| 8 :" sup i"1,...,d |x i |. If |y ´x| 8 ě 1 2 , then the left hand side of (2.21) is bounded by CB r pf q À CB r pf q|y ´x| 8 , with some constant C depending on Φλ and ψ i λ. If |y ´x| 8 ă 1 2 , then there is no pair pλ, λ 1 q such that λ ‰ λ 1 and px, yq P V λ ˆVλ 1 . Hence there exists λ P Λ such that the left hand side of (2.21) is equal to that of (B.1), so the required estimate follows.

Proof of Proposition 5 -We need to show that the component of f pyq ´x g yx f pxq on X k e is no greater than a constant multiple of B r pf q |y ´x| r´|k| . Note that x g yx f pxq is given by

x g yx f

( 2 ). 1 )

 21 One can associate to any modelled distribution f " ÿ τ PB;|τ |ăγ f τ τ P D γ pT , gq, a distribution rrf ss M P C γ pR d q such that one defines a reconstruction Rf of f setting Rf :" ÿ τ PB;|τ |ăγ P f τ rrτ ss M `rrf ss M . (1Each coefficient f τ , also has a representation f τ " ÿ τ ăµ;|µ|ăγ P f µ rrµ{τ ss g `rrf τ ss g , (1.2)

Theorem 2 .

 2 Given any family of distributions `rrτ ss P C |τ | pR d q ˘τPB;|τ|ď0 , there exists a unique admissible model M " pg, Πq on T such that one has Πτ :" ÿ σăτ P gpτ {σq rrσss `rrτ ss, (1.3) for all τ P B with |τ | ď 0.

∆

  `: T `Ñ T `b T satisfying by construction the itendity pId b ∆ `q∆ " p∆ b Idq∆ encoded in identity (2.6). Developing f ν pzq in (2.6) in terms of another reference point leads by consistency to the identity pId b ∆ `q∆ `" p∆ `b Idq∆ `.

Proposition 4 .u d i" 1 (

 41 There exists a finite set E, an open covering tU e u ePE of R d , and a family φ e , tx Þ Ñ x i e ePE of functions enjoying the following properties. (a) The functions φ e : R d Ñ r0, 8q, belong to C 8 b pR d q, φ e pxq " 0 for any x P U c e , and ř ePE φ e pxq " 1 for any x P R d . (b) The functions x Þ Ñ x i e , belong to C 8 b pR d q, and y i e ´xi e " y i ´xi for any x, y on the connected component of U e . (c) For any f P C 8 b pR d q and r ą 0, we have ˇˇf pyq ´ÿ ePE ÿ |k|ăr B k pφ e f qpxq k! py e ´xe q k ˇˇÀ B r pf q |y ´x| r , (2.21)

2 .

 2 Let then ζ : R d Þ Ñ R, stand for a smooth 'noise' and ˝stand for an element of T such that Πp˝q " ζ. We assign homogeneity α ´θ to ˝, and |τ | `θ to any Iτ , and |τ 1 ¨¨¨τ k | " |τ 1 | `¨¨¨`|τ k |, for all τ i P T . Set gpI˝q :" ΠpI˝q :" Ipζq ": Z. Denote by the unconventional sign a the resonant operator from the paraproduct decomposition of a product -see Appendix A. Then Πp˝Ip˝qq " ζZ " P Z ζ `Pζ Z `apZ, ζq, and we read on this expression that rr˝Ip˝qss M " P ζ Z `apZ, ζq.

Theorem 14 .

 14 Fix a regularity exponent γ P R, and a model M " pg, Πq on the regularity structure T . One can associate to any modelled distribution f " ÿ τ PB;|τ |ăγ f τ τ P D γ pT , gq, a distribution rrf ss M P C γ pR d q such that one defines a reconstruction Rf of f setting Rf :" ÿ τ PB;|τ |ăγ P f τ rrτ ss M `rrf ss M .

Corollary 15 .

 15 Assume we are given a map g : R d Ñ G `such that the bound (2.16) is satisfied. Let a family `rrτ ss P C |τ | pR d q ˘τPB,|τ|ď0 be given. For any τ P B with |τ | ď 0, set Πτ :" ÿ σăτ P gpτ {σq rrσss `rrτ ss. Then pg, Πq is a model on the regularity structure T ď0 , and it has a unique extension into a model on T . Proof -' Pick a basis vector τ P B with |τ | ď 0, and assume that pg, Πq is a model on the sector T ă|τ | . Set for all x P R d h τ pxq :" ÿ σăτ g x pτ {σqσ; this defines a modelled distribution in D |τ | pT , gq. Then the bound |xΠ g x τ, ϕ λ x y| À λ |τ | is equivalent to that Πτ is (one of) the reconstructions of h τ . From Theorem 14, the distribution ÿ σăτ P gpτ {σq rrσss `rrh τ ss is a reconstruction of h τ . Since Πτ ´hτ " rrτ ss ´rrh τ ss P C |τ | pR d q,

  sense.' There exist operators I ek : T Þ Ñ T `, indexed by e P E and k P N d , with positive homogeneities|X i e | " 1, ˇˇI ek τ ˇˇ" |τ | `θ ´|k|. ' One has ∆ `1 " 1 b 1, ∆ `X i e " X i e b 1 `1 b X i e, and the operators ∆ and ∆ `are related by the intertwining relations∆ `pI ek τ q " pI ek b Idq∆τ `ÿ PN d X e ! b I ek ` τ. (4.8)In addition, T satisfies the following assumptions.' There exists an operator I : T Þ Ñ T , with ˇˇIτ ˇˇ" |τ | `θ.' For any τ P B, one has∆pIτ q " pI b Idq∆τ `ÿ ePE, PN d X e ! b I e` τ.(4.9)Note that identity (4.9) identifies I ek τ as Iτ {X k e , for any k P N d , e P E. The operators I ek are the regularity structure counterparts of the operators B k pφ e Iq. Note that the restrictions on the index sets in identities (4.8) and (4.9) to indices with |k|`| | ă |τ |`θ, are redundant with the fact that I ek ` is null on T β , for β ď ´|k| ´| |. In applications to the study of stochastic PDEs with derivatives of unknown functions, such as the KPZ equation, we can also assume the existence of other operators I k : T Ñ T , associated with the integration operator B k I.Proposition 19. Let pg, Πq be a usual model on T . We assume the commutation rule ΠpIτ q " IpΠτ q.

  with |τ | " α, where J e pxq is an operator on T rather than on D γ pT , gq, defined by J e pxqσ "

( a )

 a }P f g} C β À }f } L 8 }g} C β . (b) If α ă 0, then }P f g} C α`β À }f } C α }g} C β . (c) If α `β ą 0, then › › a pf, gq › › C α`β À }f } C α }g} C β .

8 .

 8 16 , λ i `3 16 ˘. This family of bounded open subsets of R d cover R d , and are uniformly locally finite covering, i.e., sup xPR d # λ P Λ; x P U λ ( ă For x P R d and A Ă R d , set dpx, Aq :" inf |x ´y|; y P A ( .

8 b

 8 pR d q and r ą 0, we have the estimate ˇˇˇˇˇp ϕ λ f qpyq ´ÿ |k|ăr B k pϕ λ f qpxq k! `ψλ pyq ´ψλ pxq ˘kˇˇˇˇˇˇÀ B r pf q|y ´x| r . (B.1) Proof -For x, y P U λ , since pψ λ pyq ´ψλ pxqq k " py ´xq k , equation (B.1) is just a usual Taylor expansion. For x, y R U λ , the left hand side of (B.1) is equal to 0. Let y P U λ and x R U λ . Then the left hand side of (B.1) is equal to |pϕ λ f qpyq|. By assumptions, we have ˇˇpϕ λ f qpyq ˇˇÀ }f } L 8 d `y, U c λ ˘r ď }f } L 8 |y ´x| r, with an implicit constant in the first inequality depending only on r. We have the same estimate when x P U λ and y R U λ .Define now E :" 0,

  The size estimate ˇˇg yx pτ {ηq ˇˇÀ |y ´x| |τ |´|η| , then shows that h τ is a modelled distribution in D |τ | pT , gq. Here is another example.

	Lemma 6. Let f "	ř

.28) Then, it follows from identity (2.14) giving ∆ `pτ {σq, in Lemma 3, that x g yx h τ pxq " ÿ ηďσăτ g yx pσ{ηqg x pτ {σqη " ÿ ηăτ `gy pτ {ηq ´gyx pτ {ηq ˘η " h τ pyq ´ÿ ηăτ g yx pτ {ηqη. σPB f σ p¨q σ, be an element of D γ pT , gq. Then, for each τ P B, the T `-valued function f {τ :" ÿ σěτ f σ p¨q σ{τ. is an element of D γ´|τ | pT `, gq. Proof -This comes from the identity pf {τ qpyq ´x g yx `pf {τ qpxq " ÿ σěτ ´f σ pyq ´ÿ µěσ f µ pxq g yx pµ{σq ¯σ{τ, and the fact that f is a modelled distribution. Recall β 0 " min A, and fix r ą |β 0 ^0|. Theorem 7. (Hairer's reconstruction theorem) Let pg, Πq be a model over T . Fix a regularity exponent γ P R. There exists a linear continuous operator

  If γ ą 0, then there exists a unique function f Λ P C γ pR d q such that ˇˇA PpΛq ´fΛ ( ´Λx , P i px ´¨q

				EˇˇˇÀ
					2 ´iγ ,	(3.2)
	uniformly in x P R d .
	' If γ ă 0, then we have
		ˇˇ@	PpΛq ´Λx , P i px ´¨q	DˇˇÀ	2 ´iγ ,	(3.3)
	uniformly in x P R d .
	Proof -(i) We prove that one has
			ˇˇ∆ j `PpΛq ´Λx ˘pxq ˇˇÀ 2 ´jγ ,	(3.4)
	uniformly in x P R d . We write for that purpose
		ij
	PpΛqpyq ´Λx pyq "	ÿ	P j py ´uqQ j py ´vq `Λu pvq ´Λx pvq ˘dudv ´S pΛ x q
	jě´1
		ÿ	ÿ
	"
	1ď ďN

and real-valued coefficients c yx depending measurably on x and y, such that sup xPR d sup jě´1 2 jβ ˇˇ@ Θ x , P j px ´¨q Dˇˇă 8, |||c ||| C γ´β 2 ă 8, for regularity exponents β ă γ, for all 1 ď ď N . Denote PpΛq " P y,z pΛ y pzqq below. ' jě´1 ij P j py ´uqQ j py ´vqc ux Θ x pvq dudv ´S pΛ x q.

  uniformly in x, for all i ě ´1, giving indeed f 1 Λ " f Λ . If γ ă 0, we directly have from (3.4) that

	ˇˇ@	PpΛq ´Λx , P i px ´¨q	DˇˇÀ ÿ	ˇˇ@	PpΛq ´Λx , Q j px,	¨qDˇˇÀ ÿ
			jďi´2

1 Λ were another C γ function satisfying estimate (3.2), one would have ˇˇ@ f Λ ´f 1 Λ , P i px ´¨q DˇˇÀ 2 ´iγ , jďi´2 2 ´jγ À 2 ´iγ .

  The first three terms on the right hand side are smooth. To prove that the last term on the right hand side is an element of C α`β pR d q, it is sufficient, from Proposition 8, to see that ˇˇQ Condition (3.7) is reminiscent of Gubinelli's notion of controlled path[START_REF] Gubinelli | Controlling rough paths[END_REF]. Recall from Proposition 35 in[START_REF] Bailleul | Higher order paracontrolled calculus[END_REF] that for f P C α 1 and g P C α 2 , with α 1 , α 2 positive and α 1 `α2 P p0, 1q, one has ˇˇpP f gqpyq ´pP f gqpxq ´f pxq `gpyq ´gpxq ˘ˇˇÀ |y ´x| α 1 `α2 . `τ { `σ1 ˘¨¨¨g x `σn´1 { `σn ˘´g y `σn {

				gq `Pf pS gq	´N ÿ n"1	¨x gqpyq P anbn pS gq ´Px,y ´pP f 7 "
				´S pP f gq `Pf pS gq	´N ÿ n"1	¨x gqpyq P anbn pS gq ´Px,y ´pP f 7 ¯.
			ż
	À	ÿ	2 ´iβ	ˇˇP j pz ´xqQ j pz ´yq ˇˇ`| y ´x| α `2´iα ˘dxdy
	i;|i´j|ď4
		ÿ
	À 2 Identity (2.26) provides another example of a setting where Proposition 10 applies, as it i;|i´j|ď4
	states that one has for any τ, σ P B	gy
	`τ {	`σ˘´g x `τ { `σ"
	ÿ ně1 p´1q n σă `σnă `¨¨¨ă ÿ `τ g x `σ˘´g x `σn {	`σ˘ḡ
		yx `τ { `σ˘,

j ´`P f 7 ¨x g ˘pyq ¯ˇˇÀ 2 ´jpα`βq , for all j ě 1. Recall for that purpose the bound (3.6). Then we have for ˇˇQ j ´`P f 7 ¨x g ˘pyq ¯ˇť he upper bound ÿ i;|i´j|ď4 ˇˇˇż P j pz ´xqQ j pz ´yq ˆż P i py ´uqQ i py ´vqf 7 ux gpvqdudv ˙dxdy ˇˇÀ ÿ i;|i´j|ď4 ż ˇˇP j pz ´xqQ j pz ´yq ˇˇˆż ˇˇP i py ´uq ˇˇ|u ´x| α du ˙|∆ i gpyq| dxdy ´iβ `2´jα `2´iα ˘À 2 ´jpα`βq . It follows from Proposition 10 above that R ˝pf, g, hq P C α 1 `α2 `β , for any h P C β , with β P R. with ˇˇg yx `τ { `σ˘ˇÀ |y ´x| |τ |´|σ| .

  Zq `Pζ `a pZ, Zq ˘`2Z a pZ, ζq `2CpZ, Z, ζq `a`a pZ, Zq, ζ "P Z 2 ζ `PZ `2P ζ Z `apZ, ζq ˘`! 2Spζ, Z, Zq `Pζ `a pZ, Zq ˘`2P apZ,ζq Z `2 a `Z, apZ, ζq 2CpZ,so the term inside the brackets t¨¨¨u defines rr˝Ip˝q 2 ss M . As can be seen from these examples, these expressions of the brackets using the operators from[START_REF] Bailleul | Higher order paracontrolled calculus[END_REF] quickly get seemingly complicated. σθ P gpθ{ `ρq rrρss g `ÿ θPB

			Z, ζq	`a`a pZ, Zq, ζ ˘),
	Proposition 13. We have
			ÿ
		gpτ {σq "
			ηPB, σăηăτ
	gpτ {σq "	ÿ θPB `p∆τ q σθ gpθq "	ÿ θ,ρPB `,1ă `ρă	`θp∆τ q

P gpτ {ηq rrη{σss g `rrτ {σss g , for all σ, τ P B with σ ă τ .

Proof -With τ {σ " ř θPB `p∆τ q σθ θ, and θ{ `ρ " ř κPB `p∆ `θq ρκ κ, we have on the one hand, from

Proposition 12,

  , the usual property (4.4) holds for any τ P IB if and only if, for every τ P B, and x P R d , one has g x pI ek τ q "

				ÿ	g x pτ {σq I e k	`Πg x σ ˘pxq	(4.11)
				σďτ ;|k|ă|σ|`θ
	Proof -Since I ek τ " D k e Iτ ,
	g x pI ek τ q " ´Bk y	! φ e `Π g x P ą|k| b g x ˘∆Iτ	)	pyq	¯ˇˇy	"x	.
	" B k y	! φ e	´ÿ σ;|σ|`θą|k| p Πg x Iσqg x pτ {σq ¯)pyq ˇˇy	"x
	" B k y	! φ e	´ÿ σ,η;|σ|`θą|k|	pΠIηqg ´1 x pσ{ηqg x pτ {σq ¯)pyq ˇˇy	"x
	" I e k	¨ÿ σ,η;|σ|ą|k|´θ	pΠηqg ´1 x pσ{ηqg x pτ {σq 'pxq
	" I e k	¨ÿ σ;|σ|`θą|k| pΠ g x σqg x pτ {σq 'pxq.
	Definition 20. A model pg, Πq on T is said to be admissible if the identities g x pX k e q " ΠX k e pxq " x k e , the commutation rule (4.10), and (4.11) are satisfied.

  Let us show the above equalities. First we assume that (4.12) holds for any σ P B with σ ă τ . where g xx pτ {ηq " 1 η"τ , which yields (4.13). Moreover, we have (4.14) as follows.

	g x Iτ " IpΠ g x τ q	´ÿ ePE,|k|ă|τ |`θ	pp¨q e ´xe q k k!	I e k pΠ g x τ qpxq,	(4.12)
	g ´1 x pI ek τ q " ´ÿ ;|k` |ă|τ |`θ	p´x e q !	I e k` pΠ g x τ qpxq,	(4.13)
	g yx pI ek τ q "	ÿ σďτ,|k|ă|σ|`θ	g yx pτ {σqI e k pΠ g y σqpyq	´ÿ |k` |ă|τ |`θ	py e ´xe q !	I e k` pΠ g x τ qpxq. (4.14)
	Then by (4.10) and (4.11),
	Π g x Iτ " ΠIτ	´ÿ σăτ	g x pτ {σqΠ g x Iσ	´ÿ e,k	pp¨q e ´xe q k k!	g x pI ek τ q
	" IpΠτ q	´ÿ σăτ	x σq g x pτ {σq ¨IpΠ g	´ÿ ePE,|k|ă|σ|`θ	pp¨q e ´xe q k k!	I e k pΠ g x σqpxq '
											¨Ie	'
	´ÿ e,|k|ă|τ |`θ	pp¨q e ´xe q k k!	k	`Πg x τ ˘pxq	`ÿ σăτ,|k|ă|σ|`θ	g x pτ {σq I e k	`Πg x σ ˘pxq
	" I `Πτ	´ÿ σăτ	g x pτ {σqΠ g x σ	˘´ÿ e,|k|ăτ `θ pp¨q e ´xe q k k!	I e k	`Πg x τ ˘pxq,
	where by using Πτ ´řσăτ g x pτ {σqΠ g x " Π g x τ we have (4.12). On the other hand, by (4.8),
	0 " g xx pI ek	` τ q "		ÿ σďτ	g x pI ek	` σqg ´1 x pτ {σq	`ÿ m;|k` `m|ă|τ |`θ	x m e m!	g ´1 x pI ek	` `mτ q
	From this identity and (4.11),
	g ´1 x pI ek τ q "	ÿ ,m	p´x e q !	x m e m!	g ´1 x pI ek	` `mτ q "	´ÿ σďτ,	p´x e q !	g x pI ek	` σqg ´1 x pτ {σq
		"	´ÿ σďτ,	p´x e q !	g ´1 x pτ {σq	ηďσ,|k` |ă|η|`θ ÿ	g x pσ{ηq I e k`	`Πg x η ˘pxq
		"	´ÿ ηďτ,|k` |ă|η|`θ	p´x e q !	g xx pτ {ηqI e k` pΠ g x ηqpxq,
	g yx pI ek τ q "	ÿ σďτ	g y pI ek σqg ´1 x pτ {σq `ÿ	y e !	g ´1 x pI ek	` τ q
		"						ÿ	g y pσ{ηqg ´1 x pτ {σqI e k pΠ g y ηqpyq
			ηďσďτ,|k|ă|η|`θ
			´ÿ ,m;|k` `m|ă|τ |`θ	y e !	p´x e q m m!	I e k` `mpΠ	g x τ qpxq
		"	ÿ ηďτ,|k|ă|η|`θ	g yx pτ {ηqI e k pΠ g y ηqpyq	´ÿ |k` |ă|τ |`θ	py e ´xe q !	I e k` pΠ g x τ qpxq.

  1, freely generated by the set tX i e u ePE,i"1,...,d Y I ek τ ( τ PB,ePE,kPN d ,|τ |`θ´|k|ą0 , ' For any τ, σ P B, the element τ {σ P T `is contained in the subalgebra generated by the set tX i e u ePE,i"1,...,d Y I ek η

	(
	ηPB,ePE,kPN

d ;|η|ă|τ |,|η|`θ´|k|ą0 , Theorem 21. Let a regularity structure T equipped with an abstract integration map satisfy assumptions (A), (B) and (C). Given any family of distributions `rrτ ss P C |τ | pR d q ˘τPB;|τ|ď0 , there exists a unique admissible model pg, Πq on T such that one has Πτ :" ÿ σăτ P gpτ {σq rrσss `rrτ ss, (4.15) for all τ P B with |τ | ď 0. Proof -For α P A, define T pαq as the subalgebra of T `generated by tX i e u ePE,i"1..d Y I ek τ ( τ PB,ePE,kPN d ,|τ |ăα . By Assumption (C), T pαq is closed under ∆ `.

  |τ |`θ´|k| , for every k P N d and e P E. Set `Υe α h τ ˘pxq :" I e0 h τ pxq `ÿ kPN d g x pI ek τ q Proposition 19 is used to prove the following fact, proved below.

									X k e k!	.
	Lemma 22. One has Υ e α h τ P D α`θ `T ằα , g pαq	˘.
	But
				`Υe α h τ ˘pyq ´y g pαq yx	`Υe α h τ ˘pxq
	has X k e component equal to
	ˇˇˇˇg	y pI e` τ q	´ÿ ηăτ	g x pτ {ηqg yx pI e` ηq	´ÿ m	g x pI e` `mτ q	py e ´xe q m m!	ˇˇˇˇÀ |y ´x| |τ |`θ´| | ,
	from lemma 22; we recognize ˇˇg	pα	`q

  For the I e`t erm, one has the elementary estimate › › I e``h τ pyq ´x g yx h τ pxq ˘› › β À › › h τ pyq ´x g yx h τ pxq › › β´θ ď }h τ } D α |y ´x| α`θ´β . The J e and N e terms take values in the polynomial part of T `. Write τ X k e for the X k ecomponent of τ P T `. Decompose Kpy, zq " ř 8 n"0 K n py, zq, and let J e ": `φe pyqK n py, zq ˘Πg y `hτ pyq ´x g yx h τ pxq ˘βpzq dz |y ´x| α´β 2 p|k|´β´θqn À |y ´x| α`θ´|k| . `φe pyqK n py, zq ˘pΠ g y τ qpzqdz ˇˇÿ | |ăα`θ´|k| |y ´x| 2 p| |´α´θqn 'À |y ´x| α`θ´|k| ' To deal with the sum over n ď N , we use the p‹q-decomposition. For p‹q n 1 , note that since |y ´x| » 2 ´N ď 2 ´n, the function B k pφ e K n q is supported on a ball Bpx, C2 ´nq, for some positive constant C. From Taylor formula with bounded polynomials proved in Appendix B, is embedded into the space C β 0 pR d q for any β ă α -see e.g. [1, Proposition 2.74]. For α P p0, 8qzN, the norm }f } C α is equivalent to the Hölder norm }f } C α :" ÿ kPN d ,|k|ăα }B k f } L 8 `ÿ kPN d ,|k|"rαs }B k f } (α ´rαs)-Hölder , where }B k f } (α ´rαs)-Hölder is the infimum of constants C such that the property |B k f pyq ´Bk f pxq| ď C|y ´x| α´rαs , holds for any x, y P R d . For α P N, the space C α pR d q is strictly larger than the space C α b pR d q with the norm }f } C α b :" ÿ kPN d ,|k|ďα }B k f } L 8 ; see e.g. [1, page 99]. Bony's paraproduct P and resonant operator a are defined by

	N e ":	ř n N e n , be the corresponding operators. We have	ř n J e n and
	´J e n pyq `hτ pyq ´x g yx h τ pxq ˘``N e n h τ qpyq ´x g yx `pN e n h τ qpxq ¯Xk e
	" ´J e n pyq `hτ pyq ´x g yx h τ pxq ˘¯X k e	`´`N e n h τ qpyq ´x g yx `pN e n h τ qpxq ¯Xk e	": p˚q 1 n `p˚q 2 n
	and
		´J e n pyq `hτ pyq ´x g yx h τ pxq ÿ	˘``N e n h τ qpyq ´x g yx `pN e n h τ qpxq ¯Xk e
		"	ÿ βPA,|k|ăβ`θ P f g :" ż R d 1 k! B k pφ e K n q α`θ´|k| 1 k! B k y,x i,jě´1 y `żR d iďj´2	pzq pΠ g x τ qpzq dz
			`żR d y pφ R d 1 k! B k 1 k! B k pφ e K n q α`θ´|k| y,x	pzq pΠ g x τ qpzq dz
			`ÿ βPA,|k|ěβ`θ	ż R d	1 k!	B k
		": p‹q 1 n `p‹q 2 n ,
	where
	B k pφ e K n q α`θ´|k| y,x	pzq :" B k y `φe pyqK n py, zq	˘´ÿ | |ăα`θ´|k|	py e ´xe q !	B k`
			ˇˇp˚q 1 n ˇˇÀ	ÿ	ÿ
		nąN		nąN	βPA,|k|ăβ`θ
	By the definition of N , we get
	py e ´xe q ! g yx pτ {σqI e`σ `ÿ k,σ;|k|ă|σ|`θ X k e g yx pτ {σqI e I e k` pΠ g x τ qpxq k pΠ g y σqpyq. B k ÿ σ 1 k! g yx τ " pI e``J e pyqq x ÿ nąN ˇˇp˚q 2 n ˇˇď ÿ nąN ˇˇˇż R d ˇˇˇˇˇÿ y nąN py e ´xe q ! ż R d 1 B k` x `φe pxqK n px, zq ˘pΠ g x τ qpzqdz k! | |`|k|ăα`θ k! They are equal because of (4.14). We use the interwining relation (4.16) to write and `Υe ÿ nąN ¨2p|k|´α´θqn `ÿ α`θ´|k|	ˇˇˇˇÀ
						y,x

α h τ ˘pyq ´x g yx ``Υ e α h τ ˘pxq " `Υe α h τ ˘pyq ´x g yx ``I e``J e pxq ˘hτ pxq ´x g yx `pN e h τ qpxq " `Υe α h τ ˘pyq ´`I e``J e pyq ˘x g yx h τ pxq ´x g yx `pN e h τ qpxq " I e`´h τ pyq ´x g yx h τ pxq ¯`J e pyq ´hτ pyq ´x g yx h τ pxq ¯`´p N e h τ qpyq ´x g yx `pN e h τ qpxq ¯. e pyqK n py, zqq Π g y `x g yx h τ pxq ´hτ pyq ˘pzq dz " ż y `φe pyqK n py, zq ˘Πg y `x g yx h τ pxq ´hτ pyq ˘βpzq dz x `φe pxqK n px, zq ˘.

Write |y ´x| » 2 ´N . We use the p˚q-decomposition, with J n and N n separated, to estimate the sum over n ą N , and the p‹q-decomposition to estimate the sum over n ď N . Each decomposition is well-adapted to get N -independent upper bounds.

' For n ą N , we have from the bound (4.7) and its derivatives the estimate ÿ

  pxq "The 1-coefficient of f pyq ´x g yx f pxq is ˇˇˇˇˇÿ

			ÿ ePE	ÿ | `m|ăr	B `mpφ e f qpxq !m!	`xe pyq ´xe pxq	˘	X m e .
	ePE	pφ e f qpyq	´ÿ ePE	ÿ |k|ăr	B k	`φe f qpxq k!	px e pyq ´xe pxq ˘kˇˇˇˇˇˇÀ	B r pf q |y ´x| r .

This is estimate (2.21). For the X k e -coefficient of f pyq ´x g yx f pxq, with k ‰ 0, one has 1 k! ˇˇˇˇˇB k pφ e f qpxq ´ÿ | |ăr´|k| B B k pφ e f qpxq ! `xe pyq ´xe pxq ˘ ˇˇˇˇˇÀ B r pf q |y ´x| r´|k| . (B.2)

This is shown by the similar argument to the proof of the estimate (B.1).

Corollary 11. For any family ph σ q σPB `zt1u , with h σ P C |σ| , the sum Then rrτ ss g P C |τ | pR d q, for all τ P B `, and rrσss M P C |σ| pR d q, for all σ P B. Furthermore, the maps M Þ Ñ rrτ ss g P C |τ | pR d q, M Þ Ñ rrσss M P C |σ| pR d q, are continuous, for any τ P B `and σ P B.

Proof -First we construct the family rrτ ss g ; τ P B `(.

'

The proof proceeds by induction on the homogeneity |τ | of τ , starting with the case τ " 1, for which we set rr1ss g :" gp1q " 1, the constant function on R d , equal to 1. Let |τ | ą 0 and assume that the functions rrσss g P C |σ| pR d q satisfying (3.8) have been constructed for any σ P B `with |σ| ă |τ |. Applying the two-parameter extension of the paraproduct operator P to identity (2.26) with σ " 1 and ă `order, we have

`τ P gpτ { `σ1 q¨¨¨gpσ n´1 { `σnq gpσ n q `Px,y `gyx pτ q ˘. We used the fact that P f 1 " 0, for any f P S 1 pR d q, to remove the zero-contribution from the σ n " 1 term in the sum. Note that P 1 gpτ q " gpτ q ´S gpτ q, is the sum of gpτ q and a smooth term depending continuously in any Hölder topology on gpτ q P L 8 pR d q. Expanding gpσ n q by induction, we have

for any τ P BztX k e u e,k , where D k e τ :" k!pτ {X k e q.

Abstract integration operator and admissible models

Fix a positive regularity exponent θ, and let T be a concrete regularity structure. Assume for simplicity that β 0 ą ´θ, so all the elements of T have homogeneity strictly greater than ´θ. We consider in this section concrete regularity structures T equipped with an abstract integration operator I, that is a regularity structure counterpart of an operator I that is typically an integral operator given by a kernel that is singular on the diagonal, such as the Green function of a differential operator. The exponent θ quantifies the regularizing properties of the operator I in the Hölder or Besov scale.

Remark -The dynamical Φ 4 3 equation B t Φ " ∆Φ ´Φ3 `ξ seems not to satisfy the above assumption. Indeed, one should choose β 0 " ´5 2 ´ε, and θ " 2 for the heat kernel in any dimension. However, if we decompose Φ " Ψ `v, where B t Ψ " ∆Ψ `ξ and B t v " ∆v ´pv `Ψq 3 , then one can choose β 0 " 3p´1 2 ´εq instead, so the equation for v satisfies β 0 ą ´θ. A general da Prato-Debussche trick is described in Section 6 of [START_REF] Bruned | Renormalising SPDEs in regularity structures[END_REF], that allows to set the study of a generic subcritical singular partial differential equation, within the setting of regularity structures, under the assumption β 0 ą ´θ. for χ a smooth real-valued function with compact support identically equal to 1 in a neighbourhood of 0, in dimension at least d ě 3, for which one can take any θ ă 2. The associated integration map sends any C β pR d q, into C β`2 pR d q, for β R Z -these are Schauder estimates. Note however that pIζqpxq is not defined for a generic distribution ζ.

Integration operator

Let

we have ˇˇB m z B k pφ e K n q α`θ´|k| y,x pzq ˇˇÀ B α`θ pB m z K n p¨, zqq |y ´x| α`θ´|k| , (4.17)

with B r `Bm z K n p¨, zq ˘À 2 pd`|m|`r´θ´εqn , from either (4.5) or the interpolation theorem 2.80 in [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]. Hence ˇˇB m z B k pφ e K n q α`θ´|k| y,x pzq ˇˇÀ 2 pd`|m|`α´εqn |y ´x| α`θ´|k| .

It follows then from the proof of Lemma 18 that we have

˘ˇˇÀ 2 ´εn |y ´x| α`θ´|k| , so the sum over n ď N is independent of N , of order |y ´x| α`θ´|k| . As for the p‹q n 2 -terms, they involve some indices ζ with |k| ě ζ `θ, so the same elementary bounds as above give

|y ´x| α´ζ 2 p|k|´ζ´θ´εqn À 2 ´εn |y ´x| α`θ´|k| , since 2 n ď |y ´x| ´1. The sum over n ď N of the p‹q n 2 is thus independent of N , of order |y ´x| α`θ´|k| .

Remark -If pT `, T q satisfies |τ | `θ R N for any τ P B, then we can choose ε " 0 for the estimate (4.5) on the kernel K n . We need to modify the argument for the sum over n ď N . For p‹q 1 n , since B k pφ e K n q α`θ´|k| y,x " B k pφ e K n q α`θ`δ´|k| y,x in (4.17) for small δ ą 0 such that pα `θ, α `θ `δq X N " H, we have

A -Paraproducts

We summarize in this section some basic concepts and results of the Littlewood-Paley theory. Let tρ i u 8

i"´1 be a dyadic partition of unity of R d , i.e., ρ i : R d Ñ r0, 1s is a compactly supported smooth radial function with the following properties.

and supppρ 0 q Ă x P R d ; 3 4 ă |x| ă 8 3 ( . ' ρ i pxq " ρ 0 p2 ´ixq for any x P R d and i ě 0. ' ř 8 i"´1 ρ i pxq " 1 for any x P R d . We define the Littlewood-Paley blocks t∆ i u 8 i"´1 by ∆ i f :" F ´1pρ i F f q, f P S 1 pR d q, where F is a Fourier transform on R d defined by Now we define the Hölder-Besov spaces. For any α P R and f P S 1 pR d q, we define }f } C α :" sup iě´1 2 αi }∆ i f } L 8 pR d q .

We denote by C α pR d q the space of all f P S 1 pR d q with }f } C α ă 8. This definition does not ensure the separability of C α pR d q, so it may be better to consider the space C β 0 pR d q, the completion of SpR d q under the norm } ¨}C α . However, it does not matter because C α pR d q