Higher order paracontrolled calculus

Ismaël Bailleul, Frederic Bernicot

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Ismaël Bailleul, Frederic Bernicot. Higher order paracontrolled calculus. Forum of Mathematics, Sigma, 2019, 7, 10.1017/fms.2019.44 . hal-01382067

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# Higher order paracontrolled calculus

Ismaël Bailleul, Frederic Bernicot

► To cite this version:

Ismaël Bailleul, Frederic Bernicot. Higher order paracontrolled calculus. Forum of Mathematics, Sigma, 2019, 7, 10.1017/fms.2019.44. hal-01382067

HAL Id: hal-01382067

https://hal.science/hal-01382067v1

Submitted on 30 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
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Distributed under a Creative Commons Attribution 4.0 International License
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# HIGH ORDER PARACONTROLLED CALCULUS

ISMAËL BAILLEUL¹ and FRÉDÉRIC BERNICOT²

¹ Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France;
email: ismael.bailleul@univ-rennes1.fr

² Laboratoire de Mathématiques Jean Leray, CNRS – Université de Nantes,
2 Rue de la Houssinière 44322 Nantes Cedex 03, France;
email: frederic.bernicot@univ-nantes.fr

Received 14 February 2018; accepted 18 October 2019

## Abstract

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations E and F. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation

$$ (\partial_t + L)u = f(u)\zeta, $$

on a 3-dimensional closed manifold, and the generalized KPZ equation

$$ (\partial_t + L)u = f(u)\zeta + g(u)(\partial_t u)^2, $$

driven by a (1 + 1)-dimensional space/time white noise.

2010 Mathematics Subject Classification: 60H15 (primary); 35R60, 35R01 (secondary)

## Contents

1 Paracontrolled calculus 2

2 High order paracontrolled expansion 11

3 A toolkit for paracontrolled calculus 15

© The Author(s) 2019. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
<figure><img src="image_2.png" /></figure>

# HIGH ORDER PARACONTROLLED CALCULUS

ISMAËL BAILLEUL¹ and FRÉDÉRIC BERNICOT²

¹ Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France;
email: ismael.bailleul@univ-rennes1.fr

² Laboratoire de Mathématiques Jean Leray, CNRS – Université de Nantes,
2 Rue de la Houssinière 44322 Nantes Cedex 03, France;
email: frederic.bernicot@univ-nantes.fr

Received 14 February 2018; accepted 18 October 2019

## Abstract

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations E and F. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation

$$ (\partial_t + L)u = f(u)\zeta, $$

on a 3-dimensional closed manifold, and the generalized KPZ equation

$$ (\partial_t + L)u = f(u)\zeta + g(u)(\partial_t u)^2, $$

driven by a (1 + 1)-dimensional space/time white noise.

2010 Mathematics Subject Classification: 60H15 (primary); 35R60, 35R01 (secondary)

## Contents

1 Paracontrolled calculus 2

2 High order paracontrolled expansion 11

3 A toolkit for paracontrolled calculus 15

© The Author(s) 2019. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
**4** Nonlinear singular PDEs 55

**Appendix A. Details on the parabolic setting** 65

**Appendix B. Paracontrolled expansion formula** 77

**Appendix C. Examples of Tits-Weyl models** 80

**References** 93

**1. Paracontrolled calculus**

Thirty years after Lyons' seminal work on controlled differential equations [22], it is now well-understood that the construction of a robust approximation theory for continuous time stochastic systems, such as stochastic differential equations or stochastic partial differential equations, requires a twist in the notion of noise that allows to treat the resolution of such equations in a two-step process.

(a) Enhance the noise $\zeta$ into an enriched object $\hat{\zeta}$ that lives in some space of analytic objects – this is a purely *probabilistic* step.

(b) Given any such object $\hat{\zeta}$ in this space, one can introduce a Banach space $S$ such that the equation makes sense for the unknown in the image of $S$ by a simple $\hat{\zeta}$-dependent map. The equation can be formulated in $S$, and solved uniquely by a *deterministic* analytic argument, such as the contraction principle, which gives the continuity of the solution as a function of $\hat{\zeta}$.

These two steps are very different in nature and require totally different tools. The present work deals with the deterministic side of the story, point (b), for the study of singular partial differential equations (PDEs). The term singular refers here to the fact that the 'noise' in the equation is not regular enough for all the expressions in the equation to make sense analytically, given the expected regularity of the solution in terms of the regularity of the 'noise'. Recall that one can generically not make sense of the product of a distribution with a continuous function.

**1.1. Overview.** Hairer's theory of regularity structures [19] provides undoubtedly the most complete picture for the study of a whole class of singular stochastic PDEs from the above point of view – the class of the so-called singular subcritical parabolic stochastic PDEs. It comes with a very rich algebraic structure and an entirely new setting that are required to give flesh to the guiding principle
**4** Nonlinear singular PDEs 55

**Appendix A. Details on the parabolic setting** 65

**Appendix B. Paracontrolled expansion formula** 77

**Appendix C. Examples of Tits-Weyl models** 80

**References** 93

# 1. Paracontrolled calculus

Thirty years after Lyons' seminal work on controlled differential equations [22], it is now well-understood that the construction of a robust approximation theory for continuous time stochastic systems, such as stochastic differential equations or stochastic partial differential equations, requires a twist in the notion of noise that allows to treat the resolution of such equations in a two-step process.

(a) Enhance the noise $\zeta$ into an enriched object $\hat{\zeta}$ that lives in some space of analytic objects – this is a purely *probabilistic* step.

(b) Given any such object $\hat{\zeta}$ in this space, one can introduce a Banach space $S$ such that the equation makes sense for the unknown in the image of $S$ by a simple $\hat{\zeta}$-dependent map. The equation can be formulated in $S$, and solved uniquely by a *deterministic* analytic argument, such as the contraction principle, which gives the continuity of the solution as a function of $\hat{\zeta}$.

These two steps are very different in nature and require totally different tools. The present work deals with the deterministic side of the story, point (b), for the study of *singular partial differential equations* (PDEs). The term *singular* refers here to the fact that the ‘noise’ in the equation is not regular enough for all the expressions in the equation to make sense analytically, given the expected regularity of the solution in terms of the regularity of the ‘noise’. Recall that one can generically not make sense of the product of a distribution with a continuous function.

**1.1. Overview.** Hairer's theory of regularity structures [19] provides undoubtedly the most complete picture for the study of a whole class of singular stochastic PDEs from the above point of view – the class of the so-called singular subcritical parabolic stochastic PDEs. It comes with a very rich algebraic structure and an entirely new setting that are required to give flesh to the guiding principle
that a solution should be described by the datum at each point in space-time of its high order 'jet' in a basis given by the elements of the enhanced noise. Regularity structures are introduced as a tool for describing these jets. At the same time that Hairer built his theory, Gubinelli-Imkeller-Perkowski implemented in [17] this idea of giving a local/global description of a possible solution in a different way, using the language of paraproducts and avoiding the introduction of any new setting, but providing only a first order description of the objects under study. This is what we shall call from now on the *first order paracontrolled calculus*. While this kind of approach may seem far from being as powerful as Hairer's machinery, the first order paracontrolled approach to singular stochastic PDEs has been successful in recovering and extending a number of results that can be proved within the setting of regularity structures, on the parabolic Anderson model and Burgers equations [1, 2, 14, 17], the KPZ equation [18], the scalar $\Phi_3^4$ equation [9], the stochastic Navier-Stokes equation [24-26], or the study of the continuous Anderson Hamiltonian [13], to name but a few.

We develop in this work a high order version of paracontrolled calculus that allows to treat analytically within this paradigm some parabolic singular partial differential equations that are beyond the scope of the original formulation of the theory. We refer to our setting as *paracontrolled calculus*. By a 'noise' in an equation we shall simply mean a function/distribution-valued parameter $\zeta$ – realizations of a white noise are typical examples. Within our setting, and given as input a noise $\zeta$ and some initial condition, the resolution process of a typical parabolic equation

$$ \mathcal{L}u := (\partial_t + L)u = f(u, \partial u, \zeta), \qquad (1.1) $$

involves the following elementary steps. Write $\mathcal{L}^{-1} := (\partial_t + L)^{-1}$ for the resolution operator, and keep in mind that we have in hand two space-time paraproducts $P$ and $\tilde{P}$, related by the intertwining relation

$$ \mathcal{L}^{-1} \circ P = \tilde{P} \circ \mathcal{L}^{-1}; $$

all the objects are properly introduced below.

(1) **Paracontrolled ansatz.** The irregularity of the noise $\zeta$, and the form of the equation, dictate the choice of a solution space made up of functions/distributions of the form

$$ u = \sum_{i=1}^{k_0} \tilde{P}_{u_i} Z_i + u^\mu, \qquad (1.2) $$

for reference functions/distributions $Z_i$ that depend formally only on $\zeta$, to be determined later. The 'derivatives' $u_i$ of $u$ also need to satisfy a similar
that a solution should be described by the datum at each point in space-time of its high order 'jet' in a basis given by the elements of the enhanced noise. Regularity structures are introduced as a tool for describing these jets. At the same time that Hairer built his theory, Gubinelli-Imkeller-Perkowski implemented in [17] this idea of giving a local/global description of a possible solution in a different way, using the language of paraproducts and avoiding the introduction of any new setting, but providing only a first order description of the objects under study. This is what we shall call from now on the *first order paracontrolled calculus*. While this kind of approach may seem far from being as powerful as Hairer's machinery, the first order paracontrolled approach to singular stochastic PDEs has been successful in recovering and extending a number of results that can be proved within the setting of regularity structures, on the parabolic Anderson model and Burgers equations [1, 2, 14, 17], the KPZ equation [18], the scalar $\Phi_3^4$ equation [9], the stochastic Navier-Stokes equation [24-26], or the study of the continuous Anderson Hamiltonian [13], to name but a few.

We develop in this work a high order version of paracontrolled calculus that allows to treat analytically within this paradigm some parabolic singular partial differential equations that are beyond the scope of the original formulation of the theory. We refer to our setting as *paracontrolled calculus*. By a 'noise' in an equation we shall simply mean a function/distribution-valued parameter $\zeta$ – realizations of a white noise are typical examples. Within our setting, and given as input a noise $\zeta$ and some initial condition, the resolution process of a typical parabolic equation

$$ \mathcal{L}u := (\partial_t + L)u = f(u, \partial u, \zeta), \qquad (1.1) $$

involves the following elementary steps. Write $\mathcal{L}^{-1} := (\partial_t + L)^{-1}$ for the resolution operator, and keep in mind that we have in hand two space-time paraproducts $P$ and $\tilde{P}$, related by the intertwining relation

$$ \mathcal{L}^{-1} \circ P = \tilde{P} \circ \mathcal{L}^{-1}; $$

all the objects are properly introduced below.

(1) **Paracontrolled ansatz.** The irregularity of the noise $\zeta$, and the form of the equation, dictate the choice of a solution space made up of functions/distributions of the form

$$ u = \sum_{i=1}^{k_0} \tilde{P}_{u_i} Z_i + u^\sharp, \qquad (1.2) $$

for reference functions/distributions $Z_i$ that depend formally only on $\zeta$, to be determined later. The 'derivatives' $u_i$ of $u$ also need to satisfy a similar
a structure equation; their derivatives as well, and so on. See Definition 1 below. One sees the above description (1.2) of $u$ as a paracontrolled expansion at order $k_0$ for it. Denote by $\hat{u}$ the datum of $u$ and all its derivatives, and by $\hat{u}^\sharp$ the datum of all their derivatives.

(2) **Right hand side.** The use of a high order paracontrolled expansion formula, and a number of continuity results for some operators, allow together to rewrite the right hand side $f(u, \partial u, \zeta)$ of Equation (1.1) in the canonical form

$$f(u, \partial u, \zeta) = \sum_{j=1}^{k_0} \mathbf{P}_{v_j} Y_j + (b) \quad (1.3)$$

where (b) is a nice remainder and the distributions $Y_j$ depend only on $\zeta$ and the $Z_i$.

(3) **Fixed point.** Denote by $P$ the resolution of the free heat equation

$$Pu_0 := (\tau, x) \mapsto (e^{-\tau L} u_0)(x).$$

Then the fixed point relation

$$ \begin{aligned} u &= Pu_0 + \mathcal{L}^{-1}(f(u, \partial u, \zeta)) \\ &= Pu_0 + \sum_{j=1}^{k_0} \mathcal{L}^{-1}(\mathbf{P}_{v_j} Y_j) + \mathcal{L}^{-1}(b) \\ &= Pu_0 + \sum_{j=1}^{k_0} \tilde{\mathbf{P}}_{v_j} Z_j + \mathcal{L}^{-1}(b), \end{aligned} $$

imposes some consistency relations on the choice of the $Z_i = \mathcal{L}^{-1}(Y_i)$ that define them uniquely as functions of $\zeta$, and induces a fixed point relation for $\hat{u}$, or rather $\hat{u}^\sharp$.

See Proposition 35 for a justification of the name ‘derivative’ for the $u_i$ in identity (1.2). The expressions inside the $Y_i$’s that have no proper analytical sense need to be given a priori as components of the enhanced distribution $\hat{\zeta}$. As expected, the enhanced noise $\hat{\zeta}$ contains what is needed to make sense of the corresponding ill-defined products in the regularity structures setting. We shall not touch in this work on renormalization matters, and we invite the reader to read the latest developments of Hairer & co-authors on the subject; see [5, 7, 10]. In any case, we shall assume here that the enhancement $\hat{\zeta}$ of $\zeta$ is given.

We single out here the notion of paracontrolled system involved in point (1) above, in the setting that will be sufficient to deal with the generalized (PAM)
a structure equation; their derivatives as well, and so on. See Definition 1 below. One sees the above description (1.2) of $u$ as a paracontrolled expansion at order $k_0$ for it. Denote by $\hat{u}$ the datum of $u$ and all its derivatives, and by $\hat{u}^\sharp$ the datum of all their derivatives.

(2) **Right hand side.** The use of a high order paracontrolled expansion formula, and a number of continuity results for some operators, allow together to rewrite the right hand side $f(u, \partial u, \zeta)$ of Equation (1.1) in the canonical form

$$f(u, \partial u, \zeta) = \sum_{j=1}^{k_0} \mathbf{P}_{v_j} Y_j + \mathfrak{a}(b) \quad (1.3)$$

where $\mathfrak{a}(b)$ is a nice remainder and the distributions $Y_j$ depend only on $\zeta$ and the $Z_i$.

(3) **Fixed point.** Denote by $P$ the resolution of the free heat equation

$$P u_0 := (\tau, x) \mapsto (e^{-\tau L} u_0)(x).$$

Then the fixed point relation

$$\begin{align*}
u &= P u_0 + \mathcal{L}^{-1}(f(u, \partial u, \zeta)) \\
&= P u_0 + \sum_{j=1}^{k_0} \mathcal{L}^{-1}(\mathbf{P}_{v_j} Y_j) + \mathcal{L}^{-1}(b) \\
&= P u_0 + \sum_{j=1}^{k_0} \tilde{\mathbf{P}}_{v_j} Z_j + \mathcal{L}^{-1}(b),
\end{align*}$$

imposes some consistency relations on the choice of the $Z_i = \mathcal{L}^{-1}(Y_i)$ that define them uniquely as functions of $\zeta$, and induces a fixed point relation for $\hat{u}$, or rather $\hat{u}^\sharp$.

See Proposition 35 for a justification of the name ‘derivative’ for the $u_i$ in identity (1.2). The expressions inside the $Y_i$’s that have no proper analytical sense need to be given a priori as components of the enhanced distribution $\hat{\zeta}$. As expected, the enhanced noise $\hat{\zeta}$ contains what is needed to make sense of the corresponding ill-defined products in the regularity structures setting. We shall not touch in this work on renormalization matters, and we invite the reader to read the latest developments of Hairer & co-authors on the subject; see [5, 7, 10]. In any case, we shall assume here that the enhancement $\hat{\zeta}$ of $\zeta$ is given.

We single out here the notion of paracontrolled system involved in point (1) above, in the setting that will be sufficient to deal with the generalized (PAM)
and (KPZ) equations studied in Section 4. It corresponds to having $k_0 = 3$. Fix $2/5 < \alpha < 1/2$. For each $1 \le i \le 3$, we denote below by $Z_i$ a finite collection $(Z_i^{(n_i)})$ of space-time functions of parabolic regularity $i\alpha$. Given a collection $(u_i^{(n_i)})$ of bounded functions indexed by the same set $(n_i)$ as $Z_i$, we write $\tilde{P}_{u_i} Z_i$ for $\sum_{n_i} \tilde{P}_{u_i^{(n_i)}} Z_i^{(n_i)}$.

**DEFINITION 1.** A third order **paracontrolled system** is a family

$$ \hat{u} := (u, u_i, u_{ij}, u_{ijk})_{1\le i,i+j,i+j+k\le 3} $$

of collections of bounded parabolic functions defined by the datum of remainders $u_i^{\sharp} \in C^{4\alpha}$, $u_{ij}^{\sharp} \in C^{(4-i)\alpha}$, $u_{ijk}^{\sharp} \in C^{\alpha}$, $1 \le i, i+j, i+j+k \le 3$ via the identity

$$ \begin{aligned} u &= \sum_{i=1..3} \tilde{P}_{u_i} Z_i + u^{\sharp}, \\ u_i &= \sum_{i+j=1..3} \tilde{P}_{u_{ij}} Z_j + u_i^{\sharp}, \\ u_{ij} &= \sum_{i+j+k=1..3} \tilde{P}_{u_{ijk}} Z_k + u_{ij}^{\sharp}, \\ u_{ijk} &= u_{ijk}^{\sharp}. \end{aligned} $$

The above system is a shorthand notation for

$$ u_i^{(n_i)} = \sum_{i+j=1..3} \tilde{P}_{u_{ij}^{(n_i)}} Z_j + (u_i^{(n_i)})^{\sharp}, $$

and so on. There is only one function of the form $u_{ijk}$ above, namely $u_{111}$.

One sees on the above synopsis that we shall in particular obtain solutions of the (gPAM) and (gKPZ) equations under the form

$$ u = P_{f(u)} Z_1 + (2\alpha). $$

Corollary 1.11 of Hairer and Pardoux' work [21] follows then from Proposition 35, proved in Appendix C. We denote here by $[0, T]$ a time interval on which the solution of the equation is defined.

**COROLLARY 2.** Let $f$ be $C_b^5$. For $0 < t < T$, there exists a positive constant $C$ such that one has the estimate

$$ \begin{gathered} |u(e') - u(e) - f(u(e))(Z_1(e') - Z_1(e))| \le C (\sqrt{|\tau - \sigma|} + |y - x|)^{2\alpha}, \\ \text{uniformly in } e' = (\sigma, y) \text{ and } e = (\tau, x) \text{ with } |\tau - \sigma| \le T/2. \end{gathered} $$
and (KPZ) equations studied in Section 4. It corresponds to having $k_0 = 3$. Fix $2/5 < \alpha < 1/2$. For each $1 \le i \le 3$, we denote below by $Z_i$ a finite collection $(Z_i^{(n_i)})$ of space-time functions of parabolic regularity $i\alpha$. Given a collection $(u_i^{(n_i)})$ of bounded functions indexed by the same set $(n_i)$ as $Z_i$, we write $\tilde{P}_{u_i} Z_i$ for $\sum_{n_i} \tilde{P}_{u_i^{(n_i)}} Z_i^{(n_i)}$.

**DEFINITION 1.** A third order **paracontrolled system** is a family

$$ \hat{u} := (u, u_i, u_{ij}, u_{ijk})_{1\le i,i+j,i+j+k\le 3} $$

of collections of bounded parabolic functions defined by the datum of remainders $u_i^\sharp \in C^{4\alpha}$, $u_{ij}^\sharp \in C^{(4-i)\alpha}$, $u_{ijk}^\sharp \in C^\alpha$, $1 \le i, i+j, i+j+k \le 3$ via the identity

$$ u = \sum_{i=1..3} \tilde{P}_{u_i} Z_i + u^\sharp, $$

$$ u_i = \sum_{i+j=1..3} \tilde{P}_{u_{ij}} Z_j + u_i^\sharp, $$

$$ u_{ij} = \sum_{i+j+k=1..3} \tilde{P}_{u_{ijk}} Z_k + u_{ij}^\sharp, $$

$$ u_{ijk} = u_{ijk}^\sharp. $$

The above system is a shorthand notation for

$$ u_i^{(n_i)} = \sum_{i+j=1..3} \tilde{P}_{u_{ij}^{(n_i)}} Z_j + (u_i^{(n_i)})^\sharp, $$

and so on. There is only one function of the form $u_{ijk}$ above, namely $u_{111}$.

One sees on the above synopsis that we shall in particular obtain solutions of the (gPAM) and (gKPZ) equations under the form

$$ u = P_{f(u)} Z_1 + (2\alpha). $$

Corollary 1.11 of Hairer and Pardoux' work [21] follows then from Proposition 35, proved in Appendix C. We denote here by $[0, T]$ a time interval on which the solution of the equation is defined.

**COROLLARY 2.** Let $f$ be $C_b^5$. For $0 < t < T$, there exists a positive constant $C$ such that one has the estimate

$$ |u(e') - u(e) - f(u(e))(Z_1(e') - Z_1(e))| \le C \left( \sqrt{|\tau - \sigma|} + |y - x| \right)^{2\alpha}, $$

uniformly in $e' = (\sigma, y)$ and $e = (\tau, x)$ with $|\tau - \sigma| \le T/2$.
**1.2. The mechanics of computations within paracontrolled calculus.** The basics of the paracontrolled analysis of singular PDEs are easily grasped by a parallel with Itô calculus. Denote by $a, b, c$ three generic continuous martingales. The following computational rules appear as fundamental in stochastic calculus.

* The basic Itô formula

$$d(ab) = a db + b da + d\langle a, b \rangle.$$

* Itô formula

$$d(f(a)) = f'(a) da + \frac{1}{2} f''(a) d\langle a, a \rangle.$$

* Bracket rule for stochastic integrals

$$d\left(\int adb, c\right) - a d\langle b, c \rangle = 0.$$

The building blocks of the first order paracontrolled calculus devised by Gubinelli, Imkeller and Perkowski in [17] are the exact counterparts of the above three points, with the paraproduct operator in the role of the (derivative of the) stochastic integral and the diagonal operator in the role of the (derivative of the) bracket. For $a, b, c$ functions or distributions with some precise regularity properties, we have the following facts.

* Paraproduct decomposition

$$ab = P_a b + P_b a + \Pi(a, b)$$

where $P$ is the paraproduct and $\Pi$ the resonant part.

* Bony's paralinearization

$$f(a) = P_{f'(a)}a + (\text{remainder}).$$

* Fundamental corrector estimate. The operator

$$C(a, b, c) := \Pi(P_a b, c) - a\Pi(b, c) \quad (1.4)$$

is continuous for certain ranges of regularity exponents for its arguments.

The three-step resolution process of Section 1.1 for the study of a singular PDE requires that we refine these tools. The task of writing the right hand side $f(u, \partial u, \zeta)$ of a singular PDE under the form (1.3) can be divided into two subtasks. First, making sense of the ill-defined products that appear in $f(u, \partial u, \zeta)$,
**1.2. The mechanics of computations within paracontrolled calculus.** The basics of the paracontrolled analysis of singular PDEs are easily grasped by a parallel with Itô calculus. Denote by $a, b, c$ three generic continuous martingales. The following computational rules appear as fundamental in stochastic calculus.

* The basic Itô formula

$$d(ab) = a db + b da + d\langle a, b \rangle.$$

* Itô formula

$$d(f(a)) = f'(a) da + \frac{1}{2} f''(a) d\langle a, a \rangle.$$

* Bracket rule for stochastic integrals

$$d\left(\int adb, c\right) - a d\langle b, c \rangle = 0.$$

The building blocks of the first order paracontrolled calculus devised by Gubinelli, Imkeller and Perkowski in [17] are the exact counterparts of the above three points, with the paraproduct operator in the role of the (derivative of the) stochastic integral and the diagonal operator in the role of the (derivative of the) bracket. For $a, b, c$ functions or distributions with some precise regularity properties, we have the following facts.

* Paraproduct decomposition

$$ab = P_a b + P_b a + \Pi(a, b)$$

where $P$ is the paraproduct and $\Pi$ the resonant part.

* Bony's paralinearization

$$f(a) = P_{f'(a)}a + (\text{remainder}).$$

* Fundamental corrector estimate. The operator

$$C(a, b, c) := \Pi(P_a b, c) - a\Pi(b, c) \quad (1.4)$$

is continuous for certain ranges of regularity exponents for its arguments.

The three-step resolution process of Section 1.1 for the study of a singular PDE requires that we refine these tools. The task of writing the right hand side $f(u, \partial u, \zeta)$ of a singular PDE under the form (1.3) can be divided into two subtasks. First, making sense of the ill-defined products that appear in $f(u, \partial u, \zeta)$,
up to the datum of a number of formal multilinear functions of the noise $\zeta$ – the enhancement $\hat{\zeta}$. Second, writing $f(u, \partial u, \zeta)$ under the form (1.3), needed to run the fixed point argument – step (3) in Section 1.1. Ill-defined quantities appear under the form of (possibly multi-)linear maps $\mathbb{E}(\cdot)$, defined on Hölder spaces, and formally taking values in a Hölder space $C^\gamma$ of negative regularity exponent $\gamma < 0$.

* In the line of identity (1.4), to deal with the first task, we systematically use the paracontrolled structure to write the following kind of decomposition. Let $\mathbb{E}$ stand for a map that formally sends $C^\beta$ into $C^\gamma$. For $a \in C^\alpha$ and $b \in C^\beta$, with $\alpha, \beta > 0$, one has

$$ 
\begin{aligned}
\mathbb{E}(P_a b) &= a\mathbb{E}(b) + \mathbb{E}^+(a, b) \\
&= P_a \mathbb{E}(b) + \mathbb{P}_{\mathbb{E}(b)}a + \Pi(a, \mathbb{E}(b)) + \mathbb{E}^+(a, b),
\end{aligned} 
\quad (1.5) $$

for some linear operator $\mathbb{E}^+(\cdot, b)$ that formally takes values in the Hölder space $C^{\gamma+\alpha}$. The map $\Pi(\cdot, \mathbb{E}(b))$ also formally takes values in $C^{\gamma+\alpha}$, while the two paraproduct terms make perfect sense if $\mathbb{E}(b)$ does. The regularity exponents of the two possibly ill-defined terms $\Pi(a, \mathbb{E}(b))$ and $\mathbb{E}^+(a, b)$ have been increased by $\alpha$, compared to the regularity exponent of $\mathbb{E}(P_a b)$. One can iterate the kind of expansion given by (1.5) on $\Pi(a, \mathbb{E}(b))$ and $\mathbb{E}^+(a, b)$ if $a$ has a paracontrolled structure. Iterating the expansion (1.5) as many times as necessary, if possible, eventually leads to a decomposition of the initial quantity into a sum of well-defined terms, up to the *a priori* datum of a number of terms, like the above $\mathbb{E}(b)$.

* The second task involves giving a special form to the evaluation on some paraproduct terms of well-defined continuous linear functions $\mathbb{F}$. We systematically use for that purpose the following kind of decomposition. If $\mathbb{F}$ takes values in a Hölder space $C^\gamma$, whatever $\gamma \in \mathbb{R}$, and for $a \in C^\alpha$ and $b \in C^\beta$, one has

$$ \mathbb{F}(P_a b) = P_a \mathbb{F}(b) + \mathbb{F}^+(a, b), \quad (1.6) $$

for some continuous bilinear operator $\mathbb{F}^+$ that takes values in $C^{\gamma+\alpha}$.

As an example of computation, let $a_1, a_2, b$ be $\alpha$-Hölder functions, with $0 < \alpha < 1$, and let $\mathbb{E}$ send the space of $\alpha$-Hölder functions into the space of $\gamma$-Hölder functions, for some negative regularity exponent $\gamma$ with $-2\alpha < \gamma < -\alpha$. One has from formula (1.5)

$$ \mathbb{E}(P_a b) = P_a \mathbb{E}(b) + \mathbb{P}_{\mathbb{E}(b)}a + \Pi(a, \mathbb{E}(b)) + \mathbb{E}^+(a, b), $$

where the first term is $\gamma$-Hölder, the second term is $(\gamma+\alpha)$-Hölder, an elementary property of the paraproduct operator, and the third and fourth terms have formal
up to the datum of a number of formal multilinear functions of the noise $\zeta$ – the enhancement $\hat{\zeta}$. Second, writing $f(u, \partial u, \zeta)$ under the form (1.3), needed to run the fixed point argument – step (3) in Section 1.1. Ill-defined quantities appear under the form of (possibly multi-)linear maps $\mathbb{E}(\cdot)$, defined on Hölder spaces, and formally taking values in a Hölder space $C^\gamma$ of negative regularity exponent $\gamma < 0$.

* In the line of identity (1.4), to deal with the first task, we systematically use the paracontrolled structure to write the following kind of decomposition. Let $\mathbb{E}$ stand for a map that formally sends $C^\beta$ into $C^\gamma$. For $a \in C^\alpha$ and $b \in C^\beta$, with $\alpha, \beta > 0$, one has

$$ 
\begin{aligned}
\mathbb{E}(P_a b) &= a\mathbb{E}(b) + \mathbb{E}^+(a, b) \\
&= P_a \mathbb{E}(b) + \mathbb{P}_{\mathbb{E}(b)}a + \Pi(a, \mathbb{E}(b)) + \mathbb{E}^+(a, b),
\end{aligned} 
\quad (1.5) 
$$

for some linear operator $\mathbb{E}^+(\cdot, b)$ that formally takes values in the Hölder space $C^{\gamma+\alpha}$. The map $\Pi(\cdot, \mathbb{E}(b))$ also formally takes values in $C^{\gamma+\alpha}$, while the two paraproduct terms make perfect sense if $\mathbb{E}(b)$ does. The regularity exponents of the two possibly ill-defined terms $\Pi(a, \mathbb{E}(b))$ and $\mathbb{E}^+(a, b)$ have been increased by $\alpha$, compared to the regularity exponent of $\mathbb{E}(P_a b)$. One can iterate the kind of expansion given by (1.5) on $\Pi(a, \mathbb{E}(b))$ and $\mathbb{E}^+(a, b)$ if $a$ has a paracontrolled structure. Iterating the expansion (1.5) as many times as necessary, if possible, eventually leads to a decomposition of the initial quantity into a sum of well-defined terms, up to the a priori datum of a number of terms, like the above $\mathbb{E}(b)$.

* The second task involves giving a special form to the evaluation on some paraproduct terms of well-defined continuous linear functions $\mathbb{F}$. We systematically use for that purpose the following kind of decomposition. If $\mathbb{F}$ takes values in a Hölder space $C^\gamma$, whatever $\gamma \in \mathbb{R}$, and for $a \in C^\alpha$ and $b \in C^\beta$, one has

$$ \mathbb{F}(P_a b) = P_a \mathbb{F}(b) + \mathbb{F}^+(a, b), \quad (1.6) $$

for some continuous bilinear operator $\mathbb{F}^+$ that takes values in $C^{\gamma+\alpha}$.

As an example of computation, let $a_1, a_2, b$ be $\alpha$-Hölder functions, with $0 < \alpha < 1$, and let $\mathbb{E}$ send the space of $\alpha$-Hölder functions into the space of $\gamma$-Hölder functions, for some negative regularity exponent $\gamma$ with $-2\alpha < \gamma < -\alpha$. One has from formula (1.5)

$$ \mathbb{E}(P_a b) = P_a \mathbb{E}(b) + \mathbb{P}_{\mathbb{E}(b)}a + \Pi(a, \mathbb{E}(b)) + \mathbb{E}^+(a, b), $$

where the first term is $\gamma$-Hölder, the second term is $(\gamma+\alpha)$-Hölder, an elementary property of the paraproduct operator, and the third and fourth terms have formal
regularity $\gamma + \alpha$. If $a = P_{a_1}a_2$, then we have the F-type decomposition for the $(\gamma + \alpha)$-Hölder valued function $a \mapsto P_{E(b)}a$,

$$P_{E(b)}a = P_{a_1}(P_{E(b)}a_2) + \Pi_{E(b)}^{+}(a_1, a_2),$$

with the first term of regularity $(\gamma + \alpha)$ and the second term of regularity $(\gamma + 2\alpha)$. We also have the two E-type decompositions

$$\begin{aligned} \Pi(a, E(b)) &= a_1\Pi(b_1, E(b)) + \Pi(\cdot, E(b))^{+}(a_1, b_1) \\ &= P_{a_1}\Pi(b_1, E(b)) + P_{\Pi(b_1, E(b))}a_1 + \Pi(a_1, \Pi(b_1, E(b))) \\ &\quad + \Pi(\cdot, E(b))^{+}(a_1, b_1), \end{aligned}$$

and

$$\begin{aligned} E^{+}(a, b) &= a_1 E^{+}(a_2, b) + \{E^{+}(\cdot, b)\}^{+}(a_1, a_2) \\ &= P_{a_1} E^{+}(a_2, b) + P_{E^{+}(a_2, b)}a_1 + \Pi(a_1, E^{+}(a_2, b)) + \{E^{+}(\cdot, b)\}^{+}(a_1, a_2). \end{aligned}$$

In both expressions, the first term has regularity $(\gamma + \alpha)$, the second term has regularity $(\gamma + 2\alpha)$, while the third and fourth terms have formal regularity $(\gamma + 2\alpha)$. If one is only interested in having a description of $E(P_a b)$ up to terms $(\#)$ of true or formal positive regularity $(\gamma + 2\alpha)$, one then has, with $a = P_{a_1}a_2$,

$$E(P_{P_{a_1}a_2}b) = P_{P_{a_1}a_2}E(b) + \{P_{a_1}(P_{E(b)}a_2) + P_{a_1}\Pi(a_2, E(b)) + P_{a_1}E^{+}(a_2, b)\} + (\#).$$

None of the computations of Section 4, dealing with the paracontrolled analysis of concrete examples of singular PDEs, is more complicated than what we have just done. Convenient notations will be used to work with iterated operators; they are introduced in Section 3. (Note here that such expansions can be done on systems of singular PDEs as well.) On a technical level, three ingredients are used to run the three-step scheme of Section 1.1.

* The pair $(P, \tilde{P})$ of **intertwined paraproducts** introduced in [2]. It is used to define a continuous map $\Phi$ from the solution space $S$ to itself.

* A **high order paracontrolled expansion formula** generalizing Bony's paralinearization formula is used to give a paracontrolled expansion of a nonlinear function of any $\alpha$-Hölder function $u$, for $0 < \alpha < 1$. See Section 2.

* **Continuity results.** We introduce in Section 3 a number of operators and prove their continuity. These are the operators that appear in applying the decompositions (1.5) and (1.6) in the analysis of a generic right hand side $f(u, \partial u, \zeta)$ for Equation (1.1).
regularity $\gamma + \alpha$. If $a = P_{a_1}a_2$, then we have the F-type decomposition for the $(\gamma + \alpha)$-Hölder valued function $a \mapsto P_{E(b)}a$,

$$P_{E(b)}a = P_{a_1}(P_{E(b)}a_2) + \Pi_{E(b)}^{+}(a_1, a_2),$$

with the first term of regularity $(\gamma + \alpha)$ and the second term of regularity $(\gamma + 2\alpha)$. We also have the two E-type decompositions

$$\begin{aligned} \Pi(a, E(b)) &= a_1\Pi(b_1, E(b)) + \Pi(\cdot, E(b))^{+}(a_1, b_1) \\ &= P_{a_1}\Pi(b_1, E(b)) + P_{\Pi(b_1, E(b))}a_1 + \Pi(a_1, \Pi(b_1, E(b))) \\ &\quad + \Pi(\cdot, E(b))^{+}(a_1, b_1), \end{aligned}$$

and

$$\begin{aligned} E^{+}(a, b) &= a_1 E^{+}(a_2, b) + \{E^{+}(\cdot, b)\}^{+}(a_1, a_2) \\ &= P_{a_1} E^{+}(a_2, b) + P_{E^{+}(a_2, b)}a_1 + \Pi(a_1, E^{+}(a_2, b)) + \{E^{+}(\cdot, b)\}^{+}(a_1, a_2). \end{aligned}$$

In both expressions, the first term has regularity $(\gamma + \alpha)$, the second term has regularity $(\gamma + 2\alpha)$, while the third and fourth terms have formal regularity $(\gamma + 2\alpha)$. If one is only interested in having a description of $E(P_a b)$ up to terms $(\#)$ of true or formal positive regularity $(\gamma + 2\alpha)$, one then has, with $a = P_{a_1}a_2$,

$$E(P_{P_{a_1}a_2}b) = P_{P_{a_1}a_2}E(b) + \{P_{a_1}(P_{E(b)}a_2) + P_{a_1}\Pi(a_2, E(b)) + P_{a_1}E^{+}(a_2, b)\} + (\#).$$

None of the computations of Section 4, dealing with the paracontrolled analysis of concrete examples of singular PDEs, is more complicated than what we have just done. Convenient notations will be used to work with iterated operators; they are introduced in Section 3. (Note here that such expansions can be done on systems of singular PDEs as well.) On a technical level, three ingredients are used to run the three-step scheme of Section 1.1.

* The pair $(P, \tilde{P})$ of **intertwined paraproducts** introduced in [2]. It is used to define a continuous map $\Phi$ from the solution space $S$ to itself.

* A **high order paracontrolled expansion formula** generalizing Bony's paralinearization formula is used to give a paracontrolled expansion of a nonlinear function of any $\alpha$-Hölder function $u$, for $0 < \alpha < 1$. See Section 2.

* **Continuity results.** We introduce in Section 3 a number of operators and prove their continuity. These are the operators that appear in applying the decompositions (1.5) and (1.6) in the analysis of a generic right hand side $f(u, \partial u, \zeta)$ for Equation (1.1).
**1.3. Setting and results.** We adopt in this work essentially the same geometric and functional setting as in our previous work [2], slightly restricted so as not to bother here the reader with the use of weighted functional spaces. All this work could be formulated in the more general geometric/functional setting of [2]; we refrain from doing this as it may blur the simple ideas that we want to promote in this work. Let then $(M, d, \mu)$ stand for a compact smooth Riemannian manifold equipped with a doubling measure $\mu$ that may differ from the canonical Riemmanian volume measure. Let $V_1, \dots, V_{\ell_0}$ stand for some smooth vector fields on $M$, identified with first order differential operators. Given a tuple $I = (i_1, \dots, i_k)$ in $\{1, \dots, \ell_0\}^k$, we shall set $|I| := k$ and

$$V_I := V_{i_k} \cdots V_{i_1}.$$

Set

$$L := - \sum_{\ell=1}^{\ell_0} V_{\ell}^2$$

and assume that $L$ is elliptic, so that the $V_i$ span smoothly at every point of $M$ the whole tangent space. So there exist smooth functions $(\gamma_i)_{1 \le i \le \ell_0}$ such that for every function $f \in C^1(M, \mathbb{R})$ and $x \in M$ we have

$$\nabla f(x) = \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) V_{\ell}(f)(x) V_{\ell}(x).$$

The operator $L$ is then a sectorial operator in $L^2(M)$, it is injective on the quotient space of $L^2(M)$ by the space of constant functions, it has a bounded $H^\infty$-calculus on $L^2(M)$, and $-L$ generates a holomorphic semigroup $(e^{-tL})_{t>0}$ on $L^2(M)$ – see [16]. The above class of operators includes obviously the Laplacian on the flat torus. Note that under the above smoothness and ellipticity conditions, the semigroup $e^{-tL}$ has regularity estimates at any order, by which we mean that for every tuple $I$, the operators $(t^{|I|/2}V_I)e^{-tL}$ and $e^{-tL}(t^{|I|/2}V_I)$ have kernels $K_I(x, y)$ satisfying the Gaussian estimate

$$|K_I(x, y)| \lesssim \frac{1}{\mu(B(x, \sqrt{t}))} e^{-c(d(x,y)^2/\sqrt{t})}$$

and the following regularity estimate, for some constants which may depend on $|I|$. For $d(x, z) \le \sqrt{t}$

$$|K_I(x, y) - K_I(x', y)| \lesssim \frac{d(x, x')}{\sqrt{t}} \frac{1}{\mu(B(x, \sqrt{t}))} e^{-c(d(x, y)^2/\sqrt{t})}.$$
**1.3. Setting and results.** We adopt in this work essentially the same geometric and functional setting as in our previous work [2], slightly restricted so as not to bother here the reader with the use of weighted functional spaces. All this work could be formulated in the more general geometric/functional setting of [2]; we refrain from doing this as it may blur the simple ideas that we want to promote in this work. Let then (M, d, μ) stand for a compact smooth Riemannian manifold equipped with a doubling measure μ that may differ from the canonical Riemmanian volume measure. Let $V_1, \dots, V_{\ell_0}$ stand for some smooth vector fields on M, identified with first order differential operators. Given a tuple $I = (i_1, \dots, i_k)$ in $\{1, \dots, \ell_0\}^k$, we shall set $|I| := k$ and

$$V_I := V_{i_k} \cdots V_{i_1}.$$

Set

$$L := - \sum_{\ell=1}^{\ell_0} V_{\ell}^2$$

and assume that $L$ is elliptic, so that the $V_i$ span smoothly at every point of M the whole tangent space. So there exist smooth functions $(\gamma_i)_{1 \le i \le \ell_0}$ such that for every function $f \in C^1(M, \mathbb{R})$ and $x \in M$ we have

$$\nabla f(x) = \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) V_{\ell}(f)(x) V_{\ell}(x).$$

The operator $L$ is then a sectorial operator in $L^2(M)$, it is injective on the quotient space of $L^2(M)$ by the space of constant functions, it has a bounded $H^\infty$-calculus on $L^2(M)$, and $-L$ generates a holomorphic semigroup $(e^{-tL})_{t>0}$ on $L^2(M)$ – see [16]. The above class of operators includes obviously the Laplacian on the flat torus. Note that under the above smoothness and ellipticity conditions, the semigroup $e^{-tL}$ has regularity estimates at any order, by which we mean that for every tuple $I$, the operators $(t^{|I|/2}V_I)e^{-tL}$ and $e^{-tL}(t^{|I|/2}V_I)$ have kernels $K_I(x, y)$ satisfying the Gaussian estimate

$$|K_I(x, y)| \lesssim \frac{1}{\mu(B(x, \sqrt{t}))} e^{-c(d(x, y)^2/\sqrt{t})}$$

and the following regularity estimate, for some constants which may depend on $|I|$. For $d(x, z) \le \sqrt{t}$

$$|K_I(x, y) - K_I(x', y)| \lesssim \frac{d(x, x')}{\sqrt{t}} \frac{1}{\mu(B(x, \sqrt{t}))} e^{-c(d(x, y)^2/\sqrt{t})}.$$
Note again that we could equally well develop paracontrolled calculus in the more general setting adopted in our previous work [2]; we refrain from doing that here as it could obscure the simplicity of the ideas put forward here.

Given a finite time horizon $T$, we define the parabolic space $\mathcal{M}$ as

$$ \mathcal{M} := [0, T] \times \mathcal{M}, $$

and equip it with the parabolic metric

$$ \rho((\tau, x), (\sigma, y)) = \sqrt{|\tau - \sigma|} + d(x, y) $$

and the parabolic measure $\nu = dt \otimes \mu$. Then $(\mathcal{M}, \rho, \nu)$ is a doubling space of homogeneous type. Note that for $(\tau, x) \in \mathcal{M}$ and small positive radius $r$, the parabolic ball $B_{\mathcal{M}}((\tau, x), r)$ has volume

$$ \nu(B_{\mathcal{M}}((\tau, x), r)) \approx r^2 \mu(B_{\mathcal{M}}(x, r)). $$

We shall denote by $e = (\tau, x)$ a generic element of the parabolic space $\mathcal{M}$.

We have chosen to work in the scale of Hölder spaces; this makes life easier, although we could equally develop paracontrolled calculus in the functional setting of Sobolev spaces, in the line of what we did in our previous work [1]. For a real number $s$, we denote by $C^s = C^s(\mathcal{M})$, the Hölder space on $\mathcal{M}$ of order $s$, defined in terms of Besov spaces; and $C^s = C^s(\mathcal{M})$, the parabolic Hölder space. We refer the reader to Appendix A for more details on these spaces. Following our previous work [2], one can define from $L$ only parabolic paraproduct and resonant operators that have good continuity properties in the scale of parabolic Hölder spaces – see Appendix A.3. The high order paracontrolled expansion formula and the continuity results stated in Sections 2 and 3 respectively, and fully proved in Appendices B and C, make use of these operators and provide the spine of paracontrolled calculus. They are the main contributions of this work.

We illustrate our approach of the study of singular PDEs by proving well-posedness results for the 3-dimensional generalized parabolic Anderson model equation (gPAM)

$$ \mathcal{L}u := (\partial_t + L)u = f(u)\zeta, \quad (1.7) $$

and the generalized KPZ equation

$$ \mathcal{L}u = f(u)\zeta + g(u)(\partial u)^2, \quad (1.8) $$

on the one-dimensional torus, with a space/time white noise. The generalized parabolic Anderson model equation is a natural nonlinear generalization of its linear counterpart, for which $f(u) = u$. The latter equation describes the evolution of a Brownian particle in a white noise environment. The generalized (KPZ)
Note again that we could equally well develop paracontrolled calculus in the more general setting adopted in our previous work [2]; we refrain from doing that here as it could obscure the simplicity of the ideas put forward here.

Given a finite time horizon $T$, we define the parabolic space $\mathcal{M}$ as

$$ \mathcal{M} := [0, T] \times \mathcal{M}, $$

and equip it with the parabolic metric

$$ \rho((\tau, x), (\sigma, y)) = \sqrt{|\tau - \sigma|} + d(x, y) $$

and the parabolic measure $\nu = dt \otimes \mu$. Then $(\mathcal{M}, \rho, \nu)$ is a doubling space of homogeneous type. Note that for $(\tau, x) \in \mathcal{M}$ and small positive radius $r$, the parabolic ball $B_{\mathcal{M}}((\tau, x), r)$ has volume

$$ \nu(B_{\mathcal{M}}((\tau, x), r)) \approx r^2 \mu(B_{\mathcal{M}}(x, r)). $$

We shall denote by $e = (\tau, x)$ a generic element of the parabolic space $\mathcal{M}$.

We have chosen to work in the scale of Hölder spaces; this makes life easier, although we could equally develop paracontrolled calculus in the functional setting of Sobolev spaces, in the line of what we did in our previous work [1]. For a real number $s$, we denote by $C^s = C^s(\mathcal{M})$, the Hölder space on $\mathcal{M}$ of order $s$, defined in terms of Besov spaces; and $C^s = C^s(\mathcal{M})$, the parabolic Hölder space. We refer the reader to Appendix A for more details on these spaces. Following our previous work [2], one can define from $L$ only parabolic paraproduct and resonant operators that have good continuity properties in the scale of parabolic Hölder spaces – see Appendix A.3. The high order paracontrolled expansion formula and the continuity results stated in Sections 2 and 3 respectively, and fully proved in Appendices B and C, make use of these operators and provide the spine of paracontrolled calculus. They are the main contributions of this work.

We illustrate our approach of the study of singular PDEs by proving well-posedness results for the 3-dimensional generalized parabolic Anderson model equation (gPAM)

$$ \mathcal{L}u := (\partial_t + L)u = f(u)\zeta, \qquad (1.7) $$

and the generalized KPZ equation

$$ \mathcal{L}u = f(u)\zeta + g(u)(\partial u)^2, \qquad (1.8) $$

on the one-dimensional torus, with a space/time white noise. The generalized parabolic Anderson model equation is a natural nonlinear generalization of its linear counterpart, for which $f(u) = u$. The latter equation describes the evolution of a Brownian particle in a white noise environment. The generalized (KPZ)
equation appears in the study of the random motion of a string on a manifold [20]; its study in the setting of regularity structures is the object of Hairer and co-authors' works [6, 20]. The renormalization of the 70ish terms that appear in the models for this equation motivated the development of systematic renormalization procedures, such as done in the recent works of Bruned-Hairer-Zambotti [7] and Chandra-Hairer [10]. In the present work, we assume that a proper enhancement $\hat{\zeta}$ of the noise $\zeta$ is given. Defining $\hat{\zeta}$ in a stochastic setting is a very different question that is not addressed here.

We have organized this work as follows. Section 2 is dedicated to the statement and proof of a high order paracontrolled expansion formula generalizing Bony's paralinearization formula. The core results of the mechanics of computations within paracontrolled calculus are obtained in Section 3, under the form of continuity results for a number of operators. We test paracontrolled calculus on the examples of the 3-dimensional generalized parabolic Anderson model equation (1.7) (Theorems 24 and 25), and the generalized KPZ equation (1.8) (Theorem 26), in Section 4. Appendix A contains all the relevant details about the parabolic setting, approximation operators, Hölder spaces and paraproducts. Appendices B and C contain the proofs of a number of statements.

## 2. High order paracontrolled expansion

We explain in this section a simple procedure for getting an arbitrary high order expansion of a nonlinear map of a given Hölder function $u$ defined on the parabolic space $M$, in terms of its parabolic regularity properties. It provides, in the setting of Hölder spaces, a refinement of Bony's paralinearization theorem in the form of a viable alternative to the paradifferential calculus of Chemin [11]; see also [12, Theorem 2.5, page 18,] for a more readable account of [11] in the case of a second order expansion.

In its simplest form, the classical paraproduct operator $(f, g) \mapsto \Pi_f^0 g$ on the $d$-dimensional torus is defined via Fourier analysis by modulation of the high frequencies of a given 'reference' function/distribution $g$ by the low frequencies of another function/distribution $f$. For a function $f$ on the torus, we denote by $f = \sum \Delta_i(f)$ its usual Littlewood-Paley representation, where $\Delta_i(f)$ is the dyadic bloc with Fourier coefficients essentially only at the frequency scale $2^i$. Consider the Littlewood-Paley decompositions of two functions

$$f = \sum_{i} \Delta_{i}(f), \quad g = \sum_{j} \Delta_{j}(g),$$
equation appears in the study of the random motion of a string on a manifold [20]; its study in the setting of regularity structures is the object of Hairer and co-authors' works [6, 20]. The renormalization of the 70ish terms that appear in the models for this equation motivated the development of systematic renormalization procedures, such as done in the recent works of Bruned-Hairer-Zambotti [7] and Chandra-Hairer [10]. In the present work, we assume that a proper enhancement $\hat{\zeta}$ of the noise $\zeta$ is given. Defining $\hat{\zeta}$ in a stochastic setting is a very different question that is not addressed here.

We have organized this work as follows. Section 2 is dedicated to the statement and proof of a high order paracontrolled expansion formula generalizing Bony's paralinearization formula. The core results of the mechanics of computations within paracontrolled calculus are obtained in Section 3, under the form of continuity results for a number of operators. We test paracontrolled calculus on the examples of the 3-dimensional generalized parabolic Anderson model equation (1.7) (Theorems 24 and 25), and the generalized KPZ equation (1.8) (Theorem 26), in Section 4. Appendix A contains all the relevant details about the parabolic setting, approximation operators, Hölder spaces and paraproducts. Appendices B and C contain the proofs of a number of statements.

## 2. High order paracontrolled expansion

We explain in this section a simple procedure for getting an arbitrary high order expansion of a nonlinear map of a given Hölder function $u$ defined on the parabolic space $M$, in terms of its parabolic regularity properties. It provides, in the setting of Hölder spaces, a refinement of Bony's paralinearization theorem in the form of a viable alternative to the paradifferential calculus of Chemin [11]; see also [12, Theorem 2.5, page 18,] for a more readable account of [11] in the case of a second order expansion.

In its simplest form, the classical paraproduct operator $(f, g) \mapsto \Pi_f^0 g$ on the $d$-dimensional torus is defined via Fourier analysis by modulation of the high frequencies of a given 'reference' function/distribution $g$ by the low frequencies of another function/distribution $f$. For a function $f$ on the torus, we denote by $f = \sum \Delta_i(f)$ its usual Littlewood-Paley representation, where $\Delta_i(f)$ is the dyadic bloc with Fourier coefficients essentially only at the frequency scale $2^i$. Consider the Littlewood-Paley decompositions of two functions

$$f = \sum_{i} \Delta_{i}(f), \quad g = \sum_{j} \Delta_{j}(g),$$
as sums of smooth functions with localized frequencies; the paraproduct of $g$ by $f$ is defined as

$$ \Pi_f^0 g = \sum_{i<j-1} \Delta_i(f) \Delta_j(g), \qquad (2.1) $$

and the resonant part as

$$ \Pi^0(f, g) = \sum_{|i-j| \le 1} \Delta_i(f) \Delta_j(g), $$

so we have the product decomposition

$$ fg = \Pi_g^0 f + \Pi_f^0 g + \Pi^0(f, g). $$

In the parabolic setting of Section 1.3, one can define some paraproduct and resonant operators associated with the operator $L$ and its semigroup, that have the same regularity properties in the scale of parabolic Hölder spaces as the operator $\Pi^0$ in the scale of spatial Hölder spaces. We denote by $\mathbb{P}$ this paraproduct, introduced in [2], and whose definition is recalled in Appendix A.3. It depends implicitly on an integer-valued parameter $b$ that is chosen once and for all, and whose precise value is irrelevant for our purposes. It is not crucial at that stage to go into the details of the definition of $\mathbb{P}$.

The mechanics of the proof of our general Taylor expansion formula is fairly simple and better understood in the light of the proof of Bony's paralinearization theorem given by Gubinelli, Imkeller and Perkowski in [17], which we recall first.

**THEOREM 3** (Bony's paralinearization). Let $f: \mathbb{R} \to \mathbb{R}$ be a $C_b^2$ function and $u$ be a real-valued $\alpha$-Hölder function on the $d$-dimensional torus, with $0 < \alpha < 1$. Then

$$ f(u) = \Pi_{f'(u)}^0 u + f(u)^2 $$

for some remainder $f(u)^2$ of spatial Hölder regularity $2\alpha$.

*Proof.* This is a copy and paste from [17]. Denote by $K_i$ the kernels of the Fourier projectors $\Delta_i$ corresponding to the Littlewood-Paley decomposition operator, and write $K_{\le k}$ for $\sum_{i \le k} K_i$, with associated operator $S_k := \sum_{i \le k} \Delta_i$. Note that by their definition we have, for any $i \ge 1$,

$$ \int_{\mathbb{R}^d} K_i(y) dy = 0; \qquad (2.2) $$

or more properly $\int_{\mathbb{R}^d} K_i(x, y) dy = 0$, for any $x \in \mathbb{R}^d$. The trick is then simply to write

$$ f(u) - \Pi_{f'(u)}^0 (u) = \sum \Delta_i(f(u)) - S_{i-2}(f'(u))\Delta_i(u) =: \sum \epsilon_i $$
as sums of smooth functions with localized frequencies; the paraproduct of $g$ by $f$ is defined as

$$ \Pi_f^0 g = \sum_{i<j-1} \Delta_i(f) \Delta_j(g), \qquad (2.1) $$

and the resonant part as

$$ \Pi^0(f, g) = \sum_{|i-j| \le 1} \Delta_i(f) \Delta_j(g), $$

so we have the product decomposition

$$ fg = \Pi_g^0 f + \Pi_f^0 g + \Pi^0(f, g). $$

In the parabolic setting of Section 1.3, one can define some paraproduct and resonant operators associated with the operator $L$ and its semigroup, that have the same regularity properties in the scale of parabolic Hölder spaces as the operator $\Pi^0$ in the scale of spatial Hölder spaces. We denote by $\mathbb{P}$ this paraproduct, introduced in [2], and whose definition is recalled in Appendix A.3. It depends implicitly on an integer-valued parameter $b$ that is chosen once and for all, and whose precise value is irrelevant for our purposes. It is not crucial at that stage to go into the details of the definition of $\mathbb{P}$.

The mechanics of the proof of our general Taylor expansion formula is fairly simple and better understood in the light of the proof of Bony's paralinearization theorem given by Gubinelli, Imkeller and Perkowski in [17], which we recall first.

**THEOREM 3** (Bony's paralinearization). Let $f: \mathbb{R} \to \mathbb{R}$ be a $C_b^2$ function and $u$ be a real-valued $\alpha$-Hölder function on the $d$-dimensional torus, with $0 < \alpha < 1$. Then

$$ f(u) = \Pi_{f'(u)}^0 u + f(u)^2 $$

for some remainder $f(u)^2$ of spatial Hölder regularity $2\alpha$.

*Proof.* This is a copy and paste from [17]. Denote by $K_i$ the kernels of the Fourier projectors $\Delta_i$ corresponding to the Littlewood-Paley decomposition operator, and write $K_{\le k}$ for $\sum_{i \le k} K_i$, with associated operator $S_k := \sum_{i \le k} \Delta_i$. Note that by their definition we have, for any $i \ge 1$,

$$ \int_{\mathbb{R}^d} K_i(y) dy = 0; \qquad (2.2) $$

or more properly $\int_{\mathbb{R}^d} K_i(x, y) dy = 0$, for any $x \in \mathbb{R}^d$. The trick is then simply to write

$$ f(u) - \Pi_{f'(u)}^0 (u) = \sum \Delta_i(f(u)) - S_{i-2}(f'(u))\Delta_i(u) =: \sum \epsilon_i $$
with

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \{f(u(y)) - f'(u(z))u(y)\} dz dy, $$

and to take profit from the fact that $K_i$ has null mean for $i \ge 1$, as put forward in identity (2.2), to see that one also has, for $i \ge 1$,

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \{f(u(y)) - f(u(z)) - f'(u(z))(u(y) - u(z))\} dz dy. $$

One thus has

$$ |\varepsilon_i(x)| \lesssim \|f''\|_{\infty} \int |K_i(x, y) K_{\le i-2}(x, z)| |u(y) - u(z)|^2 dz dy \lesssim 2^{-2i\alpha} \|u\|_{C^\alpha}^2. $$

Since the functions $\varepsilon_i$ are frequently supported on the annulus of frequencies at the scale $2^i$, and the $\varepsilon_i$ are uniformly bounded by $2^{-2i\alpha}$, the series $\sum_i \varepsilon_i$ defines a $2\alpha$-Hölder function. $\square$

One can play exactly the same game and prove a general expansion result in a parabolic setting, with our paraproduct $\mathbf{P}$ in the role of the comparison operator.

**THEOREM 4** (Higher order Taylor expansion). Let $f: \mathbf{R} \mapsto \mathbf{R}$ be a $C_b^4$, and let $u$ be a real-valued $\alpha$-Hölder function on the parabolic space $\mathcal{M}$, with $0 < \alpha < 1$. Then

$$ f(u) = \mathbf{P}_{f'(u)}(u) + \frac{1}{2}\{\mathbf{P}_{f^{(2)}(u)}(u^2) - 2\mathbf{P}_{f^{(2)}(u)}(u)\} \\ + \frac{1}{3!}\{\mathbf{P}_{f^{(3)}(u)}(u^3) - 3\mathbf{P}_{f^{(3)}(u)}(u^2) + 3\mathbf{P}_{f^{(3)}(u)}(u^2)\} + f(u)^{\sharp} \quad (2.3) $$

for some remainder $f(u)^{\sharp}$ of parabolic Hölder regularity $4\alpha$. If $f \in C_b^5$ then the remainder term $f(u)^{\sharp}$ is a locally Lipschitz function of $u$, in the sense that

$$ \|f(u)^{\sharp} - f(v)^{\sharp}\|_{C^{4\alpha}} \lesssim (1 + \|u\|_{C^{\alpha}} + \|v\|_{C^{\alpha}})^4 \|u - v\|_{C^{\alpha}}. \quad (2.4) $$

We give here a proof of this statement in the case where $u$ is a time-independent function on the $d$-dimension torus and we can use the elementary paraproduct $\Pi^0$ instead of $\mathbf{P}$. The full proof of Theorem 4 is given in Appendix B, Theorem 36; we hope this way of proceeding will make the reasoning clear and technical-free.

*Proof.* Let us prove the second order formula in the special case where $u: \mathbf{T}^d \to \mathbf{R}$, and we use the elementary paraproduct $\Pi^0$ in place of $\mathbf{P}$. The claim amounts in that case to proving that

$$ (\star) := f(u) - \Pi_{f'(u)}^0(u) - \frac{1}{2}\{\Pi_{f^{(2)}(u)}^0(u^2) - 2\Pi_{f^{(2)}(u)}^0(u)\} $$
with

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \{f(u(y)) - f'(u(z))u(y)\} dz dy, $$

and to take profit from the fact that $K_i$ has null mean for $i \ge 1$, as put forward in identity (2.2), to see that one also has, for $i \ge 1$,

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \{f(u(y)) - f(u(z)) - f'(u(z))(u(y) - u(z))\} dz dy. $$

One thus has

$$ |\varepsilon_i(x)| \lesssim \|f''\|_{\infty} \int |K_i(x, y) K_{\le i-2}(x, z)| |u(y) - u(z)|^2 dz dy \lesssim 2^{-2i\alpha} \|u\|_{C^\alpha}^2. $$

Since the functions $\varepsilon_i$ are frequently supported on the annulus of frequencies at the scale $2^i$, and the $\varepsilon_i$ are uniformly bounded by $2^{-2i\alpha}$, the series $\sum_i \varepsilon_i$ defines a $2\alpha$-Hölder function. $\square$

One can play exactly the same game and prove a general expansion result in a parabolic setting, with our paraproduct $\mathbf{P}$ in the role of the comparison operator.

**THEOREM 4** (Higher order Taylor expansion). Let $f: \mathbf{R} \mapsto \mathbf{R}$ be a $C_b^4$, and let $u$ be a real-valued $\alpha$-Hölder function on the parabolic space $\mathcal{M}$, with $0 < \alpha < 1$. Then

$$ f(u) = \mathbf{P}_{f'(u)}(u) + \frac{1}{2}\{\mathbf{P}_{f^{(2)}(u)}(u^2) - 2\mathbf{P}_{f^{(2)}(u)}(u)\} \\ + \frac{1}{3!}\{\mathbf{P}_{f^{(3)}(u)}(u^3) - 3\mathbf{P}_{f^{(3)}(u)}(u^2) + 3\mathbf{P}_{f^{(3)}(u)}(u^2)\} + f(u)^{\sharp} \quad (2.3) $$

for some remainder $f(u)^{\sharp}$ of parabolic Hölder regularity $4\alpha$. If $f \in C_b^5$ then the remainder term $f(u)^{\sharp}$ is a locally Lipschitz function of $u$, in the sense that

$$ \|f(u)^{\sharp} - f(v)^{\sharp}\|_{C^{4\alpha}} \lesssim (1 + \|u\|_{C^{\alpha}} + \|v\|_{C^{\alpha}})^4 \|u - v\|_{C^{\alpha}}. \quad (2.4) $$

We give here a proof of this statement in the case where $u$ is a time-independent function on the $d$-dimension torus and we can use the elementary paraproduct $\Pi^0$ instead of $\mathbf{P}$. The full proof of Theorem 4 is given in Appendix B, Theorem 36; we hope this way of proceeding will make the reasoning clear and technical-free.

*Proof.* Let us prove the second order formula in the special case where $u: \mathbf{T}^d \to \mathbf{R}$, and we use the elementary paraproduct $\Pi^0$ in place of $\mathbf{P}$. The claim amounts in that case to proving that

$$ (\star) := f(u) - \Pi_{f'(u)}^0(u) - \frac{1}{2}\{\Pi_{f^{(2)}(u)}^0(u^2) - 2\Pi_{f^{(2)}(u)}^0(u)\} $$
is a 3α-Hölder function on the torus. As in the proof of Bony's paralinearization result, write (★) under the form

$$ \sum \Delta_i(f(u)) - S_{i-2}(f'(u))\Delta_i(u) - \left\{ \frac{1}{2}S_{i-2}(f^{(2)}(u))\Delta_i(u^2) - S_{i-2}(f^{(2)}(u))\Delta_i(u) \right\} =: \sum \varepsilon_i. $$

For each $i \ge 1$, we have

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \left\{ \int_0^1 f^{(2)}(u(z) + t(u(y) - u(z)))(u(y) - u(z))^2 t \, dt - \frac{1}{2} f^{(2)}(u(z))u^2(y) + f^{(2)}(u(z))u(z)u(y) \right\} dz \, dy, $$

which we can rewrite as

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \int_0^1 \int_0^1 f^{(3)}(u(z) + st(u(y) - u(z)))(u(y) - u(z))^3 ds \, t^2 \, dt \, dz \, dy, $$

using once again the fact that the kernels $K_i(x, \cdot)$ have null mean. One reads on this expression for $\varepsilon_i$ that it is of order $2^{-3i\alpha}$, uniformly in $x$. See Appendix B for a full proof of the statement in the parabolic setting.

To prove the stability estimate (2.4), write

$$ f(u)^{\sharp} := \sum_{i} \varepsilon_{i,u}, \quad f(v)^{\sharp} := \sum_{i} \varepsilon_{i,v} $$

with $\varepsilon_{i,u}$ equal to the above $\varepsilon_i$ and $\varepsilon_{i,v}$ defined similarly with $v$ in place of $u$. A uniform estimate of $\varepsilon_{i,u} - \varepsilon_{i,v}$ provides an estimate on $f(u)^{\sharp} - f(v)^{\sharp}$. The expression for $\varepsilon_{i,u} - \varepsilon_{i,v}$ involves the difference

$$ f^{(3)}(u(z) + st(u(y) - u(z)))(u(y) - u(z))^3 - f^{(3)}(v(z) + st(v(y) - v(z)))(v(y) - v(z))^3. $$

Use the fact that $f^{(3)}$ is Lipschitz continuous, that $u$ and $v$ are $C^\alpha$, to see that each of the two terms above is controlled by $d(y, z)^{3\alpha}$ and

$$ |f^{(3)}(u(z) + st(u(y) - u(z))) - f^{(3)}(v(z) + st(v(y) - v(z)))| \lesssim \|f^{(4)}\|_{\infty} \|u - v\|_{C^{\alpha}} $$
is a 3α-Hölder function on the torus. As in the proof of Bony's paralinearization result, write (★) under the form

$$ \sum \Delta_i(f(u)) - S_{i-2}(f'(u))\Delta_i(u) - \left\{ \frac{1}{2}S_{i-2}(f^{(2)}(u))\Delta_i(u^2) - S_{i-2}(f^{(2)}(u))\Delta_i(u) \right\} =: \sum \varepsilon_i. $$

For each $i \ge 1$, we have

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \left\{ \int_0^1 f^{(2)}(u(z) + t(u(y) - u(z)))(u(y) - u(z))^2 t \, dt - \frac{1}{2} f^{(2)}(u(z))u^2(y) + f^{(2)}(u(z))u(z)u(y) \right\} dz \, dy, $$

which we can rewrite as

$$ \varepsilon_i(x) = \int K_i(x, y) K_{\le i-2}(x, z) \int_0^1 \int_0^1 f^{(3)}(u(z) + st(u(y) - u(z)))(u(y) - u(z))^3 ds \, t^2 \, dt \, dz \, dy, $$

using once again the fact that the kernels $K_i(x, \cdot)$ have null mean. One reads on this expression for $\varepsilon_i$ that it is of order $2^{-3i\alpha}$, uniformly in $x$. See Appendix B for a full proof of the statement in the parabolic setting.

To prove the stability estimate (2.4), write

$$ f(u)^{\sharp} := \sum_{i} \varepsilon_{i,u}, \quad f(v)^{\sharp} := \sum_{i} \varepsilon_{i,v} $$

with $\varepsilon_{i,u}$ equal to the above $\varepsilon_i$ and $\varepsilon_{i,v}$ defined similarly with $v$ in place of $u$. A uniform estimate of $\varepsilon_{i,u} - \varepsilon_{i,v}$ provides an estimate on $f(u)^{\sharp} - f(v)^{\sharp}$. The expression for $\varepsilon_{i,u} - \varepsilon_{i,v}$ involves the difference

$$ f^{(3)}(u(z) + st(u(y) - u(z)))(u(y) - u(z))^3 - f^{(3)}(v(z) + st(v(y) - v(z)))(v(y) - v(z))^3. $$

Use the fact that $f^{(3)}$ is Lipschitz continuous, that $u$ and $v$ are $C^\alpha$, to see that each of the two terms above is controlled by $d(y, z)^{3\alpha}$ and

$$ |f^{(3)}(u(z) + st(u(y) - u(z))) - f^{(3)}(v(z) + st(v(y) - v(z)))| \lesssim \|f^{(4)}\|_{\infty} \|u - v\|_{C^{\alpha}} $$
and

$$|(u(y) - u(z))^3 - (v(y) - v(z))^3| \lesssim \|u - v\|_{C^\alpha} d(y, z)^{3\alpha} (\|u\|_{C^\alpha} + \|v\|_{C^\alpha})^2.$$

Estimate (2.4) is obtained by combining these two estimates together. □

Observe that the expansion (2.3) is exact, $f(u)^{\sharp} = 0$, for a polynomial function $f$ of degree at most 3. The above expansion formula for $f(u)$ is conveniently rewritten under the form

$$f(u) = P_{f'(u)-uf^{(2)}(u)+(1/2)u^2 f^{(3)}(u)}(u) + \frac{1}{2}P_{f^{(2)}(u)-uf^{(3)}(u)}(u^2) + \frac{1}{6}P_{f^{(3)}(u)}(u^3) + f(u)^{\sharp}.$$

Note here that the general paracontrolled expansion formula writes

$$f(u) = \sum_{n=1}^{k} \frac{1}{n!} \sum_{j=0}^{n-1} (-1)^j \binom{n}{j} P_{u^j f^{(n)}(u)}(u^{n-j}) + f(u)^{\sharp},$$

for a function $f$ of class $C^{k+1}$ with bounded $(k+1)$th derivative, and a remainder $f(u)^{\sharp}$ of parabolic Hölder regularity $(k+1)\alpha$. Note that each paraproduct $P_{u^j f^{(n)}(u)}(u^{n-j})$ is only of regularity $\alpha$, while the two brackets that appear in (2.3) are respectively $2\alpha$ and $3\alpha$-Hölder. We shall see in Proposition 17 how to write each bracket in (2.3) as a sum of terms of regularity $2\alpha$ and $3\alpha$, respectively.

### 3. A toolkit for paracontrolled calculus

As said in Section 1.2, the mechanics of computations within paracontrolled calculus lies in the analysis of how some operators behave when estimated on paraproduct terms. As a rule of thumb, an ill-defined operator $E$ satisfies an expansion of the form

$$E(P_a b) = aE(b) + E^{+}(a, b),$$

and well-defined operators $F$ satisfy an expansion of the form

$$F(P_a b) = P_a F(b) + F^{+}(a, b).$$

Both $E^+$ and $F^+$ are expected to take values in spaces of functions/distributions that are more regular than the typical elements of the spaces where $E$ and $F$ take values, respectively. Gubinelli, Imkeller and Perkowski's continuity result (1.4) is the archetypel of such an expansion. To iterate this kind of expansion, we introduce in this section a number of operators and prove their continuity properties. Together with the Taylor formula of Section 2, the results of this section are our main contribution.
and

$$|(u(y) - u(z))^3 - (v(y) - v(z))^3| \lesssim \|u - v\|_{C^\alpha} d(y, z)^{3\alpha} (\|u\|_{C^\alpha} + \|v\|_{C^\alpha})^2.$$

Estimate (2.4) is obtained by combining these two estimates together. □

Observe that the expansion (2.3) is exact, $f(u)^{\sharp} = 0$, for a polynomial function $f$ of degree at most 3. The above expansion formula for $f(u)$ is conveniently rewritten under the form

$$f(u) = P_{f'(u)-uf^{(2)}(u)+(1/2)u^2 f^{(3)}(u)}(u) + \frac{1}{2} P_{f^{(2)}(u)-uf^{(3)}(u)}(u^2) + \frac{1}{6} P_{f^{(3)}(u)}(u^3) + f(u)^{\sharp}.$$

Note here that the general paracontrolled expansion formula writes

$$f(u) = \sum_{n=1}^{k} \frac{1}{n!} \sum_{j=0}^{n-1} (-1)^j \binom{n}{j} P_{u^j f^{(n)}(u)}(u^{n-j}) + f(u)^{\sharp},$$

for a function $f$ of class $C^{k+1}$ with bounded $(k+1)$th derivative, and a remainder $f(u)^{\sharp}$ of parabolic Hölder regularity $(k+1)\alpha$. Note that each paraproduct $P_{u^j f^{(n)}(u)}(u^{n-j})$ is only of regularity $\alpha$, while the two brackets that appear in (2.3) are respectively $2\alpha$ and $3\alpha$-Hölder. We shall see in Proposition 17 how to write each bracket in (2.3) as a sum of terms of regularity $2\alpha$ and $3\alpha$, respectively.

### 3. A toolkit for paracontrolled calculus

As said in Section 1.2, the mechanics of computations within paracontrolled calculus lies in the analysis of how some operators behave when estimated on paraproduct terms. As a rule of thumb, an ill-defined operator $E$ satisfies an expansion of the form

$$E(P_a b) = aE(b) + E^{+}(a, b),$$

and well-defined operators $F$ satisfy an expansion of the form

$$F(P_a b) = P_a F(b) + F^{+}(a, b).$$

Both $E^+$ and $F^+$ are expected to take values in spaces of functions/distributions that are more regular than the typical elements of the spaces where $E$ and $F$ take values, respectively. Gubinelli, Imkeller and Perkowski's continuity result (1.4) is the archetypel of such an expansion. To iterate this kind of expansion, we introduce in this section a number of operators and prove their continuity properties. Together with the Taylor formula of Section 2, the results of this section are our main contribution.
The development of paracontrolled calculus beyond the first order calculus of [17] requires the introduction of a modified paraproduct $\tilde{\mathbf{P}}$, introduced in [2], and the use of expansion formulae of the form

$$ 
\begin{aligned}
E(\tilde{\mathbf{P}}_a b) &= aE(b) + E^+(a, b), \\
F(\tilde{\mathbf{P}}_a b) &= \mathbf{P}_a F(b) + F^+(a, b).
\end{aligned}
 $$

This technical point is needed to run the fixed point procedure described in step (3) of the three-step process of the paracontrolled analysis of a given singular PDE; see Section 1.1. Concretely, one works with $\tilde{\mathbf{P}}$ as one works with $\mathbf{P}$. The modified paraproduct $\tilde{\mathbf{P}}$ is given by the formula

$$ \tilde{\mathbf{P}}_f g := \mathcal{L}^{-1}(\mathbf{P}_f(\mathcal{L}g)), $$

where $\mathcal{L}$ stands for the parabolic differential operator $(\partial_\tau + L)$ on the parabolic space $\mathcal{M}$. See [2, Section 4.1] for a study of the continuity properties of $\tilde{\mathbf{P}}$, and Appendix C.2 for a digest. Recall from Section 1.2 the parallel between the basic rules of stochastic calculus and the fundamentals of the first order paracontrolled calculus. This integral picture of paraproduct provides a useful guide for the intuition, where the time derivative $d$ plays the role of the operator $\mathcal{L}$. In that comparison, $\tilde{\mathbf{P}}_f g$ corresponds to the formal quantity

$$ \int \left( \int f d^2 g \right) \approx \int f dg - \iint df dg, $$

after an integration by parts. So the difference between $\mathbf{P}$ and $\tilde{\mathbf{P}}$ is a kind of 'bracket' term, reminiscent of the Itô-to-Stratonovich rule for stochastic integration.

We provide in this section a number of continuity results for some operators involving the paraproduct and resonant operators, together with the modified paraproduct $\tilde{\mathbf{P}}$. We state in this section our results in their general form, in the parabolic setting of Section 1.3, and give proofs in the time-independent, space setting of the torus, of versions of each statement where we use $\Pi^0$ instead of $\tilde{\mathbf{P}}$. This should make it easier for the reader to go to the core of the machinery without fighting with some possibly overwhelming technicalities; full proofs are given in Appendix C. We hope this way of proceeding will convince the reader that the basic ideas involved here are elementary. It is not necessary, for the purpose of solving a particular singular PDEs using the paracontrolled calculus method, to get into the details of the proofs of the different results given here. We invite the reader to have a look at the results only and then go directly to Section 4 to see them on stage.
The development of paracontrolled calculus beyond the first order calculus of [17] requires the introduction of a modified paraproduct $\tilde{\mathbf{P}}$, introduced in [2], and the use of expansion formulae of the form

$$ 
\begin{aligned}
E(\tilde{\mathbf{P}}_a b) &= aE(b) + E^+(a, b), \\
F(\tilde{\mathbf{P}}_a b) &= \mathbf{P}_a F(b) + F^+(a, b).
\end{aligned}
$$

This technical point is needed to run the fixed point procedure described in step (3) of the three-step process of the paracontrolled analysis of a given singular PDE; see Section 1.1. Concretely, one works with $\tilde{\mathbf{P}}$ as one works with $\mathbf{P}$. The modified paraproduct $\tilde{\mathbf{P}}$ is given by the formula

$$ \tilde{\mathbf{P}}_f g := \mathcal{L}^{-1}(\mathbf{P}_f(\mathcal{L}g)), $$

where $\mathcal{L}$ stands for the parabolic differential operator $(\partial_\tau + L)$ on the parabolic space $\mathcal{M}$. See [2, Section 4.1] for a study of the continuity properties of $\tilde{\mathbf{P}}$, and Appendix C.2 for a digest. Recall from Section 1.2 the parallel between the basic rules of stochastic calculus and the fundamentals of the first order paracontrolled calculus. This integral picture of paraproduct provides a useful guide for the intuition, where the time derivative $d$ plays the role of the operator $\mathcal{L}$. In that comparison, $\tilde{\mathbf{P}}_f g$ corresponds to the formal quantity

$$ \int \left( \int f d^2 g \right) \approx \int f dg - \iint df dg, $$

after an integration by parts. So the difference between $\mathbf{P}$ and $\tilde{\mathbf{P}}$ is a kind of ‘bracket’ term, reminiscent of the Itô-to-Stratonovich rule for stochastic integration.

We provide in this section a number of continuity results for some operators involving the paraproduct and resonant operators, together with the modified paraproduct $\tilde{\mathbf{P}}$. We state in this section our results in their general form, in the parabolic setting of Section 1.3, and give proofs in the time-independent, space setting of the torus, of versions of each statement where we use $\Pi^0$ instead of $\tilde{\mathbf{P}}$. This should make it easier for the reader to go to the core of the machinery without fighting with some possibly overwhelming technicalities; full proofs are given in Appendix C. We hope this way of proceeding will convince the reader that the basic ideas involved here are elementary. It is not necessary, for the purpose of solving a particular singular PDEs using the paracontrolled calculus method, to get into the details of the proofs of the different results given here. We invite the reader to have a look at the results only and then go directly to Section 4 to see them on stage.
**A word of caution.** We repeatedly use below the fact that $P_1f = f$ for an arbitrary distribution. This is not true, strictly speaking, as one rather has $P_1f = f+(\text{smooth})$, for an infinitely smooth additional term that is continuous and linear with respect to $f$, for $f$ in any Hölder space, with positive or negative Hölder exponent. Using the first identity rather than the second has no effect whatsoever on the analysis below, so we prefer not to burden the reader with these somewhat irrelevant additional terms and stick to the identity $P_1f = f$.

**3.1. Corrector C, commutator D, and their iterates.** In this section, we introduce four operators C, D and R, S. The operators C and D are used to make sense of ill-defined products, while the operators R and S are used to write the right hand side $f(u, \partial u, \zeta)$ of Equation (1.1) in the form (1.3) needed to run a fixed point argument. The definitions of these operators involves the paraproduct and resonant operators and the modified paraproduct $\tilde{P}$. We define similarly operators C°, D° and R°, S° using P instead of $\tilde{P}$.

To motivate the introduction of the different operators C, D and their iterates, let set ourselves the task of making sense of the product $u\zeta$, of $u \in C^\alpha$ and $\zeta \in C^{\alpha-2}$, with $\frac{2}{5} < \alpha < \frac{1}{2}$. As a product of a $\beta$-Hölder function ($\beta$) with $\zeta$ is well defined if $\beta + \alpha - 2 > 0$, and paraproducts are always well defined in a Hölder setting, we concentrate here on the ill-defined terms that appear in the computations and write

$$ (\beta)\zeta \sim 0 $$

to mean that the product $(\beta)\zeta$ is well defined, and

$$ u\zeta \sim \Pi(u, \zeta) $$

rather than

$$ u\zeta = P_u\zeta + P_\zeta u + \Pi(u, \zeta). $$

This convention is only used in this paragraph. If the resonant term $\Pi(u, \zeta)$ were defined it would have regularity $2\alpha - 2$; we say that the term $\Pi(u, \zeta)$ has formal regularity $2\alpha - 2$. Assume

$$ u = \sum_{i=1}^{3} P_{u_i} Z_i + (4\alpha), $$

for some functions $u_i \in C^\alpha$, some functions $Z_i \in C^{i\alpha}$ depending only on the noise $\zeta$, and a remainder $(4\alpha) \in C^{4\alpha}$. We do not specify the structure of the remainder term $(4\alpha)$ as its product with $\zeta$ is well defined. In the present work, all multilinear
**A word of caution.** We repeatedly use below the fact that $P_1f = f$ for an arbitrary distribution. This is not true, strictly speaking, as one rather has $P_1f = f+(\text{smooth})$, for an infinitely smooth additional term that is continuous and linear with respect to $f$, for $f$ in any Hölder space, with positive or negative Hölder exponent. Using the first identity rather than the second has no effect whatsoever on the analysis below, so we prefer not to burden the reader with these somewhat irrelevant additional terms and stick to the identity $P_1f = f$.

**3.1. Corrector C, commutator D, and their iterates.** In this section, we introduce four operators C, D and R, S. The operators C and D are used to make sense of ill-defined products, while the operators R and S are used to write the right hand side $f(u, \partial u, \zeta)$ of Equation (1.1) in the form (1.3) needed to run a fixed point argument. The definitions of these operators involves the paraproduct and resonant operators and the modified paraproduct $\tilde{P}$. We define similarly operators C°, D° and R°, S° using P instead of $\tilde{P}$.

To motivate the introduction of the different operators C, D and their iterates, let set ourselves the task of making sense of the product $u\zeta$, of $u \in C^\alpha$ and $\zeta \in C^{\alpha-2}$, with $\frac{2}{5} < \alpha < \frac{1}{2}$. As a product of a $\beta$-Hölder function ($\beta$) with $\zeta$ is well defined if $\beta + \alpha - 2 > 0$, and paraproducts are always well defined in a Hölder setting, we concentrate here on the ill-defined terms that appear in the computations and write

$$ (\beta)\zeta \sim 0 $$

to mean that the product $(\beta)\zeta$ is well defined, and

$$ u\zeta \sim \Pi(u, \zeta) $$

rather than

$$ u\zeta = P_u\zeta + P_\zeta u + \Pi(u, \zeta). $$

This convention is only used in this paragraph. If the resonant term $\Pi(u, \zeta)$ were defined it would have regularity $2\alpha - 2$; we say that the term $\Pi(u, \zeta)$ has formal regularity $2\alpha - 2$. Assume

$$ u = \sum_{i=1}^{3} P_{u_i} Z_i + (4\alpha), $$

for some functions $u_i \in C^\alpha$, some functions $Z_i \in C^{i\alpha}$ depending only on the noise $\zeta$, and a remainder $(4\alpha) \in C^{4\alpha}$. We do not specify the structure of the remainder term $(4\alpha)$ as its product with $\zeta$ is well defined. In the present work, all multilinear
functions of the noise $\zeta$ only are assumed to be well defined, even if they have negative formal regularity. Then

$$u\zeta \sim \sum_{i=1}^{3} \Pi(\mathbf{P}_{u_i} Z_i, \zeta),$$

and setting

$$\mathcal{C}^{\circ}(a, b, c) := \Pi(\mathbf{P}_a b, c) - a\Pi(b, c),$$

for the above map $\Pi^+$, one has

$$\begin{aligned} u\zeta &\sim \sum_{i=1}^{3} (u_i \Pi(Z_i, \zeta) + \mathcal{C}^{\circ}(u_i, Z_i, \zeta)) \\ &\sim \sum_{i=1}^{3} (\Pi(u_i, \Pi(Z_i, \zeta)) + \mathcal{C}^{\circ}(u_i, Z_i, \zeta)). \end{aligned}$$

Gubinelli, Imkeller and Perkowski proved in [17] that $\mathcal{C}^{\circ}(a, b, c)$ is essentially well defined whenever the sum of the Hölder exponents of $a$, $b$ and $c$ add up to a positive number, in which case it defines an element of regularity the sum of the regularity exponents. (There is a mild restriction on the range of the different regularity exponents.) The term $\Pi(Z_i, \zeta)$ is of regularity $(i+1)\alpha - 2$, so we have

$$u\zeta \sim \sum_{i=1}^{2} (\Pi(u_i, \Pi(Z_i, \zeta)) + \mathcal{C}^{\circ}(u_i, Z_i, \zeta)).$$

The formal regularity $(i+2)\alpha - 2$ of $\mathcal{C}^{\circ}(u_i, Z_i, \zeta)$ is negative. To proceed we assume that each $u_i$ is also given in paraproduct form

$$u_i = \sum_{j=1}^{2} \mathbf{P}_{u_{ij}} Z_j + (3\alpha),$$

for $u_{ij} \in \mathcal{C}^{\alpha}$ and a remainder $(3\alpha)$ of regularity $3\alpha$. Setting

$$\mathcal{C}^{\circ}((a_1, a_2), b, c) := \mathcal{C}^{\circ}(\mathbf{P}_{a_1} a_2, b, c) - a_1 \mathcal{C}^{\circ}(a_2, b, c),$$

for the operator $(\mathcal{C}^{\circ})^{+}$, we shall prove in Theorem 8 that this map is well defined whenever the sum of the Hölder exponents of $a_1$, $a_2$, $b$ and $c$ add up to a positive number. (Here again, there is a mild restriction on the range of the regularity exponents.) So we have

$$\begin{aligned} u\zeta &\sim \sum_{i=1}^{2} (\Pi(u_{ij}, \Pi(Z_j, \Pi(Z_i, \zeta))) + \mathcal{C}^{\circ}((u_{ij}, Z_j), Z_i, \zeta)) \\ &\sim \Pi(u_{11}, \Pi(Z_1, \Pi(Z_1, \zeta))) + \mathcal{C}^{\circ}((u_{11}, Z_1), Z_1, \zeta). \end{aligned} \quad (3.1)$$
functions of the noise $\zeta$ only are assumed to be well defined, even if they have negative formal regularity. Then

$$u\zeta \sim \sum_{i=1}^{3} \Pi(\mathbf{P}_{u_i} Z_i, \zeta),$$

and setting

$$\mathbf{C}^{\circ}(a, b, c) := \Pi(\mathbf{P}_{ab}, c) - a\Pi(b, c),$$

for the above map $\Pi^+$, one has

$$\begin{aligned} u\zeta &\sim \sum_{i=1}^{3} (u_i \Pi(Z_i, \zeta) + \mathbf{C}^{\circ}(u_i, Z_i, \zeta)) \\ &\sim \sum_{i=1}^{3} (\Pi(u_i, \Pi(Z_i, \zeta)) + \mathbf{C}^{\circ}(u_i, Z_i, \zeta)). \end{aligned}$$

Gubinelli, Imkeller and Perkowski proved in [17] that $\mathbf{C}^{\circ}(a, b, c)$ is essentially well defined whenever the sum of the Hölder exponents of $a$, $b$ and $c$ add up to a positive number, in which case it defines an element of regularity the sum of the regularity exponents. (There is a mild restriction on the range of the different regularity exponents.) The term $\Pi(Z_i, \zeta)$ is of regularity $(i+1)\alpha - 2$, so we have

$$u\zeta \sim \sum_{i=1}^{2} (\Pi(u_i, \Pi(Z_i, \zeta)) + \mathbf{C}^{\circ}(u_i, Z_i, \zeta)).$$

The formal regularity $(i+2)\alpha - 2$ of $\mathbf{C}^{\circ}(u_i, Z_i, \zeta)$ is negative. To proceed we assume that each $u_i$ is also given in paraproduct form

$$u_i = \sum_{j=1}^{2} \mathbf{P}_{u_{ij}} Z_j + (3\alpha),$$

for $u_{ij} \in \mathbf{C}^{\alpha}$ and a remainder $(3\alpha)$ of regularity $3\alpha$. Setting

$$\mathbf{C}^{\circ}((a_1, a_2), b, c) := \mathbf{C}^{\circ}(\mathbf{P}_{a_1} a_2, b, c) - a_1 \mathbf{C}^{\circ}(a_2, b, c),$$

for the operator $(\mathbf{C}^{\circ})^{+}$, we shall prove in Theorem 8 that this map is well defined whenever the sum of the Hölder exponents of $a_1$, $a_2$, $b$ and $c$ add up to a positive number. (Here again, there is a mild restriction on the range of the regularity exponents.) So we have

$$\begin{aligned} u\zeta &\sim \sum_{i=1}^{2} (\Pi(u_{ij}, \Pi(Z_j, \Pi(Z_i, \zeta))) + \mathbf{C}^{\circ}((u_{ij}, Z_j), Z_i, \zeta)) \\ &\sim \Pi(u_{11}, \Pi(Z_1, \Pi(Z_1, \zeta))) + \mathbf{C}^{\circ}((u_{11}, Z_1), Z_1, \zeta). \end{aligned} \quad (3.1)$$
Assuming $u_{11} = P_{u_{11}} Z_1 + (2\alpha)$, for $u_{111} \in C^\alpha$ and a $2\alpha$ remainder $(2\alpha)$, and iterating once more the E-expansion that we have just done twice shows that $u\zeta$ is be well defined, under the assumption that all the multilinear functions of the noise $\zeta$ only that appear above are well defined and have regularity their formal regularity. This kind of computation explains why we need the corrector operator $C^\circ$ and its iterates. (The above mentioned restriction on the range of the regularity exponents in the continuity results for $C^\circ$ imposes that the term $C^\circ((3\alpha), Z_i, \zeta)$ in $C^\circ(u_i, Z_i, \zeta)$ is treated differently from the others. A *refined corrector* is introduced for that purpose in Section 3.1.1. Its use justifies identity (3.1).)

New things happen when we look at the product $u^2\zeta$, as

$$u^2\zeta \sim \Pi(2P_u u + \Pi(u, u), \zeta) \sim 2C^\circ(u, u, \zeta) + \Pi(\Pi(u, u), \zeta).$$

The $C^\circ$ term is dealt with as above. For the resonant term $\Pi(\Pi(u, u), \zeta)$ we need first to use an F-type expansion on $\Pi(u, u)$ before using an E-type expansion. We have

$$\Pi(u, u) = \sum_{i=1}^{3} (P_{u_i} \Pi(Z_i, u) + D^{\circ}(u_i, Z_i, u)),$$

with $D^{\circ}(\cdot, \cdot, u) := \Pi^{+}(\cdot, \cdot, u)$, with the notation of Section 1.2. The resonant terms

$$\Pi(P_{u_i} \Pi(Z_i, u), \zeta)$$

are dealt with as above, using the corrector operator $C^\circ$. For the resonance of $D^{\circ}(u_i, z_i, u)$ with $\zeta$, we use the paracontrolled structures of $u$ and $u_i$ and the fact that the operator $D^\circ$ satisfies an F-type expansion with respect to its first and third arguments – this is proved in the next section. This gives

$$\begin{aligned} \sum_{i=1}^{3} D^{\circ}(u_i, Z_i, u) &= D^{\circ}(u_1, Z_1, u) + (4\alpha) \\ &= P_{u_{11}}(P_{u_1} D^{\circ}(Z_1, Z_1, Z_1)) + (4\alpha)' \end{aligned}$$

for remainders $(4\alpha)$, $(4\alpha)'$ of regularity $4\alpha$, and

$$\begin{aligned} \sum_{i=1}^{3} \Pi(D^{\circ}(u_i, Z_i, u), \zeta) &\sim u_{11} \Pi(P_{u_1} D^{\circ}(Z_1, Z_1, Z_1), \zeta) \\ &\sim u_{11} u_1 \Pi(D^{\circ}(Z_1, Z_1, Z_1), \zeta) \sim 0, \end{aligned}$$

provided $\Pi(D^{\circ}(Z_1, Z_1, Z_1), \zeta)$ is given *a priori* as an element of $C^{4\alpha-2}$.
Assuming $u_{11} = P_{u_{11}} Z_1 + (2\alpha)$, for $u_{111} \in C^\alpha$ and a $2\alpha$ remainder $(2\alpha)$, and iterating once more the E-expansion that we have just done twice shows that $u\zeta$ is be well defined, under the assumption that all the multilinear functions of the noise $\zeta$ only that appear above are well defined and have regularity their formal regularity. This kind of computation explains why we need the corrector operator $C^\circ$ and its iterates. (The above mentioned restriction on the range of the regularity exponents in the continuity results for $C^\circ$ imposes that the term $C^\circ((3\alpha), Z_i, \zeta)$ in $C^\circ(u_i, Z_i, \zeta)$ is treated differently from the others. A refined corrector is introduced for that purpose in Section 3.1.1. Its use justifies identity (3.1).)

New things happen when we look at the product $u^2\zeta$, as

$$u^2\zeta \sim \Pi(2P_u u + \Pi(u, u), \zeta) \sim 2C^\circ(u, u, \zeta) + \Pi(\Pi(u, u), \zeta).$$

The $C^\circ$ term is dealt with as above. For the resonant term $\Pi(\Pi(u, u), \zeta)$ we need first to use an F-type expansion on $\Pi(u, u)$ before using an E-type expansion. We have

$$\Pi(u, u) = \sum_{i=1}^{3} (P_{u_i} \Pi(Z_i, u) + D^\circ(u_i, Z_i, u)),$$

with $D^\circ(\cdot, \cdot, u) := \Pi^+(\cdot, \cdot, u)$, with the notation of Section 1.2. The resonant terms

$$\Pi(P_{u_i} \Pi(Z_i, u), \zeta)$$

are dealt with as above, using the corrector operator $C^\circ$. For the resonance of $D^\circ(u_i, z_i, u)$ with $\zeta$, we use the paracontrolled structures of $u$ and $u_i$ and the fact that the operator $D^\circ$ satisfies an F-type expansion with respect to its first and third arguments – this is proved in the next section. This gives

$$\begin{aligned} \sum_{i=1}^{3} D^{\circ}(u_i, Z_i, u) &= D^{\circ}(u_1, Z_1, u) + (4\alpha) \\ &= P_{u_{11}}(P_{u_1} D^{\circ}(Z_1, Z_1, Z_1)) + (4\alpha)' \end{aligned}$$

for remainders $(4\alpha)$, $(4\alpha)'$ of regularity $4\alpha$, and

$$\begin{aligned} \sum_{i=1}^{3} \Pi(D^{\circ}(u_i, Z_i, u), \zeta) &\sim u_{11} \Pi(P_{u_1} D^{\circ}(Z_1, Z_1, Z_1), \zeta) \\ &\sim u_{11} u_1 \Pi(D^{\circ}(Z_1, Z_1, Z_1), \zeta) \sim 0, \end{aligned}$$

provided $\Pi(D^\circ(Z_1, Z_1, Z_1), \zeta)$ is given a priori as an element of $C^{4\alpha-2}$.
### 3.1.1. Corrector C and the outer centring operator $\mathcal{C}$. 
Define on the space $L^\infty$ of bounded measurable functions on the parabolic space $M$ the **corrector C** as the operator

$$C(f, g, h) := \Pi(\tilde{P}_f g, h) - f \Pi(g, h).$$

The next theorem is the workhorse of the first order paracontrolled calculus, such as devised in [17] by Gubinelli, Imkeller and Perkowski; we recall it here, together with its proof, as this is our starting point. Recall we denote by $C^\alpha$ the spacial Hölder spaces on the torus and by $C^\alpha$ the parabolic Hölder spaces over the compact manifold $M$.

**THEOREM 5.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha \in (0, 1)$, $\beta \in (-3, 3)$ and $\gamma \in (-\infty, 3)$. Assume $\alpha + \beta < 3$, and

$$0 < \alpha + \beta + \gamma < 1, \quad \text{while } \beta + \gamma < 0.$$

Then, the corrector $C$ extends continuously as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$.

*Proof.* As advertised above, we prove here this continuity result for a simplified version of the operator $C$, and refer the reader to Proposition 40 in Appendix C for full proofs. Assume we are working in the time-independent setting of the $d$-dimensional torus, with the operator

$$C^0(f, g, h) := \Pi^0(\Pi_f^0 g, h) - f \Pi^0(g, h).$$

We prove the continuity of the corrector $C^0$, as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$, under the above assumptions on $\alpha, \beta, \gamma$.

* We first give a heuristic proof of the statement. The resonant operator is given by

$$\Pi^0(a, b) \approx \sum_{|i-j|\le 1} \Delta_i(a) \Delta_j(b). \qquad (3.2)$$

Write

$$C^0(f, g, h) = \sum_{|i-j|\le 1} \Delta_i(\Pi_f^0 g) \Delta_j h - f \Delta_i(g) \Delta_j(h),$$

and set

$$\varepsilon'_i := \Delta_i(\Pi_f^0 g) - f \Delta_i(g),$$

so we have

$$C^0(f, g, h) = \sum_{|i-j|\le 1} \varepsilon'_i \Delta_j(h).$$
### 3.1.1. Corrector $\mathcal{C}$ and the outer centring operator $\mathcal{C}$. 
Define on the space $L^\infty$ of bounded measurable functions on the parabolic space $M$ the **corrector** $\mathcal{C}$ as the operator

$$ \mathcal{C}(f, g, h) := \Pi(\tilde{P}_f g, h) - f \Pi(g, h). $$

The next theorem is the workhorse of the first order paracontrolled calculus, such as devised in [17] by Gubinelli, Imkeller and Perkowski; we recall it here, together with its proof, as this is our starting point. Recall we denote by $C^\alpha$ the spacial Hölder spaces on the torus and by $C^\alpha$ the parabolic Hölder spaces over the compact manifold $M$.

**THEOREM 5.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha \in (0, 1)$, $\beta \in (-3, 3)$ and $\gamma \in (-\infty, 3)$. Assume $\alpha + \beta < 3$, and

$$ 0 < \alpha + \beta + \gamma < 1, \quad \text{while } \beta + \gamma < 0. $$

Then, the corrector $\mathcal{C}$ extends continuously as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$.

*Proof.* As advertised above, we prove here this continuity result for a simplified version of the operator $\mathcal{C}$, and refer the reader to Proposition 40 in Appendix C for full proofs. Assume we are working in the time-independent setting of the $d$-dimensional torus, with the operator

$$ \mathcal{C}^0(f, g, h) := \Pi^0(\Pi_f^0 g, h) - f \Pi^0(g, h). $$

We prove the continuity of the corrector $\mathcal{C}^0$, as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$, under the above assumptions on $\alpha, \beta, \gamma$.

* We first give a heuristic proof of the statement. The resonant operator is given by

$$ \Pi^0(a, b) \approx \sum_{|i-j|\le 1} \Delta_i(a) \Delta_j(b). \qquad (3.2) $$

Write

$$ \mathcal{C}^0(f, g, h) = \sum_{|i-j|\le 1} \Delta_i(\Pi_f^0 g) \Delta_j h - f \Delta_i(g) \Delta_j(h), $$

and set

$$ \varepsilon'_i := \Delta_i(\Pi_f^0 g) - f \Delta_i(g), $$

so we have

$$ \mathcal{C}^0(f, g, h) = \sum_{|i-j|\le 1} \varepsilon'_i \Delta_j(h). $$
The fact that $\epsilon_i'$ has $L^\infty$-norm of order $2^{-i(\alpha+\beta)}$ can be guessed on the expression

$$ 
\begin{aligned}
\epsilon_i'(x) &= \int K_i(x, y) \{ (\Pi_f^0 g)(y) - f(x)g(y) \} dy \\
&= \int K_i(x, y) \{ \Pi_{f-f(x)}^0 g \}(y) dy.
\end{aligned}
$$

As $y$ is concentrated near $x$, at scale $2^{-i}$, and we are looking at the $i$th Littlewood-Paley block of $P_{f-f(x)}g$, we expect

$$ |\epsilon_i'(x)| \lesssim 2^{-i\beta} \|\Pi_{f-f(x)}^0 g\|_{C^\beta} \lesssim 2^{-i\beta} \|f - f(x)\|_{L^\infty} \|g\|_{C^\beta}, $$

with a term $\|f - f(x)\|_{L^\infty}$ involving only the neighbourhood of $x$ of size $2^{-i}$, that is with

$$ \|f - f(x)\|_{L^\infty} \lesssim 2^{-i\alpha} \|f\|_{C^\alpha}, $$

since $f$ is $\alpha$-Hölder. Such an estimate would imply the continuity of the corrector $C$ as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$, if $\alpha + \beta + \gamma$ is positive, since $h$ is $\gamma$-Hölder. However, this heuristic argument does not make it clear why we need $\beta + \gamma$ to be negative to get the result.

• A mathematically correct version of the above sketch of proof is done by estimating the $L^\infty$-norm of the dyadic blocks of $\epsilon_i'$. For $k \ge i + 2$ then

$$ \Delta_k \epsilon_i' = -\Delta_k(f \Delta_i(g)) \simeq -\Delta_k(f) \Delta_i(g); $$

here and below the $\simeq$ means that the right hand side is equal to a finite sum of terms of the given form. This is here a direct consequence of frequency considerations with the frequency supports of the dyadic projections $\Delta_k$, for $k \ge i + 2$. So we have

$$ \|\Delta_k \epsilon_i'\|_{L^\infty} \lesssim 2^{-k\alpha} 2^{-i\beta} \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

For $k \le i - 2$ then

$$ \Delta_k \epsilon_i' = -\Delta_k(f \Delta_i(g)) \simeq -\Delta_k(\Delta_i(f) \Delta_i(g)) $$

hence

$$ \|\Delta_k \epsilon_i'\|_{L^\infty} \lesssim 2^{-i(\alpha+\beta)} \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

We adopt the classical notation $S_{k-2}f$ for the partial sum $\sum_{\ell \le k-2} \Delta_\ell(f)$ of the Littlewood-Paley decomposition, so for $|i-k| \le 2$ we have

$$ \Delta_k \epsilon_i' \simeq \Delta_k(\Delta_i(g) S_{i-2}(f) - S_{k+2}(f) \Delta_i(g)), $$
The fact that $\epsilon_i'$ has $L^\infty$-norm of order $2^{-i(\alpha+\beta)}$ can be guessed on the expression

$$ 
\begin{aligned}
\epsilon_i'(x) &= \int K_i(x, y) \{ (\Pi_f^0 g)(y) - f(x)g(y) \} dy \\
&= \int K_i(x, y) \{ \Pi_{f-f(x)}^0 g \}(y) dy.
\end{aligned}
$$

As $y$ is concentrated near $x$, at scale $2^{-i}$, and we are looking at the $i$th Littlewood-Paley block of $P_{f-f(x)}g$, we expect

$$ |\epsilon_i'(x)| \lesssim 2^{-i\beta} \|\Pi_{f-f(x)}^0 g\|_{C^\beta} \lesssim 2^{-i\beta} \|f - f(x)\|_{L^\infty} \|g\|_{C^\beta}, $$

with a term $\|f - f(x)\|_{L^\infty}$ involving only the neighbourhood of $x$ of size $2^{-i}$, that is with

$$ \|f - f(x)\|_{L^\infty} \lesssim 2^{-i\alpha} \|f\|_{C^\alpha}, $$

since $f$ is $\alpha$-Hölder. Such an estimate would imply the continuity of the corrector $C$ as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$, if $\alpha + \beta + \gamma$ is positive, since $h$ is $\gamma$-Hölder. However, this heuristic argument does not make it clear why we need $\beta + \gamma$ to be negative to get the result.

• A mathematically correct version of the above sketch of proof is done by estimating the $L^\infty$-norm of the dyadic blocks of $\epsilon_i'$. For $k \ge i + 2$ then

$$ \Delta_k \epsilon_i' = -\Delta_k(f \Delta_i(g)) \simeq -\Delta_k(f) \Delta_i(g); $$

here and below the $\simeq$ means that the right hand side is equal to a finite sum of terms of the given form. This is here a direct consequence of frequency considerations with the frequency supports of the dyadic projections $\Delta_k$, for $k \ge i + 2$. So we have

$$ \|\Delta_k \epsilon_i'\|_{L^\infty} \lesssim 2^{-k\alpha} 2^{-i\beta} \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

For $k \le i - 2$ then

$$ \Delta_k \epsilon_i' = -\Delta_k(f \Delta_i(g)) \simeq -\Delta_k(\Delta_i(f) \Delta_i(g)) $$

hence

$$ \|\Delta_k \epsilon_i'\|_{L^\infty} \lesssim 2^{-i(\alpha+\beta)} \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

We adopt the classical notation $S_{k-2}f$ for the partial sum $\sum_{\ell \le k-2} \Delta_\ell(f)$ of the Littlewood-Paley decomposition, so for $|i-k| \le 2$ we have

$$ \Delta_k \epsilon_i' \simeq \Delta_k(\Delta_i(g)S_{i-2}(f) - S_{k+2}(f)\Delta_i(g)), $$
hence

$$ \|\Delta_k \epsilon_i'\|_{L^\infty} \lesssim 2^{-i(\alpha+\beta)} \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

As a consequence, we always have the following estimate

$$ \|\Delta_k \epsilon_i'\|_{L^\infty} \lesssim 2^{-i\beta} 2^{-\max(i,k)\alpha} \|f\|_{C^\alpha} \|g\|_{C^\beta}. \quad (3.3) $$

We can then estimate $C^0(f, g; h)$ in some Hölder space. For a nonnegative integer $k$, we have

$$ \begin{aligned} \Delta_k(C^0(f, g, h)) &= \sum_{|i-j|\le 1} \Delta_k(\epsilon_i' \Delta_j h) \\ &\approx \sum_{\substack{i<k-2 \\ |i-j|\le 1}} \Delta_k(\epsilon_i') \Delta_j(h) + \sum_{\substack{k<i-2 \\ |i-j|\le 1}} \Delta_k(\Delta_i(\epsilon_i') \Delta_j(h)) \\ &\quad + \sum_{\substack{|k-i|\le 2 \\ |i-j|\le 1}} \Delta_k(S_i(\epsilon_i') \Delta_j(h)) \end{aligned} $$

which is then controlled, using estimate (3.3), by

$$ \begin{aligned} \|\Delta_k(C^0(f, g, h))\|_{L^\infty} &\lesssim \left( \sum_{i<k-2} 2^{-i\beta} 2^{-k\alpha} 2^{-i\gamma} + \sum_{k<i-2} 2^{-i(\alpha+\beta+\gamma)} + \sum_{|k-i|\le 2} 2^{-i(\alpha+\beta+\gamma)} \right) \\ &\quad \times \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \\ &\lesssim 2^{-k(\alpha+\beta+\gamma)} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}; \end{aligned} $$

we used the condition $\beta + \gamma < 0$ to estimate the first sum, and the condition $\alpha + \beta + \gamma > 0$, to estimate the second sum. The fact that the latter estimate holds uniformly in $k$ concludes the proof of the $(\alpha + \beta + \gamma)$-Hölder regularity of the corrector. $\square$

We emphasize the importance of the above heuristic proof of continuity of the corrector $C$ by introducing a notation.

**DEFINITION.** Given an endomorphism $A$ of some function space, we denote by $Cf$, or $C_x f$, the function

$$ (Cf)(\cdot) := f(\cdot) - f(x), $$

recentered around its value at the 'running' variable $x$, so that

$$ A(Cf)(x) = A(f - f(x))(x). $$
hence

$$ \|\Delta_k \varepsilon_i'\|_{L^\infty} \lesssim 2^{-i(\alpha+\beta)} \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

As a consequence, we always have the following estimate

$$ \|\Delta_k \varepsilon_i'\|_{L^\infty} \lesssim 2^{-i\beta} 2^{-\max(i,k)\alpha} \|f\|_{C^\alpha} \|g\|_{C^\beta}. \quad (3.3) $$

We can then estimate $C^0(f, g; h)$ in some Hölder space. For a nonnegative integer $k$, we have

$$ \begin{aligned} \Delta_k(C^0(f, g, h)) &= \sum_{|i-j|\le 1} \Delta_k(\varepsilon_i' \Delta_j h) \\ &\approx \sum_{\substack{i<k-2 \\ |i-j|\le 1}} \Delta_k(\varepsilon_i') \Delta_j(h) + \sum_{\substack{k<i-2 \\ |i-j|\le 1}} \Delta_k(\Delta_i(\varepsilon_i') \Delta_j(h)) \\ &\quad + \sum_{\substack{|k-i|\le 2 \\ |i-j|\le 1}} \Delta_k(S_i(\varepsilon_i') \Delta_j(h)) \end{aligned} $$

which is then controlled, using estimate (3.3), by

$$ \begin{aligned} \|\Delta_k(C^0(f, g, h))\|_{L^\infty} &\lesssim \left( \sum_{i<k-2} 2^{-i\beta} 2^{-k\alpha} 2^{-i\gamma} + \sum_{k<i-2} 2^{-i(\alpha+\beta+\gamma)} + \sum_{|k-i|\le 2} 2^{-i(\alpha+\beta+\gamma)} \right) \\ &\quad \times \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \\ &\lesssim 2^{-k(\alpha+\beta+\gamma)} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}; \end{aligned} $$

we used the condition $\beta + \gamma < 0$ to estimate the first sum, and the condition $\alpha + \beta + \gamma > 0$, to estimate the second sum. The fact that the latter estimate holds uniformly in $k$ concludes the proof of the $(\alpha + \beta + \gamma)$-Hölder regularity of the corrector. $\square$

We emphasize the importance of the above heuristic proof of continuity of the corrector $C$ by introducing a notation.

**DEFINITION.** Given an endomorphism $A$ of some function space, we denote by $Cf$, or $C_x f$, the function

$$ (Cf)(\cdot) := f(\cdot) - f(x), $$

recentered around its value at the 'running' variable $x$, so that

$$ A(Cf)(x) = A(f - f(x))(x). $$
Strictly speaking, the operator $\mathcal{C}$ is an operator on the space of operators A. The choice of letter $\mathcal{C}$ for this operator is for ‘centring’, and we call $\mathcal{C}$ the **outer centring operator**.

In those terms, we have

$$ \mathcal{C}(f, g, h) = \Pi(\tilde{\mathbb{P}}_{\mathcal{C}f} g, h), \qquad (3.4) $$

and

$$ \Pi(\mathbb{P}_{\mathcal{C}f, \mathcal{C}g} g, h)(x) = \Pi(\mathbb{P}_{\mathbb{P}_{b-b(x)}c-(\mathbb{P}_{b-b(x)}c)(x)} g, h)(x), $$

for instance. The main property of this operator is the following. For a function $f \in C^\alpha(\mathbf{T}^d)$ with $\alpha$ positive, we have

$$ \begin{aligned} S_k(\mathcal{C}f)(x) &= S_k(f - f(x))(x) = S_k(f)(x) - f(x) \\ &= \sum_{\ell \ge k+1} \Delta_\ell(f)(x). \end{aligned} $$

Since $f$ has positive regularity, the $L^\infty$ size of the dyadic blocks $\Delta_\ell f$ are exponentially decreasing with $\ell$, so

$$ \|S_k(\mathcal{C}f)\|_{\infty} + \|\Delta_k f\|_{\infty} \lesssim 2^{-\alpha k}. \qquad (3.5) $$

A very similar property holds in the parabolic setting, which is used in the proofs of the continuity results of this section given in Appendix C.

### 3.1.2. Dealing with remainders: refined corrector $C_{(1)}$.
The outer centring operator allows to take profit from the Hölder continuity regularity property of a function. To take further profit from the $\gamma$-Lipschitz regularity property of a function, with $1 < \gamma < 2$, one introduces a **refined corrector**, defined as follows in the model setting of the flat torus; see Definition 37 for the definition in the parabolic setting. This operator is used to take care of a number of remainder terms in the paracontrolled analysis of ill-defined terms. Set

$$ C_{(1)}^0(f, g, h) := \Pi^0(\mathbb{P}_f g, h) - f \Pi^0(g, h) - f' \Pi_{(1)}^0(g, h), $$

where

$$ \Pi_{(1)}^0(g, h)(x) := \sum_{|i-j| \le 1} \Delta_i((\cdot - x)g)(x) (\Delta_j h)(x), $$

for $x \in \mathbf{T}$. An elementary refinement of the proof of Theorem 5 gives the following statement. We refer the reader to Definition 37 for the definition of the parabolic counterpart $C_{(1)}$ of $C_{(1)}^0$.
Strictly speaking, the operator $\mathcal{C}$ is an operator on the space of operators A. The choice of letter $\mathcal{C}$ for this operator is for ‘centring’, and we call $\mathcal{C}$ the **outer centring operator**.

In those terms, we have

$$ \mathcal{C}(f, g, h) = \Pi(\tilde{\mathbb{P}}_{\mathcal{C}f} g, h), \quad (3.4) $$

and

$$ \Pi(\mathbb{P}_{\mathcal{C}f\mathcal{C}b} g, h)(x) = \Pi(\mathbb{P}_{P_{b-b(x)}c-(P_{b-b(x)}c)(x)} g, h)(x), $$

for instance. The main property of this operator is the following. For a function $f \in C^\alpha(\mathbf{T}^d)$ with $\alpha$ positive, we have

$$ \begin{aligned} S_k(\mathcal{C}f)(x) &= S_k(f - f(x))(x) = S_k(f)(x) - f(x) \\ &= \sum_{\ell \ge k+1} \Delta_\ell(f)(x). \end{aligned} $$

Since $f$ has positive regularity, the $L^\infty$ size of the dyadic blocks $\Delta_\ell f$ are exponentially decreasing with $\ell$, so

$$ \|S_k(\mathcal{C}f)\|_{\infty} + \|\Delta_k f\|_{\infty} \lesssim 2^{-\alpha k}. \quad (3.5) $$

A very similar property holds in the parabolic setting, which is used in the proofs of the continuity results of this section given in Appendix C.

### 3.1.2. Dealing with remainders: refined corrector $C_{(1)}$.
The outer centring operator allows to take profit from the Hölder continuity regularity property of a function. To take further profit from the $\gamma$-Lipschitz regularity property of a function, with $1 < \gamma < 2$, one introduces a **refined corrector**, defined as follows in the model setting of the flat torus; see Definition 37 for the definition in the parabolic setting. This operator is used to take care of a number of remainder terms in the paracontrolled analysis of ill-defined terms. Set

$$ C_{(1)}^0(f, g, h) := \Pi^0(\mathbb{P}_f g, h) - f \Pi^0(g, h) - f' \Pi_{(1)}^0(g, h), $$

where

$$ \Pi_{(1)}^0(g, h)(x) := \sum_{|i-j| \le 1} \Delta_i((\cdot - x)g)(x) (\Delta_j h)(x), $$

for $x \in \mathbf{T}$. An elementary refinement of the proof of Theorem 5 gives the following statement. We refer the reader to Definition 37 for the definition of the parabolic counterpart $C_{(1)}$ of $C_{(1)}^0$.
**THEOREM 6.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha \in (1, 2)$, $\beta \in (-3, 3)$ and $\gamma \in (-\infty, 3)$. Assume $\alpha + \beta < 3$, and

$$0 < \alpha + \beta + \gamma < 1, \quad \text{while } \beta + \gamma < 0.$$

Then, the corrector $C_{(1)}$ extends continuously as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$.

The proof of the next statement is identical to the proof of the continuity result for $C$ from Theorem 5; it is left to the reader. Recall from Definition 37 that $C_{(1)}$ is defined using the operators $\Pi_{(1)}^\ell$.

**THEOREM 7.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha, \beta \in (0, 1)$, and $\gamma \in (-\infty, 3)$. Assume

$$\beta + \gamma + 1 < 0 < \alpha + \beta + \gamma + 1 < 1.$$

Then all the operators

$$(f, g, h) \mapsto \Pi_{(1)}^\ell(\tilde{P}_f g, h) - f \Pi_{(1)}^\ell(g, h)$$

extend continuously as functions from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma+1}$, for $1 \le \ell \le \ell_0$.

**3.1.3. Iterated correctors.** Given a tuple of bounded functions $(a, b, c, d)$, set

$$\tilde{P}^{\circ 2}(a, b, c) := \tilde{P}_{\tilde{P}_{ab}c}$$

and

$$\tilde{P}^{\circ 3}(a, b, c, d) := \tilde{P}_{\tilde{P}^{\circ 2}(a,b,c)d},$$

and give similar definitions of $\Pi^{\circ 2}(a, b, c)$ and $\Pi^{\circ 3}(a, b, c, d)$, using only $P$ operators, and $(\Pi^0)^{\circ 2}(a, b, c)$ and $(\Pi^0)^{\circ 3}(a, b, c, d)$, using only $\Pi^0$ operators, respectively. Depending on whether or not such a paraproduct appears in the low frequency, in place of $f$, or high frequency, in place of $g$, in the formulae for the corrector $C(f, g, h)$ or the commutator $D(f, g, h)$, we shall talk about **lower** or **upper iterated** operators.

• We define the 4 and 5-linear **lower iterated correctors** by the formulae

$$ \begin{aligned} C((a_1, a_2), g, h) &:= C(\tilde{P}_{a_1} a_2, g, h) - a_1 C(a_2, g, h) \\ &= \Pi(\tilde{P}^{\circ 2}(a_1, a_2, g), h) - \{(\tilde{P}_{a_1} a_2) \Pi(g, h) + a_1 \Pi(\tilde{P}_{c a_2} g, h)\}, \end{aligned} \quad (3.6) $$
**THEOREM 6.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha \in (1, 2)$, $\beta \in (-3, 3)$ and $\gamma \in (-\infty, 3)$. Assume $\alpha + \beta < 3$, and

$$0 < \alpha + \beta + \gamma < 1, \quad \text{while } \beta + \gamma < 0.$$

Then, the corrector $C_{(1)}$ extends continuously as a function from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$.

The proof of the next statement is identical to the proof of the continuity result for $C$ from Theorem 5; it is left to the reader. Recall from Definition 37 that $C_{(1)}$ is defined using the operators $\Pi_{(1)}^\ell$.

**THEOREM 7.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha, \beta \in (0, 1)$, and $\gamma \in (-\infty, 3)$. Assume

$$\beta + \gamma + 1 < 0 < \alpha + \beta + \gamma + 1 < 1.$$

Then all the operators

$$(f, g, h) \mapsto \Pi_{(1)}^\ell(\tilde{P}_f g, h) - f \Pi_{(1)}^\ell(g, h)$$

extend continuously as functions from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma+1}$, for $1 \le \ell \le \ell_0$.

**3.1.3. Iterated correctors.** Given a tuple of bounded functions $(a, b, c, d)$, set

$$\tilde{P}^{\circ 2}(a, b, c) := \tilde{P}_{\tilde{P}_{ab}c}$$

and

$$\tilde{P}^{\circ 3}(a, b, c, d) := \tilde{P}_{\tilde{P}^{\circ 2}(a,b,c)d},$$

and give similar definitions of $\Pi^{\circ 2}(a, b, c)$ and $\Pi^{\circ 3}(a, b, c, d)$, using only $P$ operators, and $(\Pi^0)^{\circ 2}(a, b, c)$ and $(\Pi^0)^{\circ 3}(a, b, c, d)$, using only $\Pi^0$ operators, respectively. Depending on whether or not such a paraproduct appears in the low frequency, in place of $f$, or high frequency, in place of $g$, in the formulae for the corrector $C(f, g, h)$ or the commutator $D(f, g, h)$, we shall talk about **lower** or **upper iterated** operators.

* We define the 4 and 5-linear **lower iterated correctors** by the formulae

$$
\begin{align}
C((a_1, a_2), g, h) &:= C(\tilde{P}_{a_1} a_2, g, h) - a_1 C(a_2, g, h) \nonumber \\
&= \Pi(\tilde{P}^{\circ 2}(a_1, a_2, g), h) - \{(\tilde{P}_{a_1} a_2) \Pi(g, h) + a_1 \Pi(\tilde{P}_{c a_2} g, h)\}, \tag{3.6}
\end{align}
$$
and

$$ C(((a_1, a_2), a_3), g, h) := C((\tilde{P}_{a_1} a_2, a_3), g, h) - a_1 C((a_2, a_3), g, h), \quad (3.7) $$

also equal to

$$ \Pi(\tilde{P}^{\circ 3}(a_1, a_2, a_3, g), h) - \{(\tilde{P}^{\circ 2}(a_1, a_2, a_3))\Pi(g, h) + (\tilde{P}_{a_1} a_2)\Pi(\tilde{P}_{\mathscr{C}a_3} g, h) + a_1\Pi(\tilde{P}_{\mathscr{C}\tilde{P}_{\mathscr{C}a_2}a_3} g, h)\}. $$

The conditions $(\cdots) < 3$ that appear in Theorem 8 below are purely technical; a choice of implicit constant $b$ in the definition of the paraproduct operator $P = P^{(b)}$ would change the bound 3 for any other bound. In any concrete situation, one can assume that such a good choice of parameter $b$ has been done and forget about that condition.

**THEOREM 8.** Let $\alpha_1, \alpha_2, \mu$ be regularity exponents in $(0, 1)$, and $\alpha_3 \in (-3, 3)$. Let $\nu \in (-\infty, 3]$ be another regularity exponent.

* Assume $(\alpha_1 + \alpha_2 + \mu) < 3$, and

$$ \begin{gathered} \alpha_2 + \mu + \nu < 0, \quad \alpha_1 + \mu + \nu < 0 \\ (\alpha_1 + \alpha_2 + \mu + \nu) \in (0, 1). \end{gathered} $$

Then the 4-linear lower iterated corrector

$$ \begin{gathered} C^{\alpha_1} \times C^{\alpha_2} \times C^{\mu} \times C^{\nu} \rightarrow C^{\alpha_1+\alpha_2+\mu+\nu} \\ (a_1, a_2, g, h) \mapsto C((a_1, a_2), g, h) \end{gathered} $$

is continuous.

* Assume that $(\alpha_1 + \alpha_2 + \alpha_3 + \mu) < 3$, and

$$ \begin{gathered} (\alpha_3 + \mu + \nu) < 0, \quad \alpha_2 + \alpha_3 + \mu + \nu < 0 \\ (\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu) \in (0, 1). \end{gathered} $$

The 5-linear lower iterated corrector

$$ \begin{gathered} C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\mu} \times C^{\nu} \rightarrow C^{\alpha_1+\alpha_2+\alpha_3+\mu+\nu} \\ (a_1, a_2, a_3, g, h) \mapsto C(((a_1, a_2), a_3), g, h) \end{gathered} $$

is continuous.
and

$$ C(((a_1, a_2), a_3), g, h) := C((\tilde{P}_{a_1} a_2, a_3), g, h) - a_1 C((a_2, a_3), g, h), \quad (3.7) $$

also equal to

$$ \Pi(\tilde{P}^{\circ 3}(a_1, a_2, a_3, g), h) - \{ (\tilde{P}^{\circ 2}(a_1, a_2, a_3))\Pi(g, h) + (\tilde{P}_{a_1} a_2)\Pi(\tilde{P}_{\mathscr{C}a_3} g, h) + a_1\Pi(\tilde{P}_{\mathscr{C}\tilde{P}_{\mathscr{C}a_2} a_3} g, h) \}. $$

The conditions $(\cdots) < 3$ that appear in Theorem 8 below are purely technical; a choice of implicit constant $b$ in the definition of the paraproduct operator $P = P^{(b)}$ would change the bound 3 for any other bound. In any concrete situation, one can assume that such a good choice of parameter $b$ has been done and forget about that condition.

**THEOREM 8.** Let $\alpha_1, \alpha_2, \mu$ be regularity exponents in $(0, 1)$, and $\alpha_3 \in (-3, 3)$. Let $\nu \in (-\infty, 3]$ be another regularity exponent.

* Assume $(\alpha_1 + \alpha_2 + \mu) < 3$, and

$$ \begin{gathered} \alpha_2 + \mu + \nu < 0, \quad \alpha_1 + \mu + \nu < 0 \\ (\alpha_1 + \alpha_2 + \mu + \nu) \in (0, 1). \end{gathered} $$

Then the 4-linear lower iterated corrector

$$ C^{\alpha_1} \times C^{\alpha_2} \times C^{\mu} \times C^{\nu} \rightarrow C^{\alpha_1+\alpha_2+\mu+\nu} \\ (a_1, a_2, g, h) \mapsto C((a_1, a_2), g, h) $$

is continuous.

* Assume that $(\alpha_1 + \alpha_2 + \alpha_3 + \mu) < 3$, and

$$ \begin{gathered} (\alpha_3 + \mu + \nu) < 0, \quad \alpha_2 + \alpha_3 + \mu + \nu < 0 \\ (\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu) \in (0, 1). \end{gathered} $$

The 5-linear lower iterated corrector

$$ C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\mu} \times C^{\nu} \rightarrow C^{\alpha_1+\alpha_2+\alpha_3+\mu+\nu} \\ (a_1, a_2, a_3, g, h) \mapsto C(((a_1, a_2), a_3), g, h) $$

is continuous.
*Proof.* To get a clear idea of the mechanics at play, we only prove here the analogue statement of the time-independent setting of the flat torus. That means that we aim to prove that the formula

$$ \Pi^0((\Pi^0)^{\circ3}(a_1, a_2, a_3, g), h) - \{(\Pi^0)^{\circ2}(a_1, a_2, a_3)\Pi^0(g, h) + (\Pi_{a_1}^0 a_2)\Pi^0(\Pi_{\mathscr{C}a_3}^0 g, h) + a_1\Pi^0(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_2}^0 a_3}^0 g, h)\} $$

defines a continuous map from $C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\mu} \times C^{\nu}$ to $C^{\alpha_1+\alpha_2+\alpha_3+\mu+\nu}$, under the above conditions on the regularity exponents. We then let the reader to complete and adapt the proof in the full parabolic setting.

To see how the second term in the expansion arises, use formula (3.4) for the corrector and write

$$ \{\Pi^0((\Pi^0)^{\circ3}(a_1, a_2, a_3, g), h) - (\Pi^0)^{\circ2}(a_1, a_2, a_3)\Pi^0(g, h)\}(x) \\ = C^0((\Pi^0)^{\circ2}(a_1, a_2, a_3), g; h)(x) \\ = \Pi^0(\Pi_{\mathscr{C}(\Pi^0)^{\circ2}(a_1, a_2, a_3)}^0 g, h)(x). $$

Note that since

$$ \Pi_{a_1}^0 a_2 = (\Pi_{a_1}^0 a_2)(x) + \mathscr{C} \Pi_{a_1}^0 a_2, $$

we have the identity

$$ \mathscr{C}(\Pi^0)^{\circ2}(a_1, a_2, a_3) = (\Pi_{a_1}^0 a_2)(x)\mathscr{C}a_3 + \mathscr{C}\Pi_{\mathscr{C}\Pi_{a_1}^0 a_2}^0 a_3. $$

It follows that $\Pi^0((\Pi^0)^{\circ3}(a_1, a_2, a_3, g), h)$ is equal to

$$ (\Pi^0)^{\circ2}(a_1, a_2, a_3)\Pi^0(g, h) + (\Pi_{a_1}^0 a_2)\Pi^0(\Pi_{\mathscr{C}a_3}^0 g, h) + \Pi^0(\Pi_{\mathscr{C}\Pi_{a_1}^0 a_2}^0 g, h). $$

Writing $a_1 = a_1(x) + \mathscr{C}a_1$, in the above expression for the remainder yields the formula

$$ C(a_1, a_2, a_3, g, h) = \Pi^0(\Pi_{\mathscr{C}\Pi_{a_1}^0 a_2}^0 g, h). \qquad (3.8) $$

The fact that it defines a $(\alpha_1 + \alpha_2 + \alpha_3 + \mu + \nu)$-Hölder function if this exponent is positive can be seen as follows. For every $x$ we have

$$ \begin{align*} \Pi^0(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_1}^0 a_2}^0 g, h)(x) &\approx \sum_k \Delta_k(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_1}^0 a_2}^0 g)(x)\Delta_k(h)(x) \\ &\approx \sum_k S_{k-2}(\mathscr{C}\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_2}^0 a_3}^0)(x)\Delta_k(g)(x)\Delta_k(h)(x) \\ &\approx \sum_k \Delta_{k-2}(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_1}^0 a_2}^0 g)(x)\Delta_k(g)(x)\Delta_k(h)(x), \end{align*} $$
*Proof.* To get a clear idea of the mechanics at play, we only prove here the analogue statement of the time-independent setting of the flat torus. That means that we aim to prove that the formula

$$ \Pi^0((\Pi^0)^{\circ3}(a_1, a_2, a_3, g), h) - \{(\Pi^0)^{\circ2}(a_1, a_2, a_3)\Pi^0(g, h) + (\Pi_{a_1}^0 a_2)\Pi^0(\Pi_{\mathscr{C}a_3}^0 g, h) + a_1\Pi^0(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_2}^0 a_3}^0 g, h)\} $$

defines a continuous map from $C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\mu} \times C^{\nu}$ to $C^{\alpha_1+\alpha_2+\alpha_3+\mu+\nu}$, under the above conditions on the regularity exponents. We then let the reader to complete and adapt the proof in the full parabolic setting.

To see how the second term in the expansion arises, use formula (3.4) for the corrector and write

$$ \{\Pi^0((\Pi^0)^{\circ3}(a_1, a_2, a_3, g), h) - (\Pi^0)^{\circ2}(a_1, a_2, a_3)\Pi^0(g, h)\}(x) \\ = C^0((\Pi^0)^{\circ2}(a_1, a_2, a_3), g; h)(x) \\ = \Pi^0(\Pi_{\mathscr{C}(\Pi^0)^{\circ2}(a_1, a_2, a_3)}^0 g, h)(x). $$

Note that since

$$ \Pi_{a_1}^0 a_2 = (\Pi_{a_1}^0 a_2)(x) + \mathscr{C} \Pi_{a_1}^0 a_2, $$

we have the identity

$$ \mathscr{C}(\Pi^0)^{\circ2}(a_1, a_2, a_3) = (\Pi_{a_1}^0 a_2)(x) \mathscr{C} a_3 + \mathscr{C} \Pi_{\mathscr{C}\Pi_{a_1}^0 a_2}^0 a_3. $$

It follows that $\Pi^0((\Pi^0)^{\circ3}(a_1, a_2, a_3, g), h)$ is equal to

$$ (\Pi^0)^{\circ2}(a_1, a_2, a_3)\Pi^0(g, h) + (\Pi_{a_1}^0 a_2)\Pi^0(\Pi_{\mathscr{C}a_3}^0 g, h) + \Pi^0(\Pi_{\mathscr{C}\Pi_{a_1}^0 a_2}^0 g, h). $$

Writing $a_1 = a_1(x) + \mathscr{C}a_1$, in the above expression for the remainder yields the formula

$$ C(a_1, a_2, a_3, g, h) = \Pi^0(\Pi_{\mathscr{C}\Pi_{a_1}^0 a_2}^0 g, h). \quad (3.8) $$

The fact that it defines a $(\alpha_1 + \alpha_2 + \alpha_3 + \mu + \nu)$-Hölder function if this exponent is positive can be seen as follows. For every $x$ we have

$$ \begin{align*} \Pi^0(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_1}^0 a_2}^0 g, h)(x) &\approx \sum_k \Delta_k(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_1}^0 a_2}^0 g)(x)\Delta_k(h)(x) \\ &\approx \sum_k S_{k-2}(\mathscr{C}\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_2}^0 a_3}^0)(x)\Delta_k(g)(x)\Delta_k(h)(x) \\ &\approx \sum_k \Delta_{k-2}(\Pi_{\mathscr{C}\Pi_{\mathscr{C}a_1}^0 a_2}^0 g)(x)\Delta_k(g)(x)\Delta_k(h)(x), \end{align*} $$
where we used (3.5). Iterating the reasoning, we get

$$ \Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}^0 a_2}^0}^0 a_3 g, h)(x) \\ \approx \sum_k \Delta_{k-6}(a_1)(x) \Delta_{k-4}(a_2)(x) \Delta_{k-2}(a_3)(x) \Delta_k(g)(x) \Delta_k(h)(x) \quad (3.9) $$

and so since $(\alpha_1 + \alpha_2 + \alpha_3 + \mu + \nu)$ is nonnegative, and setting

$$ m := \|a_1\|_{C^{\alpha_1}} \|a_2\|_{C^{\alpha_2}} \|a_3\|_{C^{\beta}} \|g\|_{C^{\alpha_3}} \|h\|_{C^{\nu}}, $$

we conclude that

$$ \begin{aligned} |\Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}^0 a_2}^0}^0 a_3 g, h)(x)| & \lesssim m \sum_k 2^{-k(\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu)} \\ & \lesssim m, \end{aligned} $$

uniformly in $x$, which yields that the main quantity defines a bounded function. Using (3.9), we can also obtain its Hölder character. For $x \neq y$, we have

$$ \begin{aligned} |\Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}^0 a_2}^0}^0 a_3 g, h)(x) - \Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}^0 a_2}^0}^0 a_3 g, h)(y)| \\ & \lesssim \sum_k |\Delta_{k-6}(a_1)(x)\Delta_{k-4}(a_2)(x)\Delta_{k-2}(a_3)(x)\Delta_k(g)(x)\Delta_k(h)(x) \\ & \qquad - \Delta_{k-6}(a_1)(y)\Delta_{k-4}(a_2)(y)\Delta_{k-2}(a_3)(y)\Delta_k(g)(y)\Delta_k(h)(y)| \\ & \lesssim m \left( \sum_{1 \le 2^k |x-y|} 2^{-k(\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu)} + \sum_{1 \ge 2^k |x-y|} |x-y| 2^{k-k(\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu)} \right) \\ & \lesssim m |x-y|^{\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu}; \end{aligned} $$

in the second sum, over $1 \ge 2^k |x - y|$, we have used the finite increment theorem together with the fact that differentiating one operator $\Delta_k$ is equivalent to multiplying it by $2^k$, together with the condition $(\alpha_1 + \alpha_2 + \alpha_3 + \mu + \nu) \in (0, 1)$. $\square$

It is also necessary and possible to 'expand' the corrector $C$ simultaneously on its first two arguments. Define

$$ C((a_1, a_2), (b_1, b_2), h) := C((a_1, a_2), P_{b_1} b_2, h) - P_{b_1} C((a_1, a_2), b_2, h). $$

We let the reader write this operator in terms of the outer centring operator $\mathcal{C}$, like in identity (3.8).
where we used (3.5). Iterating the reasoning, we get

$$ \Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}a_2}^0}^0 a_3 g, h)(x) \\ \approx \sum_k \Delta_{k-6}(a_1)(x)\Delta_{k-4}(a_2)(x)\Delta_{k-2}(a_3)(x)\Delta_k(g)(x)\Delta_k(h)(x) \quad (3.9) $$

and so since $(\alpha_1 + \alpha_2 + \alpha_3 + \mu + \nu)$ is nonnegative, and setting

$$ m := \|a_1\|^{C\alpha_1} \|a_2\|^{C\alpha_2} \|a_3\|^{C\beta} \|g\|^{C\alpha_3} \|h\|^{C\nu}, $$

we conclude that

$$ \begin{aligned} |\Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}a_2}^0}^0 a_3 g, h)(x)| & \lesssim m \sum_k 2^{-k(\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu)} \\ & \lesssim m, \end{aligned} $$

uniformly in $x$, which yields that the main quantity defines a bounded function. Using (3.9), we can also obtain its Hölder character. For $x \neq y$, we have

$$ \begin{aligned} |\Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}a_2}^0}^0 a_3 g, h)(x) - \Pi^0(\Pi_{\mathcal{C}\Pi_{\mathcal{C}\Pi_{\mathcal{C}a_1}a_2}^0}^0 a_3 g, h)(y)| \\ & \lesssim \sum_k |\Delta_{k-6}(a_1)(x)\Delta_{k-4}(a_2)(x)\Delta_{k-2}(a_3)(x)\Delta_k(g)(x)\Delta_k(h)(x) \\ & \qquad - \Delta_{k-6}(a_1)(y)\Delta_{k-4}(a_2)(y)\Delta_{k-2}(a_3)(y)\Delta_k(g)(y)\Delta_k(h)(y)| \\ & \lesssim m \left( \sum_{1 \le 2^k |x-y|} 2^{-k(\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu)} + \sum_{1 \ge 2^k |x-y|} |x-y| 2^{-k(\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu)} \right) \\ & \lesssim m |x-y|^{\alpha_1 + \alpha_2 + \beta + \alpha_3 + \nu}; \end{aligned} $$

in the second sum, over $1 \ge 2^k |x-y|$, we have used the finite increment theorem together with the fact that differentiating one operator $\Delta_k$ is equivalent to multiplying it by $2^k$, together with the condition $(\alpha_1 + \alpha_2 + \alpha_3 + \mu + \nu) \in (0, 1)$. $\square$

It is also necessary and possible to 'expand' the corrector $C$ simultaneously on its first two arguments. Define

$$ C((a_1, a_2), (b_1, b_2), h) := C((a_1, a_2), P_{b_1} b_2, h) - P_{b_1} C((a_1, a_2), b_2, h). $$

We let the reader write this operator in terms of the outer centring operator $\mathcal{C}$, like in identity (3.8).
**PROPOSITION 9.** Assume that $(\alpha_1 + \alpha_2 + \mu_1 + \mu_2) < 3$, and

$$(\alpha_2 + \mu_2 + \nu) < 0, \quad \alpha_1 + \mu_2 + \nu < 0$$

$$(\alpha_1 + \alpha_2 + \mu_1 + \mu_2 + \nu) \in (0, 1) \quad \text{and} \quad \mu_2 \in (0, 1).$$

Then the 5-linear iterated corrector

$$C^{\alpha_1} \times C^{\alpha_2} \times C^{\mu_1} \times C^{\mu_2} \times C^{\nu} \rightarrow C^{\alpha_1+\alpha_2+\mu_1+\mu_2+\nu} \\ (a_1, a_2, b_1, b_2, h) \mapsto C((a_1, a_2), (b_1, b_2), h)$$

is continuous.

*Proof.* As in the proof of Theorem 8, observe that (in the flat torus setting) we have

$$\mathcal{C}((a_1, a_2), \mathbf{P}_{b_1} b_2, h) \simeq \sum_k \Delta_{k-4}(a_1) \Delta_{k-2}(a_2) \Delta_k(\mathbf{P}_{b_1} b_2) \Delta_k(h) \\ \simeq \sum_k \Delta_{k-4}(a_1) \Delta_{k-2}(a_2) S_{k-2}(b_1) \Delta_k(b_2) \Delta_k(h).$$

At the same time,

$$\mathbf{P}_{b_1} \mathcal{C}((a_1, a_2), b_2, h) = \sum_k S_{k-2}(b_1) \Delta_k(\mathcal{C}((a_1, a_2), b_2, h)) \\ = \sum_k S_{k-2}(b_1) \Delta_k \left( \sum_j \Delta_{j-4}(a_1) \Delta_{j-2}(a_2) \Delta_j(b_2) \Delta_j(h) \right).$$

In the previous sum, the parameters $k, j$ have to be equivalent in order to have a nonvanishing contribution. With the normalization $\sum \Delta_k = \text{Id}$, we obtain

$$(\star) := \mathcal{C}((a_1, a_2), \mathbf{P}_{b_1} b_2, h)(x) - \mathbf{P}_{b_1} \mathcal{C}((a_1, a_2), b_2, h)(x) \\ \simeq \sum_{k,j} \Delta_k(\Delta_{j-4}(a_1)\Delta_{j-2}(a_2)(S_{j-2}(b_1) - S_{k-2}(b_1)(x))\Delta_j(b_2)\Delta_j(h))(x).$$

From this decomposition, it is easy to check that for $\ell \ge 0$ and because $\alpha_1 + \alpha_2 + \mu_1 + \mu_2 + \nu \in (0, 1)$, and $\mu_1 \in (0, 1)$, we have

$$\|\Delta_{\ell}(\star)\|_{L^{\infty}} \lesssim \sum_{k,j \gtrsim \ell} 2^{-j(\alpha_1+\alpha_2+\mu_2+\nu)} (2^{-j} + 2^{-k})^{\mu_1} m \\ \lesssim 2^{-\ell(\alpha_1+\alpha_2+\mu_1+\mu_2+\nu)} m,$$

with

$$m := \|a_1\|_{C^{\alpha_1}} \|a_2\|_{C^{\alpha_2}} \|b_1\|_{C^{\mu_1}} \|b_2\|_{C^{\mu_2}} \|h\|_{C^{\nu}}.$$

□
**PROPOSITION 9.** Assume that $(\alpha_1 + \alpha_2 + \mu_1 + \mu_2) < 3$, and

$$ (\alpha_2 + \mu_2 + \nu) < 0, \quad \alpha_1 + \mu_2 + \nu < 0 $$

$$ (\alpha_1 + \alpha_2 + \mu_1 + \mu_2 + \nu) \in (0, 1) \quad \text{and} \quad \mu_2 \in (0, 1). $$

Then the 5-linear iterated corrector

$$ C^{\alpha_1} \times C^{\alpha_2} \times C^{\mu_1} \times C^{\mu_2} \times C^v \rightarrow C^{\alpha_1+\alpha_2+\mu_1+\mu_2+v} \\ (a_1, a_2, b_1, b_2, h) \mapsto C((a_1, a_2), (b_1, b_2), h) $$

is continuous.

*Proof.* As in the proof of Theorem 8, observe that (in the flat torus setting) we have

$$ \mathcal{C}((a_1, a_2), \mathbf{P}_{b_1} b_2, h) \approx \sum_k \Delta_{k-4}(a_1) \Delta_{k-2}(a_2) \Delta_k(\mathbf{P}_{b_1} b_2) \Delta_k(h) \\ \approx \sum_k \Delta_{k-4}(a_1) \Delta_{k-2}(a_2) S_{k-2}(b_1) \Delta_k(b_2) \Delta_k(h). $$

At the same time,

$$ \mathbf{P}_{b_1} \mathcal{C}((a_1, a_2), b_2, h) = \sum_k S_{k-2}(b_1) \Delta_k(\mathcal{C}((a_1, a_2), b_2, h)) \\ = \sum_k S_{k-2}(b_1) \Delta_k \left( \sum_j \Delta_{j-4}(a_1) \Delta_{j-2}(a_2) \Delta_j(b_2) \Delta_j(h) \right). $$

In the previous sum, the parameters $k, j$ have to be equivalent in order to have a nonvanishing contribution. With the normalization $\sum \Delta_k = \text{Id}$, we obtain

$$ (\star) := \mathcal{C}((a_1, a_2), \mathbf{P}_{b_1} b_2, h)(x) - \mathbf{P}_{b_1} \mathcal{C}((a_1, a_2), b_2, h)(x) \\ \approx \sum_{k,j} \Delta_k(\Delta_{j-4}(a_1)\Delta_{j-2}(a_2)(S_{j-2}(b_1) - S_{k-2}(b_1)(x))\Delta_j(b_2)\Delta_j(h))(x). $$

From this decomposition, it is easy to check that for $\ell \ge 0$ and because $\alpha_1 + \alpha_2 + \mu_1 + \mu_2 + \nu \in (0, 1)$, and $\mu_1 \in (0, 1)$, we have

$$ \|\Delta_{\ell}(\star)\|_{L^{\infty}} \lesssim \sum_{k,j \gtrsim \ell} 2^{-j(\alpha_1+\alpha_2+\mu_2+\nu)} (2^{-j} + 2^{-k})^{\mu_1} m \\ \lesssim 2^{-\ell(\alpha_1+\alpha_2+\mu_1+\mu_2+\nu)} m, $$

with

$$ m := \|a_1\|_{C^{\alpha_1}} \|a_2\|_{C^{\alpha_2}} \|b_1\|_{C^{\mu_1}} \|b_2\|_{C^{\mu_2}} \|h\|_{C^v}. $$

□
The 5-linear iterated corrector will never appear explicitly in our computations as it will provide a remainder term. The 4- and 5-linear **upper iterated correctors** are defined by the formulae

$$C(f, (a_1, a_2), h) := C(f, \tilde{P}_{a_1} a_2, h) - a_1 C(f, a_2, h)$$

and

$$C(f, (a_1, (a_2, a_3)), h) := C(f, (a_1, \tilde{P}_{a_2} a_3), h) - a_2 C(f, (a_1, a_3), h).$$

**THEOREM 10.** (i) Assume $\mu, \alpha_1 \in (0, 1)$, and the exponents $(\mu + \alpha_1 + \nu)$ and $(\mu + \alpha_2 + \nu)$ are negative, and

$$(\mu + \alpha_1 + \alpha_2 + \nu) \in (0, 1).$$

Then the 4-linear upper iterated corrector

$$C^{\mu} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\nu} \rightarrow C^{\mu+\alpha_1+\alpha_2+\nu} \\ (f, a_1, a_2, h) \mapsto C(f, (a_1, a_2), h)$$

is continuous.

(ii) Assume $\mu, \alpha_1, \alpha_2 \in (0, 1)$, and the exponents $(\mu + \alpha_i + \nu)$ are all negative, for $1 \le i \le 3$, and

$$(\mu + \alpha_1 + \alpha_2 + \alpha_3 + \nu) \in (0, 1).$$

Then the 5-linear upper iterated corrector

$$C^{\mu} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\nu} \rightarrow C^{\mu+\alpha_1+\alpha_2+\alpha_3+\nu} \\ (f, a_1, a_2, a_3, h) \mapsto C(f, (a_1, (a_2, a_3)), h)$$

is continuous.

*Proof.* We only sketch the proof of the continuity result of the 4-linear operator in the model case of the time-independent setting of the flat torus, and rely on formula (3.2) for the diagonal operator $\Pi^0(\cdot, \cdot)$ for that purpose. See Proposition 44 in Appendix C.2 for a fully detailed proof in the parabolic setting. In the present setting, the quantity $C^0(f; a_1, a_2; g)$ is then given by a sum of the form

$$C^0(f, (a_1, a_2), h) = \sum_{|i-j| \le 1} \varepsilon_i' \Delta_j h,$$
The 5-linear iterated corrector will never appear explicitly in our computations as it will provide a remainder term. The 4- and 5-linear **upper iterated correctors** are defined by the formulae

$$C(f, (a_1, a_2), h) := C(f, \tilde{P}_{a_1} a_2, h) - a_1 C(f, a_2, h)$$

and

$$C(f, (a_1, (a_2, a_3)), h) := C(f, (a_1, \tilde{P}_{a_2} a_3), h) - a_2 C(f, (a_1, a_3), h).$$

**THEOREM 10.** (i) Assume $\mu, \alpha_1 \in (0, 1)$, and the exponents $(\mu + \alpha_1 + \nu)$ and $(\mu + \alpha_2 + \nu)$ are negative, and

$$(\mu + \alpha_1 + \alpha_2 + \nu) \in (0, 1).$$

Then the 4-linear upper iterated corrector

$$C^{\mu} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\nu} \rightarrow C^{\mu+\alpha_1+\alpha_2+\nu} \\ (f, a_1, a_2, h) \mapsto C(f, (a_1, a_2), h)$$

is continuous.

(ii) Assume $\mu, \alpha_1, \alpha_2 \in (0, 1)$, and the exponents $(\mu + \alpha_i + \nu)$ are all negative, for $1 \le i \le 3$, and

$$(\mu + \alpha_1 + \alpha_2 + \alpha_3 + \nu) \in (0, 1).$$

Then the 5-linear upper iterated corrector

$$C^{\mu} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\nu} \rightarrow C^{\mu+\alpha_1+\alpha_2+\alpha_3+\nu} \\ (f, a_1, a_2, a_3, h) \mapsto C(f, (a_1, (a_2, a_3)), h)$$

is continuous.

*Proof.* We only sketch the proof of the continuity result of the 4-linear operator in the model case of the time-independent setting of the flat torus, and rely on formula (3.2) for the diagonal operator $\Pi^0(\cdot, \cdot)$ for that purpose. See Proposition 44 in Appendix C.2 for a fully detailed proof in the parabolic setting. In the present setting, the quantity $C^0(f; a_1, a_2; g)$ is then given by a sum of the form

$$C^0(f, (a_1, a_2), h) = \sum_{|i-j| \le 1} \varepsilon_i' \Delta_j h,$$
with

$$ \varepsilon_i' := \{ \Delta_i(\mathbf{P}_f(\mathbf{P}_{a_1}a_2)) - a_1\Delta_i(\mathbf{P}_f a_2) \} + f\{a_1\Delta_i a_2 - \Delta_i(\mathbf{P}_{a_1}a_2) \}. $$

We read on the expression

$$ \begin{aligned} \varepsilon_i'(x) &= \int K_i(x, y) \{ \mathbf{P}_f(\mathbf{P}_{a_1}a_2)(y) - a_1(x)(\mathbf{P}_f a_2)(y) \\ &\quad + (fa_1)(x)a_2(y) - f(x)(\mathbf{P}_{a_1}a_2)(y) \} dy \\ &= \int K_i(x, y) \mathbf{P}_{f-f(x)1}(\mathbf{P}_{a_1-a_1(x)1}a_2)(y) dy, \end{aligned} $$

that

$$ \varepsilon_i' = \Delta_i(\mathbf{P}_{\mathscr{C}_f}(\mathbf{P}_{\mathscr{C}_{a_1}}a_2)) $$

has $L^\infty$-norm of order $2^{-i(\mu+\alpha_1+\alpha_2)}$, as a consequence of (3.5). The proof is then not fully completed, since the block $\varepsilon_i'\Delta_i h$ is not perfectly localized in frequency at scale $2^i$, so an extra decomposition is necessary. The same thing happens in the proof of Theorem 5. We do not give the details here and refer the reader to the proof of Proposition 44 in Appendix C. □

3.1.4. **Commutator D.** Define on the space $L^\infty$ of bounded measurable functions on the parabolic space $\mathcal{M}$ the **commutator D** as the operator

$$ \mathbf{D}(f, g, h) := \Pi(\tilde{\mathbf{P}}_f g, h) - \mathbf{P}_f(\Pi(g, h)), $$

**THEOREM 11.** (a) For positive regularity exponents $\alpha, \beta$ and $\gamma$, the commutator $\mathbf{D}$ is continuous from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$.

(b) The commutator $\mathbf{D}$ is bounded from $C^\alpha \times C^\beta \times C^\gamma$ into $C^{\alpha+\beta+\gamma}$ for $\alpha \in (-1, 0)$ as soon as $\alpha + \beta + \gamma > 0$.

*Proof.* Assume we are working in the time-independent setting of the $d$-dimensional torus, with the operator

$$ \mathbf{D}^0(f, g, h) := \Pi^0(\Pi_f^0 g, h) - \Pi_f^0(\Pi^0(g, h)). $$

(a) We refer the reader to Proposition 40, in Appendix C.1, for a full proof of the regularity statement for the commutator $\mathbf{D}$. We simply mention here that in the
with

$$ \varepsilon'_i := \{ \Delta_i(\mathbf{P}_f(\mathbf{P}_{a_1}a_2)) - a_1\Delta_i(\mathbf{P}_f a_2) \} + f\{a_1\Delta_i a_2 - \Delta_i(\mathbf{P}_{a_1}a_2) \}. $$

We read on the expression

$$ \begin{aligned} \varepsilon'_{i}(x) &= \int K_{i}(x, y) \{ \mathbf{P}_{f}(\mathbf{P}_{a_{1}}a_{2})(y) - a_{1}(x)(\mathbf{P}_{f}a_{2})(y) \\ &\quad + (fa_{1})(x)a_{2}(y) - f(x)(\mathbf{P}_{a_{1}}a_{2})(y) \} dy \\ &= \int K_{i}(x, y) \mathbf{P}_{f-f(x)1}(\mathbf{P}_{a_{1}-a_{1}(x)1}a_{2})(y) dy, \end{aligned} $$

that

$$ \varepsilon'_{i} = \Delta_{i}(\mathbf{P}_{\mathscr{C}_{f}}(\mathbf{P}_{\mathscr{C}_{a_{1}}a_{2}})) $$

has $L^\infty$-norm of order $2^{-i(\mu+\alpha_1+\alpha_2)}$, as a consequence of (3.5). The proof is then not fully completed, since the block $\varepsilon'_i \Delta_i h$ is not perfectly localized in frequency at scale $2^i$, so an extra decomposition is necessary. The same thing happens in the proof of Theorem 5. We do not give the details here and refer the reader to the proof of Proposition 44 in Appendix C. □

**3.1.4. Commutator D.** Define on the space $L^\infty$ of bounded measurable functions on the parabolic space $\mathcal{M}$ the **commutator** D as the operator

$$ \mathbf{D}(f, g, h) := \Pi(\tilde{\mathbf{P}}_f g, h) - \mathbf{P}_f(\Pi(g, h)), $$

**THEOREM 11.** (a) For positive regularity exponents $\alpha, \beta$ and $\gamma$, the commutator D is continuous from $C^\alpha \times C^\beta \times C^\gamma$ to $C^{\alpha+\beta+\gamma}$.

(b) The commutator D is bounded from $C^\alpha \times C^\beta \times C^\gamma$ into $C^{\alpha+\beta+\gamma}$ for $\alpha \in (-1, 0)$ as soon as $\alpha + \beta + \gamma > 0$.

*Proof.* Assume we are working in the time-independent setting of the $d$-dimensional torus, with the operator

$$ \mathbf{D}^0(f, g, h) := \Pi^0(\Pi_f^0 g, h) - \Pi_f^0(\Pi^0(g, h)). $$

(a) We refer the reader to Proposition 40, in Appendix C.1, for a full proof of the regularity statement for the commutator D. We simply mention here that in the
special case of $D^0$, the regularity estimate comes from the following identity

$$ 
\begin{aligned}
\Delta_k(D^0(f, g, h)) &= \sum_{\ell \ge k-2} \Delta_k(\Delta_\ell(g)S_{\ell-2}(f)\Delta_\ell(h)) - S_{k-2}(f)\Delta_k(\Delta_\ell(g)\Delta_\ell(h)) \\
&\approx \sum_{\ell \ge k-2} \Delta_k(\Delta_\ell(g)S_{\ell-2}(f)\Delta_\ell(h) - S_{k-2}(f)\Delta_\ell(g)\Delta_\ell(h)).
\end{aligned}
\quad (3.10) $$

(b) This case is easy and do not use the ‘difference’ structure in the commutator. Indeed since $\alpha < 0$ then $\alpha + \beta + \gamma > 0$ implies $\beta + \gamma > 0$ and so by using the boundedness of the paraproducts and those of the resonant part, it directly comes

$$ \Pi(\tilde{P}_f g, h) \in C^{\alpha+\beta+\gamma} \quad \text{and} \quad P_f(\Pi(g, h)) \in C^{\alpha+\beta+\gamma}. \quad \square $$

**3.1.5. Back to the high order paracontrolled expansion (2.3).** The high order paracontrolled expansion (2.3) might seem far from a possibly expected ordinary Taylor-type expansion, such as it appears for instance in regularity structures [19]. The difference is not that big, as we see in this section by looking at the resonant term $\Pi(f(u), \zeta)$. This intermezzo has a formal character. Recall we denote by $C^\circ$, $D^\circ$ the corrector, commutator and their iterates, built from the resonant operator and the usual paraproduct $\Pi$, $P$ instead of the modified paraproduct $\tilde{P}$; they enjoy the same continuity properties as $C$ and $D$. We consider in this section the case where $u \in C^\alpha$, with $\frac{2}{5} < \alpha < \frac{1}{2}$, and for a function $g \in C_b^3$, we write the second order version of formula (2.3), giving the paracontrolled expansion of $g(u)$, under the form

$$ g(u) := P_{g'(u)}u + \frac{1}{2}\{P_{g^{(2)}(u)}u^2 - P_{g^{(2)}(u)}u\} + g(u)^{(1)}; $$

the exponent (1) referring to the fact that $g(u)^{(1)} \in C^{3\alpha} \subset C^1$, so it can be differentiated in the space direction. We work in our general parabolic setting over a Riemannian manifold so the refined corrector involves the additional term $\sum_{\ell=1}^{\ell_0} \gamma_\ell(V_\ell f) \Pi_{(1)}^\ell(g, h)$, from Definition 37. The reader can think of the time-independent flat torus setting, where the additional term in the definition of the refined corrector is simply $f'\Pi_{(1)}^0(g, h)$, with ' denoting space derivative.

**Notation.** Given any $\beta \in \mathbb{R}$, denote by $(\beta)$ an element of $C^\beta$ whose precise definition is unimportant for the reasoning, and whose only noticeable property is its regularity. Its definition may change from line to line.

Pick now a function $f \in C_b^4$. Writing

$$ u^2 = 2P_u u + \Pi(u, u), $$

$$ u^3 = P_{u^2}u + 2P_u(P_u u) + P_u(\Pi(u, u)) + 2\Pi(u, P_u u) + \Pi(u, \Pi(u, u)), $$
special case of $D^0$, the regularity estimate comes from the following identity

$$ 
\begin{aligned}
\Delta_k(D^0(f, g, h)) &= \sum_{\ell \ge k-2} \Delta_k(\Delta_\ell(g)S_{\ell-2}(f)\Delta_\ell(h)) - S_{k-2}(f)\Delta_k(\Delta_\ell(g)\Delta_\ell(h)) \\
&\approx \sum_{\ell \ge k-2} \Delta_k(\Delta_\ell(g)S_{\ell-2}(f)\Delta_\ell(h) - S_{k-2}(f)\Delta_\ell(g)\Delta_\ell(h)).
\end{aligned}
\quad (3.10) $$

(b) This case is easy and do not use the 'difference' structure in the commutator. Indeed since $\alpha < 0$ then $\alpha + \beta + \gamma > 0$ implies $\beta + \gamma > 0$ and so by using the boundedness of the paraproducts and those of the resonant part, it directly comes

$$ \Pi(\tilde{P}_f g, h) \in C^{\alpha+\beta+\gamma} \quad \text{and} \quad P_f(\Pi(g, h)) \in C^{\alpha+\beta+\gamma}. \quad \square $$

**3.1.5. Back to the high order paracontrolled expansion (2.3).** The high order paracontrolled expansion (2.3) might seem far from a possibly expected ordinary Taylor-type expansion, such as it appears for instance in regularity structures [19]. The difference is not that big, as we see in this section by looking at the resonant term $\Pi(f(u), \zeta)$. This intermezzo has a formal character. Recall we denote by $C^\circ$, $D^\circ$ the corrector, commutator and their iterates, built from the resonant operator and the usual paraproduct $\Pi$, $P$ instead of the modified paraproduct $\tilde{P}$; they enjoy the same continuity properties as $C$ and $D$. We consider in this section the case where $u \in C^\alpha$, with $\frac{2}{5} < \alpha < \frac{1}{2}$, and for a function $g \in C_b^3$, we write the second order version of formula (2.3), giving the paracontrolled expansion of $g(u)$, under the form

$$ g(u) := P_{g'(u)}u + \frac{1}{2}\{P_{g^{(2)}(u)}u^2 - P_{g^{(2)}(u)}u\} + g(u)^{(1)}; $$

the exponent (1) referring to the fact that $g(u)^{(1)} \in C^{3\alpha} \subset C^1$, so it can be differentiated in the space direction. We work in our general parabolic setting over a Riemannian manifold so the refined corrector involves the additional term $\sum_{\ell=1}^{\ell_0} \gamma_\ell(V_\ell f) \Pi_{(1)}^\ell(g, h)$, from Definition 37. The reader can think of the time-independent flat torus setting, where the additional term in the definition of the refined corrector is simply $f'\Pi_{(1)}^0(g, h)$, with ' denoting space derivative.

**Notation.** Given any $\beta \in \mathbb{R}$, denote by $(\beta)$ an element of $C^\beta$ whose precise definition is unimportant for the reasoning, and whose only noticeable property is its regularity. Its definition may change from line to line.

Pick now a function $f \in C_b^4$. Writing

$$ u^2 = 2P_u u + \Pi(u, u), $$

$$ u^3 = P_{u^2}u + 2P_u(P_u u) + P_u(\Pi(u, u)) + 2\Pi(u, P_u u) + \Pi(u, \Pi(u, u)), $$
the third order paracontrolled expansion formula (2.3) writes

$$f(u) = P_{a_1}u + P_{a_2}P_uu + P_{a_3}(P_u(P_uu)) + P_{a_4}(P_u^2u) \\ + P_{b_1}\Pi(u, u) + P_{b_2}(P_u\Pi(u, u)) + P_{b_3}(\Pi(u, P_uu)) \\ + P_c(\Pi(u, \Pi(u, u))) + f(u)^{\sharp},$$

where $f(u)^{\sharp} \in C^{4\alpha}$, and

$$a_1 := f'(u) - uf^{(2)}(u) + \frac{1}{2}u^2 f^{(3)}(u),$$

$$a_2 := f^{(2)}(u) - uf^{(3)}(u),$$

$$a_3 := \frac{1}{3} f^{(3)}(u),$$

$$a_4 := \frac{1}{6} f^{(3)}(u),$$

$$b_1 := \frac{1}{2}(f^{(2)}(u) - f^{(3)}(u)),$$

$$b_2 := \frac{1}{6} f^{(3)}(u),$$

$$b_3 := \frac{1}{3} f^{(3)}(u),$$

$$c := \frac{1}{6} f^{(3)}(u).$$

Plugging this formula inside $\Pi(f(u), \zeta)$ gives for it the expression

$$\begin{aligned} & a_1 \Pi(u, \zeta) + C^\circ(a_1, u, \zeta) \\ & + a_2 u \Pi(u, \zeta) + a_2 C^\circ(u, u, \zeta) + u C^\circ(a_2, u, \zeta) + C^\circ(a_2, u, u, \zeta) \\ & + a_3 u^2 \Pi(u, \zeta) + 2a_3 u C^\circ(u, u, \zeta) + u^2 C^\circ(a_3, u, \zeta) \\ & + 2u C^\circ(a_3, u, u, \zeta) + a_3 C^\circ(u, u, u, \zeta) \\ & + a_4 u^2 \Pi(u, \zeta) + a_4 C^\circ(u^2, u, \zeta) + u^2 C^\circ(a_4, u, \zeta) + C^\circ(a_4, u^2, u, \zeta) \\ & + b_1 \Pi(\Pi(u, u), \zeta) + C^\circ(b_1, \Pi(u, u), \zeta) \\ & + b_2 u \Pi(\Pi(u, u), \zeta) + b_2 C^\circ(u, \Pi(u, u), \zeta) + u C^\circ(b_2, \Pi(u, u), \zeta) \\ & + b_3 u \Pi(\Pi(u, u), \zeta) \\ & + b_3 C^\circ(u, \Pi(u, u), \zeta) + b_3 \Pi(D^\circ(u, u, u), \zeta) + u C^\circ(b_3, \Pi(u, u), \zeta) \\ & + c \Pi(\Pi(u, \Pi(u, u)), \zeta) + (5\alpha - 2). \end{aligned}$$

To get this expression in terms only of primitive quantities involving ill-defined terms where $u$ appears rather than a function of $u$, use the identity

$$\begin{aligned} C^\circ(a_i(u), u, \zeta) = & a_i'(u) C^\circ(u, u, \zeta) + a_i^{(2)}(u) C^\circ((u, u), u, \zeta) \\ & + \frac{1}{2} a_i^{(2)}(u) C^\circ(\Pi(u, u), u, \zeta) \\ & + (5\alpha - 2) + C^\circ(a_i(u)^{(1)}, u, \zeta) \end{aligned}$$
the third order paracontrolled expansion formula (2.3) writes

$$f(u) = P_{a_1}u + P_{a_2}P_uu + P_{a_3}(P_u(P_uu)) + P_{a_4}(P_u^2u) \\ + P_{b_1}\Pi(u, u) + P_{b_2}(P_u\Pi(u, u)) + P_{b_3}(\Pi(u, P_uu)) \\ + P_c(\Pi(u, \Pi(u, u))) + f(u)^{\sharp},$$

where $f(u)^{\sharp} \in C^{4\alpha}$, and

$$a_1 := f'(u) - uf^{(2)}(u) + \frac{1}{2}u^2 f^{(3)}(u),$$

$$a_2 := f^{(2)}(u) - uf^{(3)}(u),$$

$$a_3 := \frac{1}{3} f^{(3)}(u),$$

$$a_4 := \frac{1}{6} f^{(3)}(u),$$

$$b_1 := \frac{1}{2}(f^{(2)}(u) - f^{(3)}(u)),$$

$$b_2 := \frac{1}{6} f^{(3)}(u),$$

$$b_3 := \frac{1}{3} f^{(3)}(u),$$

$$c := \frac{1}{6} f^{(3)}(u).$$

Plugging this formula inside $\Pi(f(u), \zeta)$ gives for it the expression

$$\begin{aligned} & a_1 \Pi(u, \zeta) + C^\circ(a_1, u, \zeta) \\ & + a_2 u \Pi(u, \zeta) + a_2 C^\circ(u, u, \zeta) + u C^\circ(a_2, u, \zeta) + C^\circ(a_2, u, u, \zeta) \\ & + a_3 u^2 \Pi(u, \zeta) + 2a_3 u C^\circ(u, u, \zeta) + u^2 C^\circ(a_3, u, \zeta) \\ & + 2u C^\circ(a_3, u, u, \zeta) + a_3 C^\circ(u, u, u, \zeta) \\ & + a_4 u^2 \Pi(u, \zeta) + a_4 C^\circ(u^2, u, \zeta) + u^2 C^\circ(a_4, u, \zeta) + C^\circ(a_4, u^2, u, \zeta) \\ & + b_1 \Pi(\Pi(u, u), \zeta) + C^\circ(b_1, \Pi(u, u), \zeta) \\ & + b_2 u \Pi(\Pi(u, u), \zeta) + b_2 C^\circ(u, \Pi(u, u), \zeta) + u C^\circ(b_2, \Pi(u, u), \zeta) \\ & + b_3 u \Pi(\Pi(u, u), \zeta) \\ & + b_3 C^\circ(u, \Pi(u, u), \zeta) + b_3 \Pi(D^\circ(u, u, u), \zeta) + u C^\circ(b_3, \Pi(u, u), \zeta) \\ & + c \Pi(\Pi(u, \Pi(u, u)), \zeta) + (5\alpha - 2). \end{aligned}$$

To get this expression in terms only of primitive quantities involving ill-defined terms where $u$ appears rather than a function of $u$, use the identity

$$\begin{aligned} C^\circ(a_i(u), u, \zeta) = & a_i'(u) C^\circ(u, u, \zeta) + a_i^{(2)}(u) C^\circ((u, u), u, \zeta) \\ & + \frac{1}{2} a_i^{(2)}(u) C^\circ(\Pi(u, u), u, \zeta) \\ & + (5\alpha - 2) + C^\circ(a_i(u)^{(1)}, u, \zeta) \end{aligned}$$
$$ = a'_{i}(u) \mathbf{C}^{\circ}(u, u, \zeta) + a_{i}^{(2)}(u) \mathbf{C}^{\circ}((u, u), u, \zeta) \\ + \frac{1}{2} a_{i}^{(2)}(u) \mathbf{C}^{\circ}(\Pi(u, u), u, \zeta) \\ + (5\alpha - 2) + \sum_{\ell=1}^{\ell_0} \gamma_{\ell} V_{\ell}(a_{i}(u))^{(1)} \Pi_{(1)}^{\ell}(u, \zeta). $$

We expand $\mathbf{C}^{\circ}(u^{2}, u, \zeta)$ using the identity $u^{2} = 2\mathbf{P}_{u}u + \Pi(u, u)$ and the iterated corrector. After simplification, one gets the following multiplicative decomposition for $\Pi(f(u), \zeta)$

$$ \begin{aligned} & f'(u) \Pi(u, \zeta) + f^{(2)}(u) \left\{ \mathbf{C}^{\circ}(u, u, \zeta) + \frac{1}{2} \Pi(\Pi(u, u), \zeta) \right\} \\ & + f^{(3)}(u) \left\{ \frac{1}{3} \mathbf{C}^{\circ}(u, (u, u), \zeta) + \frac{1}{3} \mathbf{C}^{\circ}((u, u), u, \zeta) \right. \\ & \qquad + \frac{1}{6} \mathbf{C}^{\circ}(u, \Pi(u, u), \zeta) + \frac{1}{6} \mathbf{C}^{\circ}(\Pi(u, u), u, \zeta) \\ & \qquad \left. + \frac{1}{3} \Pi(\mathbf{D}^{\circ}(u, u, u), \zeta) + \frac{1}{6} \Pi(\Pi(u, \Pi(u, u)), \zeta) \right\} \\ & + \sum_{\ell=1}^{\ell_0} (\star)_{\ell} \Pi_{(1)}^{\ell}(u, \zeta) + (5\alpha - 2), \end{aligned} \quad (3.11) $$

with

$$ (\star)_{\ell} := \gamma_{\ell} V_{\ell}(a_{1}(u))^{(1)} + u \gamma_{\ell} V_{\ell}(a_{2}(u))^{(1)} + u^{2} \gamma_{\ell} V_{\ell}(a_{3}(u))^{(1)} + a_{4}(u))^{(1)} \in \mathcal{C}^{3\alpha-1}. \quad (3.12) $$

In the time-independent setting of the flat torus, this reduces to

$$ \sum_{\ell=1}^{\ell_0} (\star)_{\ell} \Pi_{(1)}^{\ell}(u, \zeta) \\ = \{(a_1(u))^{(1)}' + u(a_2(u))^{(1)}' + u^2(a_3(u))^{(1)} + a_4(u))^{(1)}'\} \Pi_{(1)}(u, \zeta). $$

The term $\Pi_{(1)}^{\ell}(u, \zeta)$ has formal regularity $2\alpha - 1$, slightly less than 0, and the product with the above bracket term $\{\cdot\}$ is well-posed provided $\Pi_{(1)}^{\ell}(u, \zeta)$ can be given meaning as an element of regularity $2\alpha - 1$.

Write each expression $\Pi(u^k, \zeta)$ in multiplicative form and assign to a product the regularity of its term of lowest regularity. For an expression $B$ in multiplicative form, we can then denote by $\lfloor B \rfloor_{k\alpha-2}$ the part of $B$ that is of formal regularity $(k\alpha - 2)$. So we have for instance

$$ \lfloor \Pi(u, \zeta) \rfloor_{2\alpha-2} = \Pi(u, \zeta), $$
and

$$ \Pi(\mathbf{P}_u u, \zeta) = u\Pi(u, \zeta) + \mathbf{C}^{\circ}(u, u, \zeta), $$

with

$$ \lfloor \Pi(\mathbf{P}_u u, \zeta) \rfloor_{2\alpha-2} = u\Pi(u, \zeta) $$

and

$$ \lfloor \Pi(\mathbf{P}_u u, \zeta) \rfloor_{3\alpha-2} = \mathbf{C}^{\circ}(u, u, \zeta). $$

In those terms, formula (3.11) for $\Pi(f(u), \zeta)$ takes the following Taylor-type form

$$ \begin{aligned} \Pi(f(u), \zeta) = f'(u)\lfloor\Pi(u, \zeta)\rfloor_{2\alpha-2} &+ f^{(2)}(u)\lfloor\Pi\left(\frac{u^2}{2!}, \zeta\right)\rfloor_{3\alpha-2} \\ &+ f^{(3)}(u)\lfloor\Pi\left(\frac{u^3}{3!}, \zeta\right)\rfloor_{4\alpha-2} \\ &+ \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(V_{\ell}(a_i(u)^{(1)}))\Pi_{(1)}^{\ell}(u, \zeta) + (5\alpha - 2). \end{aligned} \quad (3.13) $$

**3.1.6. Iterated commutators.** In addition to the above continuity properties for the iterated correctors, we also need the following continuity result on iterated commutator operators D.

**PROPOSITION 12.** • Given positive regularity exponents $\alpha_1, \alpha_2, \gamma, \delta$, the formulae

$$ \begin{aligned} & C^{\alpha_1} \times C^{\alpha_2} \times C^{\gamma} \times C^{\delta} \rightarrow C^{\alpha_1+\alpha_2+\gamma+\delta} \\ & (a_1, a_2, g, h) \mapsto D(\tilde{P}_{a_1} a_2, g, h) - P_{a_1} D(a_2, g, h), \end{aligned} $$

for the lower iterated commutator, and

$$ \begin{aligned} & C^{\gamma} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\delta} \rightarrow C^{\gamma+\alpha_1+\alpha_2+\delta} \\ & (f, a_1, a_2, h) \mapsto D(f, \tilde{P}_{a_1} a_2, h) - P_{a_1} D(f, a_2, h), \end{aligned} $$

for the upper iterated commutator, define continuous operators. The result also holds true if $\alpha_1 \in (0, 1)$ and $-1 < \alpha_2 < 0$, with $\alpha_2 + \gamma + \delta > 0$.

• Fix $\alpha, \gamma, \delta_1 > 0$ and $\delta_2 \in (0, 1)$. The high frequency commutator

$$ \begin{aligned} & C^{\alpha} \times C^{\gamma} \times C^{\delta_1} \times C^{\delta_2} \rightarrow C^{\alpha+\gamma+\delta_1+\delta_2} \\ & (f, g, h_1, h_2) \mapsto D(f, g, \tilde{P}_{h_1} h_2) - P_{h_1} D(f, g, h_2), \end{aligned} $$

is bounded. This continuity result also holds true if $\alpha \in (-1, 0)$, provided $\alpha + \gamma + \delta_2 > 0$.
and

$$ \Pi(\mathbf{P}_u u, \zeta) = u\Pi(u, \zeta) + \mathbf{C}^{\circ}(u, u, \zeta), $$

with

$$ \lfloor \Pi(\mathbf{P}_u u, \zeta) \rfloor_{2\alpha-2} = u\Pi(u, \zeta) $$

and

$$ \lfloor \Pi(\mathbf{P}_u u, \zeta) \rfloor_{3\alpha-2} = \mathbf{C}^{\circ}(u, u, \zeta). $$

In those terms, formula (3.11) for $\Pi(f(u), \zeta)$ takes the following Taylor-type form

$$ \begin{aligned} \Pi(f(u), \zeta) = f'(u)\lfloor\Pi(u, \zeta)\rfloor_{2\alpha-2} &+ f^{(2)}(u)\lfloor\Pi\left(\frac{u^2}{2!}, \zeta\right)\rfloor_{3\alpha-2} \\ &+ f^{(3)}(u)\lfloor\Pi\left(\frac{u^3}{3!}, \zeta\right)\rfloor_{4\alpha-2} \\ &+ \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(V_{\ell}(a_i(u)^{(1)}))\Pi_{(1)}^{\ell}(u, \zeta) + (5\alpha - 2). \end{aligned} \quad (3.13) $$

**3.1.6. Iterated commutators.** In addition to the above continuity properties for the iterated correctors, we also need the following continuity result on iterated commutator operators $\mathbf{D}$.

**PROPOSITION 12.** • Given positive regularity exponents $\alpha_1, \alpha_2, \gamma, \delta$, the formulae

$$ \begin{aligned} & C^{\alpha_1} \times C^{\alpha_2} \times C^{\gamma} \times C^{\delta} \rightarrow C^{\alpha_1+\alpha_2+\gamma+\delta} \\ & (a_1, a_2, g, h) \mapsto \mathbf{D}(\tilde{\mathbf{P}}_{a_1} a_2, g, h) - \mathbf{P}_{a_1} \mathbf{D}(a_2, g, h), \end{aligned} $$

for the lower iterated commutator, and

$$ \begin{aligned} & C^{\gamma} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\delta} \rightarrow C^{\gamma+\alpha_1+\alpha_2+\delta} \\ & (f, a_1, a_2, h) \mapsto \mathbf{D}(f, \tilde{\mathbf{P}}_{a_1} a_2, h) - \mathbf{P}_{a_1} \mathbf{D}(f, a_2, h), \end{aligned} $$

for the upper iterated commutator, define continuous operators. The result also holds true if $\alpha_1 \in (0, 1)$ and $-1 < \alpha_2 < 0$, with $\alpha_2 + \gamma + \delta > 0$.

• Fix $\alpha, \gamma, \delta_1 > 0$ and $\delta_2 \in (0, 1)$. The high frequency commutator

$$ \begin{aligned} & C^{\alpha} \times C^{\gamma} \times C^{\delta_1} \times C^{\delta_2} \rightarrow C^{\alpha+\gamma+\delta_1+\delta_2} \\ & (f, g, h_1, h_2) \mapsto \mathbf{D}(f, g, \tilde{\mathbf{P}}_{h_1} h_2) - \mathbf{P}_{h_1} \mathbf{D}(f, g, h_2), \end{aligned} $$

is bounded. This continuity result also holds true if $\alpha \in (-1, 0)$, provided $\alpha + \gamma + \delta_2 > 0$.
*Proof.* As in the proof of Theorem 5, we analyse in the present proof what happens in the time-independent setting of the *d*-dimensional torus, in the case where we also use $Π^0$ instead of $\tilde{P}$. So we set

$$D^0(a_1, a_2, g, h) := D^0(\Pi_{a_1}^0 a_2, g, h) - \Pi_{a_1}^0 D^0(a_2, g, h)$$

and have a look at its continuity properties on the spacial Hölder spaces. Using formula (3.10), it follows that we have

$$\begin{aligned} \Delta_k(D^0(a_1, a_2, g, h)) &\approx \Delta_k(D^0(P_{a_1}a_2, g, h)) - S_{k-2}(a_1) \Delta_k(D^0(a_2, g, h)) \\ &\approx \sum_{\ell \ge k-2} \Delta_k\{\Delta_\ell(g)\Delta_\ell(h)(S_{\ell-2}(P_{a_1}a_2) - S_{k-2}(P_{a_1}a_2) \\ &\qquad - S_{k-2}(a_1)(S_{\ell-2}a_2 - S_{k-2}a_2))\}. \end{aligned}$$

The quantity inside the brackets is equal to

$$\begin{aligned} &S_{\ell-2}(P_{a_1}a_2) - S_{k-2}(P_{a_1}a_2) - S_{k-2}(a_1)(S_{\ell-2}(a_2) - S_{k-2}(a_2)) \\ &= \sum_{j=k-1}^{\ell-2} \Delta_j(P_{a_1}a_2) - S_{k-2}(a_1)\Delta_j(a_2) \\ &\approx \sum_{j=k-1}^{\ell-2} S_{j-2}(a_1)\Delta_j(a_2) - S_{k-2}(a_1)\Delta_j(a_2) \\ &\approx \sum_{j=k-1}^{\ell-2} (S_{j-2}a_1 - S_{k-2}a_1)\Delta_j(a_2), \end{aligned}$$

which is bounded in $L^\infty$ by

$$\sum_{j=k+1}^{\ell} 2^{-k\alpha} \|a_1\|_{C^{a_1}} 2^{-j\beta} \|a_2\|_{C^{a_2}} \lesssim 2^{-k(\alpha_1+\alpha_2)} \|a_1\|_{C^{a_1}} \|a_2\|_{C^{a_2}}.$$

This estimate allows us to conclude that

$$\|\Delta_k(D^0(a_1, a_2, g, h))\|_{\infty} \lesssim 2^{-k(\alpha_1+\alpha_2+\gamma+\delta)} \|a_1\|_{C^{a_1}} \|a_2\|_{C^{a_2}} \|g\|_{C^\gamma} \|h\|_{C^\delta},$$

uniformly in $k$, which proves the continuity result for the 4-linear operator $D^0$. A very similar proof gives the continuity of the simplified version of the upper iterated commutator; we leave the details to the reader.

For the second statement of the first item of the proposition, with $\alpha_2 \in (-1, 0)$ we follow the same computations and since we have now

$$\sum_{j=k-1}^{\ell-2} 2^{-k\alpha} \|a_1\|_{C^{a_1}} 2^{-j\alpha_2} \|a_2\|_{C^{a_2}} \lesssim 2^{-k\alpha_1 - \ell\alpha_2} \|a_1\|_{C^{a_1}} \|a_2\|_{C^{a_2}},$$
*Proof.* As in the proof of Theorem 5, we analyse in the present proof what happens in the time-independent setting of the *d*-dimensional torus, in the case where we also use $\Pi^0$ instead of $\tilde{\mathbb{P}}$. So we set

$$ \mathbb{D}^0(a_1, a_2, g, h) := \mathbb{D}^0(\Pi_{a_1}^0 a_2, g, h) - \Pi_{a_1}^0 \mathbb{D}^0(a_2, g, h) $$

and have a look at its continuity properties on the spacial Hölder spaces. Using formula (3.10), it follows that we have

$$ \begin{aligned} \Delta_k(\mathbb{D}^0(a_1, a_2, g, h)) &\approx \Delta_k(\mathbb{D}^0(\mathbb{P}_{a_1} a_2, g, h)) - S_{k-2}(a_1) \Delta_k(\mathbb{D}^0(a_2, g, h)) \\ &\approx \sum_{\ell \ge k-2} \Delta_k\{\Delta_\ell(g) \Delta_\ell(h)(S_{\ell-2}(\mathbb{P}_{a_1} a_2) - S_{k-2}(\mathbb{P}_{a_1} a_2) \\ &\qquad - S_{k-2}(a_1)(S_{\ell-2}a_2 - S_{k-2}a_2))\}. \end{aligned} $$

The quantity inside the brackets is equal to

$$ \begin{aligned} &S_{\ell-2}(\mathbb{P}_{a_1} a_2) - S_{k-2}(\mathbb{P}_{a_1} a_2) - S_{k-2}(a_1)(S_{\ell-2}(a_2) - S_{k-2}(a_2)) \\ &= \sum_{j=k-1}^{\ell-2} \Delta_j(\mathbb{P}_{a_1} a_2) - S_{k-2}(a_1)\Delta_j(a_2) \\ &\approx \sum_{j=k-1}^{\ell-2} S_{j-2}(a_1)\Delta_j(a_2) - S_{k-2}(a_1)\Delta_j(a_2) \\ &\approx \sum_{j=k-1}^{\ell-2} (S_{j-2}a_1 - S_{k-2}a_1)\Delta_j(a_2), \end{aligned} $$

which is bounded in $L^\infty$ by

$$ \sum_{j=k+1}^{\ell} 2^{-k\alpha} \|a_1\|_{C^{a_1}} 2^{-j\beta} \|a_2\|_{C^{a_2}} \lesssim 2^{-k(\alpha_1+\alpha_2)} \|a_1\|_{C^{a_1}} \|a_2\|_{C^{a_2}}. $$

This estimate allows us to conclude that

$$ \|\Delta_k(\mathbb{D}^0(a_1, a_2, g, h))\|_{\infty} \lesssim 2^{-k(\alpha_1+\alpha_2+\gamma+\delta)} \|a_1\|_{C^{a_1}} \|a_2\|_{C^{a_2}} \|g\|_{C^\gamma} \|h\|_{C^\delta}, $$

uniformly in $k$, which proves the continuity result for the 4-linear operator $\mathbb{D}^0$. A very similar proof gives the continuity of the simplified version of the upper iterated commutator; we leave the details to the reader.

For the second statement of the first item of the proposition, with $\alpha_2 \in (-1, 0)$ we follow the same computations and since we have now

$$ \sum_{j=k-1}^{\ell-2} 2^{-k\alpha} \|a_1\|_{C^{a_1}} 2^{-j\alpha_2} \|a_2\|_{C^{a_2}} \lesssim 2^{-k\alpha_1 - \ell\alpha_2} \|a_1\|_{C^{a_1}} \|a_2\|_{C^{a_2}}, $$
then

$$ \begin{aligned} \| \Delta_k (D^0 (a_1, a_2, g, h)) \|_{\infty} &\lesssim \sum_{\ell \ge k} 2^{-k\alpha_1} 2^{-\ell(\alpha_2 + \gamma + \delta)} \|a_1|C^{\alpha_1}\|a_2\|C^{\alpha_2}\|g\|_{C^\gamma}\|h\|_{C^\delta} \\ &\lesssim 2^{-\ell(\alpha + \beta + \gamma + \delta)} \|a_1|C^{\alpha_1}\|a_2\|C^{\alpha_2}\|g\|_{C^\gamma}\|h\|_{C^\delta}, \end{aligned} $$

since $\beta + \gamma + \delta > 0$.

For the second item of the proposition, the same reasoning can be applied by observing that now

$$ \begin{aligned} & \Delta_k [D(f, g, \tilde{P}_{h_1} h_2) - P_{h_1} D(f, g, h_2)] \\ & \approx \sum_{\ell \ge k} \Delta_k [\Delta_j(g)(S_{\ell-2}f - S_{k-2}f) \Delta_{\ell}h_2(S_{\ell-2}h_1 - S_{k-2}h_1)]; \end{aligned} $$

we conclude by using the regularity of the four functions. □

**3.2. Iterated paraproducts.** The operators $C$ and $D$ introduced in Section 3 are used to analyse ill-defined products. The operators $\mathbb{R}$ and $\mathbb{S}$ that we introduce in this section are used to write down the different terms that appear from using the $C$ and $D$ operators in the paraproduct form required to apply the fixed point strategy for the analysis of Equation (1.1) – Step (3) in Section 1.1. As a motivating example, let us set ourselves the task of writing the paraproduct $P_{\zeta}u$ under the form $\sum_{i=1}^4 P_{v_i} Y_j + (5\alpha - 2)$, for some $v_j \in C^\alpha$, some $Y_j$ that depend only on the $Z_i$ and the noise, and a remainder $(5\alpha - 2)$ of regularity $5\alpha - 2$, assuming if necessary that the $u_i$ also have a paracontrolled expansion up to some $u_i$-dependent order. We repeatedly use for that purpose the $F$-type decomposition (1.6) on the paraproduct map and its iterates. We use here the sign $\sim$ for the equality of functions, up to some term of positive regularity. We have

$$ P_{\zeta} u \sim \sum_{i=1}^{3} P_{\zeta} P_{u_i} Z_i, $$

and setting

$$ S^{\circ}(\zeta, u_i, Z_i) := P_{\zeta} P_{u_i} Z_i - P_{u_i} P_{\zeta} Z_i, $$

we have

$$ P_{\zeta} u \sim \sum_{i=1}^{3} (P_{u_i} P_{\zeta} Z_i + S^{\circ}(\zeta, u_i, Z_i)). $$
then

$$ \begin{aligned} \| \Delta_k (D^0 (a_1, a_2, g, h)) \|_{\infty} & \lesssim \sum_{\ell \ge k} 2^{-k\alpha_1} 2^{-\ell(\alpha_2 + \gamma + \delta)} \|a_1|C^{\alpha_1}\|a_2\|C^{\alpha_2}\|g\|_{C^\gamma}\|h\|_{C^\delta} \\ & \lesssim 2^{-\ell(\alpha + \beta + \gamma + \delta)} \|a_1|C^{\alpha_1}\|a_2\|C^{\alpha_2}\|g\|_{C^\gamma}\|h\|_{C^\delta}, \end{aligned} $$

since $\beta + \gamma + \delta > 0$.

For the second item of the proposition, the same reasoning can be applied by observing that now

$$ \begin{aligned} & \Delta_k [D(f, g, \tilde{P}_{h_1} h_2) - P_{h_1} D(f, g, h_2)] \\ & \approx \sum_{\ell \ge k} \Delta_k [\Delta_j(g)(S_{\ell-2}f - S_{k-2}f) \Delta_{\ell}h_2(S_{\ell-2}h_1 - S_{k-2}h_1)]; \end{aligned} $$

we conclude by using the regularity of the four functions. □

**3.2. Iterated paraproducts.** The operators $C$ and $D$ introduced in Section 3 are used to analyse ill-defined products. The operators $\mathbb{R}$ and $\mathbb{S}$ that we introduce in this section are used to write down the different terms that appear from using the $C$ and $D$ operators in the paraproduct form required to apply the fixed point strategy for the analysis of Equation (1.1) – Step (3) in Section 1.1. As a motivating example, let us set ourselves the task of writing the paraproduct $P_{\zeta}u$ under the form $\sum_{i=1}^4 P_{v_i} Y_j + (5\alpha - 2)$, for some $v_j \in C^\alpha$, some $Y_j$ that depend only on the $Z_i$ and the noise, and a remainder $(5\alpha - 2)$ of regularity $5\alpha - 2$, assuming if necessary that the $u_i$ also have a paracontrolled expansion up to some $u_i$-dependent order. We repeatedly use for that purpose the $F$-type decomposition (1.6) on the paraproduct map and its iterates. We use here the sign $\sim$ for the equality of functions, up to some term of positive regularity. We have

$$ P_{\zeta} u \sim \sum_{i=1}^{3} P_{\zeta} P_{u_i} Z_i, $$

and setting

$$ S^{\circ}(\zeta, u_i, Z_i) := P_{\zeta} P_{u_i} Z_i - P_{u_i} P_{\zeta} Z_i, $$

we have

$$ P_{\zeta} u \sim \sum_{i=1}^{3} (P_{u_i} P_{\zeta} Z_i + S^{\circ}(\zeta, u_i, Z_i)). $$
Theorem 13 below shows that the ‘swap’ operator $S^°$ sends continuously $C^{\alpha_1} \times C^{\alpha_2}$ into $C^{\alpha_2+\alpha_1+\alpha_2}$. So we have

$$P_{\zeta} u \sim \sum_{i=1}^{3} P_{u_i} P_{\zeta} Z_i + S^{\circ}(\zeta, u_1, Z_1) + S^{\circ}(\zeta, u_2, Z_2).$$

The first three terms on the right hand side have a good form. To analyse the two $S^{\circ}(\zeta, \cdot, \cdot)$ terms, we do an F-type expansion on each of them, taking profit from the paracontrolled expansion

$$u_1 = \sum_{j=1}^{2} P_{u_{1j}} Z_j + (3\alpha), \quad u_2 = P_{u_{21}} Z_1 + (2\alpha),$$

of $u_1$ and $u_2$. The map

$$S_{\zeta}^{\circ}((a_1, a_2), b) := S^{\circ}(\zeta, P_{a_1} a_2, b) - P_{a_1} S^{\circ}(\zeta, a_2, b)$$

happens indeed to send $C^{\alpha_2-2} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\beta}$ into $C^{\alpha_2+\alpha_1+\alpha_2+\beta}$ continuously – Theorem 13, so we have

$$P_{\zeta} u \sim \sum_{i=1}^{3} P_{u_i} (P_{\zeta} Z_i) + \sum_{j=1}^{2} P_{u_{1j}} S^{\circ}(\zeta, Z_j, Z_1) + P_{u_{21}} S^{\circ}(\zeta, Z_1, Z_2) + S^{\circ}(\zeta, u_{11}, Z_1, Z_1). \quad (3.14)$$

A further F-type expansion on the last term in the above right hand side does the job.

New things happen when we look at the paraproduct $P_{\zeta} u^2$, as we have to deal with a term

$$P_{\zeta} P_{u} u = P_{u} P_{\zeta} u + S^{\circ}(\zeta, u, u).$$

Using (3.14), we end up with terms of the form $P_u P_{u_a} Y_a$. A similar thing happens in the analysis of $S^{\circ}(\zeta, u, u)$ and $P_{\zeta} \Pi(u, u)$. To deal with $P_u P_{u_a} Y_a$, we use the *merging operator* $R^°$

$$P_u P_{u_a} Y_a := P_{uu_a} Y_a + R^\circ(u, u_a, Y_a),$$

and prove some continuity results and some expansion property of $R^°$ with respect to its first two arguments.

In this section, we define and state continuity results for the swap and merging operators $S$ and $R$ defined below. We prove here some of the results in the model setting of the time-independent flat torus and refer the reader to Appendix C.2 for the description of how things work in the parabolic setting.
Theorem 13 below shows that the ‘swap’ operator $S^°$ sends continuously $C^{\alpha_1} \times C^{\alpha_2}$ into $C^{\alpha_2+\alpha_1+\alpha_2}$. So we have

$$P_{\zeta} u \sim \sum_{i=1}^{3} P_{u_i} P_{\zeta} Z_i + S^{\circ}(\zeta, u_1, Z_1) + S^{\circ}(\zeta, u_2, Z_2).$$

The first three terms on the right hand side have a good form. To analyse the two $S^{\circ}(\zeta, \cdot, \cdot)$ terms, we do an F-type expansion on each of them, taking profit from the paracontrolled expansion

$$u_1 = \sum_{j=1}^{2} P_{u_{1j}} Z_j + (3\alpha), \quad u_2 = P_{u_{21}} Z_1 + (2\alpha),$$

of $u_1$ and $u_2$. The map

$$S_{\zeta}^{\circ}((a_1, a_2), b) := S^{\circ}(\zeta, P_{a_1} a_2, b) - P_{a_1} S^{\circ}(\zeta, a_2, b)$$

happens indeed to send $C^{\alpha_2-2} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\beta}$ into $C^{\alpha_2-2+\alpha_1+\alpha_2+\beta}$ continuously – Theorem 13, so we have

$$P_{\zeta} u \sim \sum_{i=1}^{3} P_{u_i} (P_{\zeta} Z_i) + \sum_{j=1}^{2} P_{u_{1j}} S^{\circ}(\zeta, Z_j, Z_1) \\ + P_{u_{21}} S^{\circ}(\zeta, Z_1, Z_2) + S^{\circ}(\zeta, u_{11}, Z_1, Z_1). \qquad (3.14)$$

A further F-type expansion on the last term in the above right hand side does the job.

New things happen when we look at the paraproduct $P_{\zeta} u^2$, as we have to deal with a term

$$P_{\zeta} P_{u} u = P_{u} P_{\zeta} u + S^{\circ}(\zeta, u, u).$$

Using (3.14), we end up with terms of the form $P_u P_{u_a} Y_a$. A similar thing happens in the analysis of $S^{\circ}(\zeta, u, u)$ and $P_{\zeta} \Pi(u, u)$. To deal with $P_u P_{u_a} Y_a$, we use the *merging operator* $R^{\circ}$

$$P_u P_{u_a} Y_a := P_{uu_a} Y_a + R^{\circ}(u, u_a, Y_a),$$

and prove some continuity results and some expansion property of $R^{\circ}$ with respect to its first two arguments.

In this section, we define and state continuity results for the swap and merging operators $S$ and $R$ defined below. We prove here some of the results in the model setting of the time-independent flat torus and refer the reader to Appendix C.2 for the description of how things work in the parabolic setting.
### 3.2.1. Swap operator S.
The result stated below in Theorem 13 is fully proved in Appendix C.2 – see Propositions 38 and 42. Given Hölder distributions $f, g_1, g_2, g_3, g, h$, we define the **modified commutator on paraproducts**, and its iterates, by the formulae

$$S(f, g, h) := P_f(\tilde{P}_g h) - P_g(P_f h),$$

and

$$S(f, (g_1, g_2), h) := S(f, \tilde{P}_{g_1} g_2, h) - P_{g_1}(S(f, g_2, h)) \quad (3.15)$$

and

$$S(f, ((g_1, g_2), g_3), h) := S(f, (\tilde{P}_{g_1} g_2, g_3), h) - P_{g_1}(S(f, (g_2, g_3), h)). \quad (3.16)$$

The continuity properties of these operators are given in the following statement. The range $(-3, 3)$ in the next statement is purely technical and can be replaced by any other interval by an adequate choice of constant $b$ in the definition of the paraproduct $P = P^{(b)}$. Note that no gain of regularity comes from the first argument in the regularity statements below.

**THEOREM 13.** *Pick $\gamma \in \mathbb{R}$.*

(a) *Pick $\alpha \in (0, 1)$ and $\beta \in \mathbb{R}$ such that $\gamma + \alpha + \beta \in (-3, 3)$. The map*

$$C^{\gamma} \times C^{\alpha} \times C^{\beta} \rightarrow C^{(\gamma \wedge 0) + \alpha + \beta} \\ (f, g, h) \mapsto S(f, g, h)$$

is continuous.

(b) *Pick $\alpha_1, \alpha_2 \in (0, 1)$ and $\beta \in \mathbb{R}$ such that $\gamma + \alpha_1 + \alpha_2 + \beta \in (-3, 3)$. The map*

$$C^{\gamma} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\beta} \rightarrow C^{(\gamma \wedge 0) + \alpha_1 + \alpha_2 + \beta} \\ (f, g_1, g_2, h) \mapsto S(f, (g_1, g_2), h)$$

is continuous.

(c) *Pick $\alpha_1, \alpha_2, \alpha_3 \in (0, 1)$ and $\beta \in \mathbb{R}$ such that $\gamma + \alpha_1 + \alpha_2 + \alpha_3 + \beta \in (-3, 3)$. The map*

$$C^{\gamma} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\beta} \rightarrow C^{(\gamma \wedge 0) + \alpha_1 + \alpha_2 + \alpha_3 + \beta} \\ (f, g_1, g_2, g_3, h) \mapsto S(f, ((g_1, g_2), g_3), h)$$

is continuous.
### 3.2.1. Swap operator S.
The result stated below in Theorem 13 is fully proved in Appendix C.2 – see Propositions 38 and 42. Given Hölder distributions $f, g_1, g_2, g_3, g, h$, we define the **modified commutator on paraproducts**, and its iterates, by the formulae

$$S(f, g, h) := P_f(\tilde{P}_g h) - P_g(P_f h),$$

and

$$S(f, (g_1, g_2), h) := S(f, \tilde{P}_{g_1} g_2, h) - P_{g_1}(S(f, g_2, h)) \quad (3.15)$$

and

$$S(f, ((g_1, g_2), g_3), h) := S(f, (\tilde{P}_{g_1} g_2, g_3), h) - P_{g_1}(S(f, (g_2, g_3), h)). \quad (3.16)$$

The continuity properties of these operators are given in the following statement. The range $(-3, 3)$ in the next statement is purely technical and can be replaced by any other interval by an adequate choice of constant $b$ in the definition of the paraproduct $P = P^{(b)}$. Note that no gain of regularity comes from the first argument in the regularity statements below.

**THEOREM 13.** *Pick $\gamma \in \mathbb{R}$.*

(a) *Pick $\alpha \in (0, 1)$ and $\beta \in \mathbb{R}$ such that $\gamma + \alpha + \beta \in (-3, 3)$. The map*

$$C^{\gamma} \times C^{\alpha} \times C^{\beta} \rightarrow C^{(\gamma \wedge 0) + \alpha + \beta} \\ (f, g, h) \mapsto S(f, g, h)$$

*is continuous.*

(b) *Pick $\alpha_1, \alpha_2 \in (0, 1)$ and $\beta \in \mathbb{R}$ such that $\gamma + \alpha_1 + \alpha_2 + \beta \in (-3, 3)$. The map*

$$C^{\gamma} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\beta} \rightarrow C^{(\gamma \wedge 0) + \alpha_1 + \alpha_2 + \beta} \\ (f, g_1, g_2, h) \mapsto S(f, (g_1, g_2), h)$$

*is continuous.*

(c) *Pick $\alpha_1, \alpha_2, \alpha_3 \in (0, 1)$ and $\beta \in \mathbb{R}$ such that $\gamma + \alpha_1 + \alpha_2 + \alpha_3 + \beta \in (-3, 3)$. The map*

$$C^{\gamma} \times C^{\alpha_1} \times C^{\alpha_2} \times C^{\alpha_3} \times C^{\beta} \rightarrow C^{(\gamma \wedge 0) + \alpha_1 + \alpha_2 + \alpha_3 + \beta} \\ (f, g_1, g_2, g_3, h) \mapsto S(f, ((g_1, g_2), g_3), h)$$

*is continuous.*
### 3.2.2. Merging operator $\mathbb{R}$ and inner difference operator $\mathcal{D}$.
The value at $x \in \mathbb{T}^d$ of some paraproduct $P_u v$ is a sum over the integers $i$ of terms of the form

$$ (\Pi_u^{0,(i)} v)(x) := \iint K_i(x, y) K_{\le i-1}(x, z) u(z)v(y) dz dy. $$

We thus have for $f \in L^\infty$, $g \in C^\alpha$ with $\alpha \in (0, 1)$, and $h \in C^\nu$,

$$ \begin{align} (\Pi_f^{0,(i)}(\Pi_g^0 h) - \Pi_{fg}^{0,(i)}h)(x) & = \iint K_i(x, y) K_{\le i-1}(x, z) f(z)(\Pi_{g-g(z)}^0 h)(y) dz dy \nonumber \\ & =: \iint K_i(x, y) K_{\le i-1}(x, z) f(z)(\Pi_{\mathcal{D}g}^0 h)(y) dz dy. \tag{3.17} \end{align} $$

The above identity defines the **inner difference operator** $\mathcal{D}(=\mathcal{D}_z)$. In those terms, we have

$$ \mathbb{R}^0(f, g, h) := \Pi_f^0(\Pi_a^0 g) - \Pi_{fa}^0 g = \Pi_f^0(\Pi_{\mathcal{D}a}^0 g) $$

and, given the definition of the inner difference operator in the parabolic setting of Section C, we have more generally,

$$ \mathbb{R}(f, g, h) := \mathbb{P}_f(\tilde{\mathbb{P}}_g h) - \mathbb{P}_{fg}h = \mathbb{P}_f(\tilde{\mathbb{P}}_{\mathcal{D}g}h). \tag{3.18} $$

We use the same letter $\mathcal{D}$ in the spatial and parabolic settings. Compare this expression with the formal multiple integral, where we use the same letters to make it more striking,

$$ \int f(z) d\left(\int g dh\right)(z) = \int fg dh + \int f(z) d\left(\int (g-g(z)) dh\right)(z). $$

A similar reasoning as in the proof of continuity of $C(f, g, h) = \Pi(\mathbb{P}_{\mathcal{C}fg}, h)$, shows that $\mathbb{R}$ sends continuously $L^\infty \times C^\alpha \times C^\nu$ into $C^{\alpha+\nu}$, as soon as $\alpha \in (0, 1)$. A subtle thing happens here, though, as one has actually a refined continuity estimate on $\mathbb{R}^\circ$ that also takes into account the regularity of $f$ as well. It is related to the fact that $\mathbb{R}^\circ(\mathbf{1}, g, h) = \mathbb{R}^\circ(f, \mathbf{1}, h) = 0$, for all $f, g, h$, whereas $\mathbb{R}(\mathbf{1}, g, h) \neq 0$.

**PROPOSITION 14.** • For $\alpha, \beta \in [0, 1)$ and $\gamma \in (-3, 3)$, we have

$$ \|\mathbb{R}^\circ(f, g, h)\|_{C^{\alpha+\beta+\gamma}} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}. \tag{3.19} $$

• For $a_1, a_2, h \in C^\alpha$, one has

$$ \mathbb{R}(1, \mathbb{P}_{a_1} a_2, h) - \mathbb{P}_{a_1} \mathbb{R}(1, a_2, h) \in C^{3\alpha}. $$
### 3.2.2. Merging operator $\mathbb{R}$ and inner difference operator $\mathcal{D}$.
The value at $x \in \mathbb{T}^d$ of some paraproduct $P_u v$ is a sum over the integers $i$ of terms of the form

$$ (\Pi_u^{0,(i)} v)(x) := \iint K_i(x, y) K_{\le i-1}(x, z) u(z)v(y) dz dy. $$

We thus have for $f \in L^\infty$, $g \in C^\alpha$ with $\alpha \in (0, 1)$, and $h \in C^\nu$,

$$ \begin{align} (\Pi_f^{0,(i)}(\Pi_g^0 h) - \Pi_{fg}^{0,(i)}h)(x) & = \iint K_i(x, y) K_{\le i-1}(x, z) f(z)(\Pi_{g-g(z)}^0 h)(y) dz dy \nonumber \\ & =: \iint K_i(x, y) K_{\le i-1}(x, z) f(z)(\Pi_{\mathcal{D}g}^0 h)(y) dz dy. \tag{3.17} \end{align} $$

The above identity defines the **inner difference operator** $\mathcal{D}(=\mathcal{D}_z)$. In those terms, we have

$$ \mathbb{R}^0(f, g, h) := \Pi_f^0(\Pi_a^0 g) - \Pi_{fa}^0 g = \Pi_f^0(\Pi_{\mathcal{D}a}^0 g) $$

and, given the definition of the inner difference operator in the parabolic setting of Section C, we have more generally,

$$ \mathbb{R}(f, g, h) := \mathbb{P}_f(\tilde{\mathbb{P}}_g h) - \mathbb{P}_{fg}h = \mathbb{P}_f(\tilde{\mathbb{P}}_{\mathcal{D}g}h). \tag{3.18} $$

We use the same letter $\mathcal{D}$ in the spatial and parabolic settings. Compare this expression with the formal multiple integral, where we use the same letters to make it more striking,

$$ \int f(z) d\left(\int g dh\right)(z) = \int fg dh + \int f(z) d\left(\int (g-g(z)) dh\right)(z). $$

A similar reasoning as in the proof of continuity of $C(f, g, h) = \Pi(\mathbb{P}_{\mathcal{C}fg}, h)$, shows that $\mathbb{R}$ sends continuously $L^\infty \times C^\alpha \times C^\nu$ into $C^{\alpha+\nu}$, as soon as $\alpha \in (0, 1)$. A subtle thing happens here, though, as one has actually a refined continuity estimate on $\mathbb{R}^\circ$ that also takes into account the regularity of $f$ as well. It is related to the fact that $\mathbb{R}^\circ(\mathbf{1}, g, h) = \mathbb{R}^\circ(f, \mathbf{1}, h) = 0$, for all $f, g, h$, whereas $\mathbb{R}(\mathbf{1}, g, h) \ne 0$.

**PROPOSITION 14.** • For $\alpha, \beta \in [0, 1)$ and $\gamma \in (-3, 3)$, we have

$$ \|\mathbb{R}^\circ(f, g, h)\|_{C^{\alpha+\beta+\gamma}} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}. \tag{3.19} $$

• For $a_1, a_2, h \in C^\alpha$, one has

$$ \mathbb{R}(1, \mathbb{P}_{a_1} a_2, h) - \mathbb{P}_{a_1} \mathbb{R}(1, a_2, h) \in C^{3\alpha}. $$
* If $a_2, g \in C^\alpha$, with $0 < \alpha < 1/2$, and $a_1 \in C^\nu$, with $\nu \in (0, 1/2)$, and $h \in C^\gamma$ for $\gamma \in (-3, 3)$, then we have

$$ \mathbf{R}^{\circ}(\mathbf{P}_{a_1}a_2, g, h) - \mathbf{P}_{a_1}\mathbf{R}^{\circ}(a_2, g, h) \in C^{2\alpha+\gamma+\nu}. \quad (3.20) $$

Note that we need both $f$ and $g$ to be in $C^\alpha$ in (3.19); this is required by the method of proof, based on an interpolation argument.

*Proof.* The first statement (3.19) is proved in details in the third statement, inequality (C.3), of Proposition 38 in Appendix C. The second statement is a particular ($u = 1$) case of Proposition 43, also in the same appendix. Let us check the last statement (3.20). Because of the symmetry character that the first two arguments of $\mathbf{R}^\circ$ play in the proof of its regularity properties, Proposition 43 implies the boundedness of the operator

$$ (a_1, a_2, g, h) \to \mathbf{R}^{\circ}(\mathbf{P}_{a_1}a_2, g, h) - \mathbf{P}_{a_1}\mathbf{R}^{\circ}(a_2, g, h) $$

from $C^{2\nu} \times L^\infty \times C^{2\alpha} \times C^\gamma$ into $C^{2\nu+2\alpha+\gamma}$, because $2\nu, 2\alpha < 1$. We also have the boundedness of that operator from $L^\infty \times C^{2\alpha} \times L^\infty \times C^\gamma$ into $C^{2\alpha+\gamma}$, because $2\alpha < 1$ – we only use (3.19) to estimate the two quantities and do not use the difference structure in the commutator. We conclude by interpolating between these two estimates. $\square$

**THEOREM 15.** *Let $f, g \in L^\infty$ and $h \in C^\nu$ be given for $\nu \in (-3, 3)$. Let also $a_1 \in C^{\alpha_1}$ and $a_2 \in C^{\alpha_2}$ be given, with $\alpha_1, \alpha_2 \in (0, 1)$. Then both*

$$ \mathbf{R}^{\circ}(f, (a_1, a_2), h) := \mathbf{R}^{\circ}(f, \mathbf{P}_{a_1}a_2, h) - \mathbf{R}^{\circ}(f a_1, a_2, h) \quad (3.21) $$

and

$$ \mathbf{R}^{\circ}((a_1, a_2), g, h) := \mathbf{R}^{\circ}(\tilde{\mathbf{P}}_{a_1}a_2, g, h) - \mathbf{P}_{a_1}\mathbf{R}^{\circ}(a_2, g, h) \quad (3.22) $$

are elements of $C^{\alpha_1+\alpha_2+\nu}$.

* If $f \in C^\mu$, with $\mu \in (0, 1)$, and $a_1 \in C^{\alpha_1}$ and $a_2 \in C^{\alpha_2}$, and $h \in C^\nu$ with $\nu \in (-3, 3)$, then we have

$$ \mathbf{R}^{\circ}(f, (a_1, a_2), h) - \mathbf{P}_f(\mathbf{R}^{\circ}(\mathbf{1}, (a_1, a_2), h)) \in C^{\mu+\alpha_1+\alpha_2+\nu}. \quad (3.23) $$

The range $(0, 1)$ for the exponent $\alpha, \beta$ and $\gamma$, is dictated by the operator $\mathcal{D}$, which involves a first order increment and so can only encode regularity at order at most 1.
* If $a_2, g \in C^\alpha$, with $0 < \alpha < 1/2$, and $a_1 \in C^\nu$, with $\nu \in (0, 1/2)$, and $h \in C^\gamma$ for $\gamma \in (-3, 3)$, then we have

$$ \mathbf{R}^{\circ}(\mathbf{P}_{a_1}a_2, g, h) - \mathbf{P}_{a_1}\mathbf{R}^{\circ}(a_2, g, h) \in C^{2\alpha+\gamma+\nu}. \quad (3.20) $$

Note that we need both $f$ and $g$ to be in $C^\alpha$ in (3.19); this is required by the method of proof, based on an interpolation argument.

*Proof.* The first statement (3.19) is proved in details in the third statement, inequality (C.3), of Proposition 38 in Appendix C. The second statement is a particular ($u = 1$) case of Proposition 43, also in the same appendix. Let us check the last statement (3.20). Because of the symmetry character that the first two arguments of $\mathbf{R}^\circ$ play in the proof of its regularity properties, Proposition 43 implies the boundedness of the operator

$$ (a_1, a_2, g, h) \rightarrow \mathbf{R}^{\circ}(\mathbf{P}_{a_1}a_2, g, h) - \mathbf{P}_{a_1}\mathbf{R}^{\circ}(a_2, g, h) $$

from $C^{2\nu} \times L^\infty \times C^{2\alpha} \times C^\gamma$ into $C^{2\nu+2\alpha+\gamma}$, because $2\nu, 2\alpha < 1$. We also have the boundedness of that operator from $L^\infty \times C^{2\alpha} \times L^\infty \times C^\gamma$ into $C^{2\alpha+\gamma}$, because $2\alpha < 1$ – we only use (3.19) to estimate the two quantities and do not use the difference structure in the commutator. We conclude by interpolating between these two estimates. $\square$

**THEOREM 15.** • Let $f, g \in L^\infty$ and $h \in C^\nu$ be given for $\nu \in (-3, 3)$. Let also $a_1 \in C^{\alpha_1}$ and $a_2 \in C^{\alpha_2}$ be given, with $\alpha_1, \alpha_2 \in (0, 1)$. Then both

$$ \mathbf{R}^{\circ}(f, (a_1, a_2), h) := \mathbf{R}^{\circ}(f, \mathbf{P}_{a_1}a_2, h) - \mathbf{R}^{\circ}(f a_1, a_2, h) \quad (3.21) $$

and

$$ \mathbf{R}^{\circ}((a_1, a_2), g, h) := \mathbf{R}^{\circ}(\tilde{\mathbf{P}}_{a_1}a_2, g, h) - \mathbf{P}_{a_1}\mathbf{R}^{\circ}(a_2, g, h) \quad (3.22) $$

are elements of $C^{\alpha_1+\alpha_2+\nu}$.

* If $f \in C^\mu$, with $\mu \in (0, 1)$, and $a_1 \in C^{\alpha_1}$ and $a_2 \in C^{\alpha_2}$, and $h \in C^\nu$ with $\nu \in (-3, 3)$, then we have

$$ \mathbf{R}^{\circ}(f, (a_1, a_2), h) - \mathbf{P}_f(\mathbf{R}^{\circ}(\mathbf{1}, (a_1, a_2), h)) \in C^{\mu+\alpha_1+\alpha_2+\nu}. \quad (3.23) $$

The range $(0, 1)$ for the exponent $\alpha, \beta$ and $\gamma$, is dictated by the operator $\mathcal{D}$, which involves a first order increment and so can only encode regularity at order at most 1.
*Proof.* The second estimate (3.22) is proved in the appendix, see Proposition 43, because the first two arguments of $\mathbb{R}^\circ$ play a 'symmetric' role in the proof of the continuity estimates; see Remark 16.

We prove the two other corresponding statements (3.21) and (3.23) in the model time-independent setting of the flat torus. We prove first (3.21). We have

$$ \Pi_f^0(\Pi_{a_1}^0 a_2 g) - \Pi_f^0(\Pi_{a_1}^0 a_2 g) - \Pi_{f a_1}^0(\Pi_{a_2}^0 g) = \Pi_f^0(\Pi_{a_1}^0 a_2 g) - \Pi_{f a_1}^0(\Pi_{a_2}^0 g) $$

is a sum over $i$ of double integrals

$$ \iint K_i(x, y) K_{\le i-2}(x, z) f(z) (\Pi_{\mathcal{D}(\Pi_{a_1}^0 a_2 - a_1(z)a_2)}^0 g)(y) dz dy \\ = \iint K_i(x, y) K_{\le i-2}(x, z) f(z) (\Pi_{\mathcal{D}(\Pi_{a_1}^0 a_2)}^0 g)(y) dz dy $$

on which we read off that their $L^\infty$ norm is of order $2^{-i(\alpha_1 + \alpha_2 + \nu)}$. This point finishes the proof since the last quantity corresponds to the dyadic blocks $\Delta_i(\mathbb{R}(f, (a_1, a_2), g))$.

We prove (3.23). On the other hand, we have that

$$ \Delta_i(\mathbb{P}_f(\mathbb{R}(\mathbf{1}; (a_1, a_1); g)))(x) \\ = \iiint K_i(x, y) K_{\le i-2}(x, u) f(u) K_{\le i-2}(x, z) (\Pi_{\mathcal{D}\Pi_{a_1}^0 a_2}^0 g)(y) dz dy du. $$

So using that the kernel $K_{\le i-1}(x, \cdot)$ has an integral equal to 1, we deduce that the difference of the two dyadic blocks is given by

$$ \iiint K_i(x, y) K_{\le i-2}(x, z) K_{\le i-2}(x, u) (f(z) - f(u)) (\Pi_{\mathcal{D}\Pi_{a_1}^0 a_2}^0 g)(y) dz dy du, $$

on which we read off that their $L^\infty$ norm is of order $2^{-i(\mu + \alpha_1 + \alpha_2 + \nu)}$. $\square$

**REMARK 16.** The operator $\mathbb{R}^\circ$ is not symmetric in a proper sense, but rather in terms of how the regularity properties of its first two arguments are taken into account. The frequency analysis is exactly the same, if we assume a regularity on $f$ or on $g$. Since $\mathbb{R}^\circ$ involves only $\mathbb{P}$-paraproducts and not $\tilde{\mathbb{P}}$-paraproducts, we can exploit the difference structure between $\mathbb{P}_{fg}$ and $\mathbb{P}_f$ or $\mathbb{P}_g$. In the operator $\mathbb{R}$, one can only exploit the regularity on $g$ in $\mathbb{R}(f, g, h)$ because of the $\tilde{\mathbb{P}}$-operator on $g$. This phenomenon is illustrated by the observation that

$$ \mathbb{R}^\circ(1, g, h) = \mathbb{R}^\circ(f, 1, h) = 0 = \mathbb{R}(f, 1, h), $$

while $\mathbb{R}(1, g, h) = \tilde{\mathbb{P}}_g h - \mathbb{P}_g h \neq 0$.
*Proof.* The second estimate (3.22) is proved in the appendix, see Proposition 43, because the first two arguments of $\mathbb{R}^\circ$ play a 'symmetric' role in the proof of the continuity estimates; see Remark 16.

We prove the two other corresponding statements (3.21) and (3.23) in the model time-independent setting of the flat torus. We prove first (3.21). We have

$$ \Pi_f^0(\Pi_{a_1}^0 a_2 g) - \Pi_f^0(\Pi_{a_1}^0 a_2 g) - \Pi_{f a_1}^0(\Pi_{a_2}^0 g) = \Pi_f^0(\Pi_{a_1}^0 a_2 g) - \Pi_{f a_1}^0(\Pi_{a_2}^0 g) $$

is a sum over $i$ of double integrals

$$ \iint K_i(x, y) K_{\le i-2}(x, z) f(z) (\Pi_{\mathcal{D}(\Pi_{a_1}^0 a_2 - a_1(z)a_2)}^0 g)(y) dz dy \\ = \iint K_i(x, y) K_{\le i-2}(x, z) f(z) (\Pi_{\mathcal{D}(\Pi_{a_1}^0 a_2)}^0 g)(y) dz dy $$

on which we read off that their $L^\infty$ norm is of order $2^{-i(\alpha_1 + \alpha_2 + v)}$. This point finishes the proof since the last quantity corresponds to the dyadic blocks $\Delta_i(\mathbb{R}(f, (a_1, a_2), g))$.

We prove (3.23). On the other hand, we have that

$$ \Delta_i(\mathbb{P}_f(\mathbb{R}(\mathbf{1}; (a_1, a_1); g)))(x) \\ = \iiint K_i(x, y) K_{\le i-2}(x, u) f(u) K_{\le i-2}(x, z) (\Pi_{\mathcal{D}\Pi_{a_1}^0 a_2}^0 g)(y) dz dy du. $$

So using that the kernel $K_{\le i-1}(x, \cdot)$ has an integral equal to 1, we deduce that the difference of the two dyadic blocks is given by

$$ \iiint K_i(x, y) K_{\le i-2}(x, z) K_{\le i-2}(x, u) (f(z) - f(u)) (\Pi_{\mathcal{D}\Pi_{a_1}^0 a_2}^0 g)(y) dz dy du, $$

on which we read off that their $L^\infty$ norm is of order $2^{-i(\mu + \alpha_1 + \alpha_2 + v)}$. $\square$

**REMARK 16.** The operator $\mathbb{R}^\circ$ is not symmetric in a proper sense, but rather in terms of how the regularity properties of its first two arguments are taken into account. The frequency analysis is exactly the same, if we assume a regularity on $f$ or on $g$. Since $\mathbb{R}^\circ$ involves only $\mathbb{P}$-paraproducts and not $\tilde{\mathbb{P}}$-paraproducts, we can exploit the difference structure between $\mathbb{P}_{fg}$ and $\mathbb{P}_f$ or $\mathbb{P}_g$. In the operator $\mathbb{R}$, one can only exploit the regularity on $g$ in $\mathbb{R}(f, g, h)$ because of the $\tilde{\mathbb{P}}$-operator on $g$. This phenomenon is illustrated by the observation that

$$ \mathbb{R}^\circ(1, g, h) = \mathbb{R}^\circ(f, 1, h) = 0 = \mathbb{R}(f, 1, h), $$

while $\mathbb{R}(1, g, h) = \tilde{\mathbb{P}}_g h - \mathbb{P}_g h \neq 0$.
3.2.3. Back again to the high order paracontrolled expansion (2.3). Recall from Proposition 14 that the operator $\mathbb{R}^\circ(a, b, c) = P_a P_b c - P_{ab} c$, has better regularity properties than the merging operator $\mathbb{R}$. We can use the continuity properties of $\mathbb{R}^\circ$ to rewrite the high order paracontrolled expansion (2.3) and clarify the cancellations that happen in each of its brackets. We have first that

$$ 
\begin{aligned}
P_{f^{(2)}(u)} u^2 - 2P_{f^{(2)}(u)} u &= P_{f^{(2)}(u)} \Pi(u, u) + 2P_{f^{(2)}(u)} (P_u u) - 2P_{f^{(2)}(u)} u \\
&= P_{f^{(2)}(u)} \Pi(u, u) + 2\mathbb{R}^\circ(f^{(2)}(u), u, u) \\
&= P_{f^{(2)}(u)} \Pi(u, u) + 2P_{f^{(3)}(u)} \mathbb{R}^\circ(u, u, u) + (2\alpha \wedge 1 + 2\alpha),
\end{aligned}
$$

from Bony's paralinearization $f^{(2)}(u) = P_{f^{(3)}(u)} u + (2\alpha)$ and the continuity properties of the merging operator $\mathbb{R}$. Write $\mathbb{D}^\circ(a, b, c)$ for

$$ \Pi(P_a b, c) - P_a \Pi(b, c); $$

this operator has the same regularity properties as $\mathbb{D}$. The third order expansion formula is only needed when $\alpha < 1/2$. In that case, an elementary computation shows that

$$ 
\begin{aligned}
P_{f^{(3)}(u)} u^3 - 3P_{uf^{(3)}(u)} u^2 + 3P_{u^2 f^{(3)}(u)} u \\
&= (3\mathbb{R}^\circ(f^{(3)}(u), u^2, u) + 2P_{f^{(3)}(u)} \mathbb{R}^\circ(u, u, u) \\
&\quad + P_{f^{(3)}(u)}(2\mathbb{D}^\circ(u, u, u) + \Pi(u, \Pi(u, u))) \\
&\quad - 6\mathbb{R}^\circ(u f^{(3)}(u), u, u)) + 3\mathbb{R}^\circ(f^{(3)}(u), u, \Pi(u, u)) \\
&= P_{f^{(3)}(u)}(2\mathbb{R}^\circ(u, u, u) + 2\mathbb{D}^\circ(u, u, u) + \Pi(u, \Pi(u, u))) \\
&\quad + \{3\mathbb{R}^\circ(f^{(3)}(u), \Pi(u, u), u) + 6\mathbb{R}^\circ(f^{(3)}(u), (u, u), u) \\
&\quad\quad + 3\mathbb{R}^\circ(f^{(3)}(u), u, \Pi(u, u))\}
\end{aligned}
$$

is the sum of 3$\alpha$-terms and a 4$\alpha$-term, as one can use in that case the refined continuity estimates (3.20) on $\mathbb{R}$.

COROLLARY 17. Let $f \in C_b^5$. If $\alpha < 1/2$, one has

$$ 
\begin{aligned}
f(u) = P_{f'(u)} u + \frac{1}{2!} P_{f^{(2)}(u)} \Pi(u, u) \\
+ \frac{1}{3!} P_{f^{(3)}(u)} (8 \mathbb{R}^\circ(u, u, u) + 2 \mathbb{D}^\circ(u, u, u) + \Pi(u, \Pi(u, u))) \\
+ f(u)^2,
\end{aligned}
\quad (3.24)
$$

for a remainder $f(u)^2 \in C^{4\alpha}$.

PROPOSITION 18. Let a noise $\zeta \in C^{\alpha-2}$ be given, together with a function $u$ given by a paracontrolled system with reference functions $\mathbf{Z}$, and $f \in C_b^5$. Then one can
3.2.3. Back again to the high order paracontrolled expansion (2.3). Recall from Proposition 14 that the operator $\mathbb{R}^\circ(a, b, c) = P_a P_b c - P_{ab} c$, has better regularity properties than the merging operator $\mathbb{R}$. We can use the continuity properties of $\mathbb{R}^\circ$ to rewrite the high order paracontrolled expansion (2.3) and clarify the cancellations that happen in each of its brackets. We have first that

$$
\begin{align*}
P_{f^{(2)}(u)} u^2 - 2P_{f^{(2)}(u)} u &= P_{f^{(2)}(u)} \Pi(u, u) + 2P_{f^{(2)}(u)} (P_u u) - 2P_{f^{(2)}(u)} u \\
&= P_{f^{(2)}(u)} \Pi(u, u) + 2\mathbb{R}^\circ(f^{(2)}(u), u, u) \\
&= P_{f^{(2)}(u)} \Pi(u, u) + 2P_{f^{(3)}(u)} \mathbb{R}^\circ(u, u, u) + (2\alpha \wedge 1 + 2\alpha),
\end{align*}
$$

from Bony's paralinearization $f^{(2)}(u) = P_{f^{(3)}(u)}u + (2\alpha)$ and the continuity properties of the merging operator $\mathbb{R}$. Write $\mathbb{D}^\circ(a, b, c)$ for

$$ \Pi(P_a b, c) - P_a \Pi(b, c); $$

this operator has the same regularity properties as $\mathbb{D}$. The third order expansion formula is only needed when $\alpha < 1/2$. In that case, an elementary computation shows that

$$
\begin{align*}
& P_{f^{(3)}(u)} u^3 - 3P_{uf^{(3)}(u)} u^2 + 3P_{u^2 f^{(3)}(u)} u \\
&= (3\mathbb{R}^\circ(f^{(3)}(u), u^2, u) + 2P_{f^{(3)}(u)} \mathbb{R}^\circ(u, u, u) \\
&\quad + P_{f^{(3)}(u)} (2\mathbb{D}^\circ(u, u, u) + \Pi(u, \Pi(u, u))) \\
&\quad - 6\mathbb{R}^\circ(u f^{(3)}(u), u, u)) + 3\mathbb{R}^\circ(f^{(3)}(u), u, \Pi(u, u)) \\
&= P_{f^{(3)}(u)} (2\mathbb{R}^\circ(u, u, u) + 2\mathbb{D}^\circ(u, u, u) + \Pi(u, \Pi(u, u))) \\
&\quad + \{3\mathbb{R}^\circ(f^{(3)}(u), \Pi(u, u), u) + 6\mathbb{R}^\circ(f^{(3)}(u), (u, u), u) \\
&\quad\quad + 3\mathbb{R}^\circ(f^{(3)}(u), u, \Pi(u, u))\}
\end{align*}
$$

is the sum of 3$\alpha$-terms and a 4$\alpha$-term, as one can use in that case the refined continuity estimates (3.20) on $\mathbb{R}$.

COROLLARY 17. Let $f \in C_b^5$. If $\alpha < 1/2$, one has

$$
\begin{align}
f(u) = P_{f'(u)} u &+ \frac{1}{2!} P_{f^{(2)}(u)} \Pi(u, u) \nonumber \\
&+ \frac{1}{3!} P_{f^{(3)}(u)} (8 \mathbb{R}^{\circ}(u, u, u) + 2 \mathbb{D}^{\circ}(u, u, u) + \Pi(u, \Pi(u, u))) \nonumber \\
&+ f(u)^{\sharp}, \tag{3.24}
\end{align}
$$

for a remainder $f(u)^{\sharp} \in C^{4\alpha}$.

PROPOSITION 18. Let a noise $\zeta \in C^{\alpha-2}$ be given, together with a function $u$ given by a paracontrolled system with reference functions $\mathbf{Z}$, and $f \in C_b^5$. Then one can
write the product $f(u)\zeta$ under the form

$$f(u)\zeta = \mathbf{P}_{f(u)}\zeta + \sum_{i=1}^{2} \mathbf{P}_{f'(u)u_i}Y_i + \mathbf{P}_{f'(u)u_{11}}Y_{11} + \mathbf{P}_{f^{(2)}(u)u_1^2}Y_{(1,1)} + (\sharp) \quad (3.25)$$

for explicit reference distributions $Y$'s that depend only on the noise $\zeta$ and $Z$, and a remainder term

$$(\sharp) \in \mathcal{L}(C^{4\alpha}) \subset C^{4\alpha-2},$$

that depends continuously on $\hat{u}$ and the $Y$'s.

The proper interpretation of the above statement is that the remainder $(\sharp) \in \mathcal{L}(C^{4\alpha})$, provided the noise $\zeta$ is enhanced into $\hat{\zeta}$, with components belonging to $\mathcal{L}(C^{k\alpha-2})$-spaces; the remainder $(\sharp)$ also depends on $\hat{\zeta}$; see Remark 1 after the proof. We first give a detailed proof of the statement, in which the reader will see that we keep repeating the same kind of computations. The mechanics at play here will be emphasized after the proof; this is nothing but the mechanics of Section 1.2.

*Proof.* We start from the identity

$$f(u)\zeta = \mathbf{P}_{f(u)}\zeta + \mathbf{P}_{\zeta}f(u) + \Pi(f(u), \zeta),$$

and treat the second and third terms separately. We shall denote by (b) a term in $\mathcal{L}(C^{4\alpha})$ that may change from place to place. A term of the form $\mathbf{P}_{\zeta}b$, with $b \in C^{3\alpha}$, is a (b)-term; so is a term $\mathbf{P}_a(b)$, if $a$ has nonnegative parabolic Hölder regularity. We use below the following fact proved in Proposition 38 – (C.1) and Proposition 42 of Appendix C.

LEMMA 19. Let $a_1 \in C^{\alpha_1}$, $a_2 \in C^{\alpha_2}$, $a_3 \in C^{\alpha_3}$ be given, with $\alpha_i \in (0, 1)$. The quantities

$$S^{\circ}(\zeta, a_1, a_2), \quad S(\zeta, a_1, a_2), \quad S(\zeta, (a_1, a_2), a_3),$$

belong to $\mathcal{L}(C^{4\alpha})$, if $\alpha_1 + \alpha_2 \ge 3\alpha$ for the first two quantities, and $\alpha_1 + \alpha_2 + \alpha_3 \ge 3\alpha$, for the third quantity. These operators are continuous functions of the $a_i$ under these assumptions.

(1) We have

$$\begin{aligned} \mathbf{P}_{\zeta} f(u) &= \mathbf{P}_{\zeta} \mathbf{P}_{f'(u)} u + \frac{1}{2} \mathbf{P}_{\zeta} \mathbf{P}_{f^{(2)}(u)} \Pi(u, u) + (b) \\ &= \mathbf{P}_{f'(u)} \mathbf{P}_{\zeta} u + S^{\circ}(\zeta, f'(u), u) + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)} \mathbf{P}_{\zeta} \Pi(u, u) + (b), \end{aligned}$$
write the product $f(u)\zeta$ under the form

$$f(u)\zeta = \mathbf{P}_{f(u)}\zeta + \sum_{i=1}^{2} \mathbf{P}_{f'(u)u_i}Y_i + \mathbf{P}_{f'(u)u_{11}}Y_{11} + \mathbf{P}_{f^{(2)}(u)u_1^2}Y_{(1,1)} + (\sharp) \quad (3.25)$$

for explicit reference distributions $Y$'s that depend only on the noise $\zeta$ and $Z$, and a remainder term

$$(\sharp) \in \mathcal{L}(C^{4\alpha}) \subset C^{4\alpha-2},$$

that depends continuously on $\hat{u}$ and the $Y$'s.

The proper interpretation of the above statement is that the remainder $(\sharp) \in \mathcal{L}(C^{4\alpha})$, provided the noise $\zeta$ is enhanced into $\hat{\zeta}$, with components belonging to $\mathcal{L}(C^{k\alpha-2})$-spaces; the remainder $(\sharp)$ also depends on $\hat{\zeta}$; see Remark 1 after the proof. We first give a detailed proof of the statement, in which the reader will see that we keep repeating the same kind of computations. The mechanics at play here will be emphasized after the proof; this is nothing but the mechanics of Section 1.2.

*Proof.* We start from the identity

$$f(u)\zeta = \mathbf{P}_{f(u)}\zeta + \mathbf{P}_{\zeta}f(u) + \Pi(f(u), \zeta),$$

and treat the second and third terms separately. We shall denote by (b) a term in $\mathcal{L}(C^{4\alpha})$ that may change from place to place. A term of the form $\mathbf{P}_{\zeta}b$, with $b \in C^{3\alpha}$, is a (b)-term; so is a term $\mathbf{P}_a(b)$, if $a$ has nonnegative parabolic Hölder regularity. We use below the following fact proved in Proposition 38 – (C.1) and Proposition 42 of Appendix C.

LEMMA 19. Let $a_1 \in C^{\alpha_1}$, $a_2 \in C^{\alpha_2}$, $a_3 \in C^{\alpha_3}$ be given, with $\alpha_i \in (0, 1)$. The quantities

$$S^{\circ}(\zeta, a_1, a_2), \quad S(\zeta, a_1, a_2), \quad S(\zeta, (a_1, a_2), a_3),$$

belong to $\mathcal{L}(C^{4\alpha})$, if $\alpha_1 + \alpha_2 \ge 3\alpha$ for the first two quantities, and $\alpha_1 + \alpha_2 + \alpha_3 \ge 3\alpha$, for the third quantity. These operators are continuous functions of the $a_i$ under these assumptions.

(1) We have

$$\begin{aligned} \mathbf{P}_{\zeta} f(u) &= \mathbf{P}_{\zeta} \mathbf{P}_{f'(u)} u + \frac{1}{2} \mathbf{P}_{\zeta} \mathbf{P}_{f^{(2)}(u)} \Pi(u, u) + (b) \\ &= \mathbf{P}_{f'(u)} \mathbf{P}_{\zeta} u + S^{\circ}(\zeta, f'(u), u) + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)} \mathbf{P}_{\zeta} \Pi(u, u) + (b), \end{aligned}$$
from Lemma 19. So $P_{\zeta} f(u)$ is equal to

$$
\begin{align*}
& P_{f'(u)} P_{u_i} (P_{\zeta} Z_i) + P_{f'(u)} S(\zeta, u_i, Z_i) \\
& \qquad + S^{\circ}(\zeta, f'(u), u) + \frac{1}{2} P_{f^{(2)}(u)} P_{\zeta} P_{u_i}^2 \Pi(Z_1, Z_1) + \text{(b)} \\
&= (P_{f'(u)} u_i P_{\zeta} Z_i + R^{\circ}(f'(u), u_i, P_{\zeta} Z_i)) + (P_{f'(u)} P_{u_{ij}} S(\zeta, Z_j, Z_i) + \text{(b)}) \\
& \qquad + (P_{f^{(2)}(u)} u_i^2 S^{\circ}(\zeta, Z_1, Z_1) + \text{(b)}) \\
& \qquad + \left(\frac{1}{2} P_{f^{(2)}(u)} P_{u_i}^2 (P_{\zeta} \Pi(Z_1, Z_1)) + \text{(b)}\right) + \text{(b)},
\end{align*}
$$

from Lemma 19 again. Note that it follows from the sharp continuity estimate (14)
from Proposition 14 that $\mathbf{R}^\circ(f'(u), u_i, \mathbf{P}_\xi Z_i)$ is a (b)-term. Also, one has from
Lemma 19 that $\mathbf{S}(\xi, Z_j, Z_i) \in \mathcal{L}(\mathcal{C}^{(i+j+1)\alpha})$ and $\mathbf{S}^\circ(\xi, Z_j, Z_i) \in \mathcal{L}(\mathcal{C}^{(i+j+1)\alpha})$,
while $\mathbf{P}_\xi \Pi(Z_1, Z_1) \in \mathcal{L}(\mathcal{C}^{3\alpha})$. This allows us to write $\mathbf{P}_\xi f(u)$ under the required
form

$$
\mathbf{P}_{\zeta} f(u) = \mathbf{P}_{f'(u)u_i}(\mathbf{P}_{\zeta} Z_i) + \mathbf{P}_{f'(u)u_{11}}\mathbf{S}(\zeta, Z_1, Z_1) + \mathbf{P}_{f^{(2)}(u)u_1^2}\mathbf{S}^{\circ}(\zeta, Z_1, Z_1) \\
\qquad + \frac{1}{2}\mathbf{P}_{f^{(2)}(u)u_1^2}(\mathbf{P}_{\zeta}\Pi(Z_1, Z_1)) + \mathbf{P}_{f^{(2)}(u)u_1^2}\mathbf{S}^{\circ}(\zeta, Z_1, Z_1)
$$

(2) Consider now the resonant term $\Pi(f(u), \zeta)$, and start for that purpose from
the multiplicative formula (3.11)

$$
\begin{align*}
& f'(u) \Pi(u, \zeta) + f^{(2)}(u) \left\{ \mathbf{C}^{\circ}(u, u, \zeta) + \frac{1}{2} \Pi(\Pi(u, u), \zeta) \right\} \\
& \quad + f^{(3)}(u) \left\{ \begin{aligned}
& \frac{1}{3} \mathbf{C}^{\circ}(u, (u, u), \zeta) + \frac{1}{3} \mathbf{C}^{\circ}((u, u), u, \zeta) \\
& + \frac{1}{6} \mathbf{C}^{\circ}(u, \Pi(u, u), \zeta) + \frac{1}{6} \mathbf{C}^{\circ}(\Pi(u, u), u, \zeta) \\
& + \frac{1}{3} \Pi(\mathbf{D}^{\circ}(u, u, u), \zeta) + \frac{1}{6} \Pi(\Pi(u, \Pi(u, u)), \zeta)
\end{aligned} \right\} \\
& \quad + \sum_{\ell=1}^{\ell_0} (\star)_{\ell} \Pi^{\ell}_{(1)}(u, \zeta) + (5\alpha - 2)
\end{align*}
$$

for $\Pi(f(u), \zeta)$ – recall $(\star)_{\ell}$ is defined in (3.12). It suffices to see that each term
in this decomposition has the form (3.25) of the statement; we proceed from the
more to the less regular terms.

(2.1) Use Theorem **7** on the expansion rule for the operator $\Pi_{(1)}$, and the first order paracontrolled structure of $u$, to write

$$
\Pi_{(1)}^{\ell}(u, \zeta) = u_1 \Pi_{(1)}^{\ell}(Z_1, \zeta) + (3\alpha - 1).
$$
This gives the paracontrolled representation

$$ (\star)_{\ell} \Pi_{(1)}^{\ell}(u, \zeta) = \mathbf{P}_{(\star)_{\ell}u_1} \Pi_{(1)}^{\ell}(Z_1, \zeta) + (5\alpha - 2), \quad (3.26) $$

since $\alpha < 1/2$.

(2.2) Write {$u$} for the bracket term after $f^{(3)}(u)$. The first order paracontrolled structure of $u$ and the continuity properties of the correctors and commutators give

$$ f^{(3)}(u)\{u\} = \mathbf{P}_{f^{(3)}(u)u_1^3}\{Z_1\} + (5\alpha - 2), $$

for a remainder term $(5\alpha - 2) \in C^{5\alpha-2}$. The fact that $\{Z_1\} \in \mathcal{L}(C^{4\alpha})$ is part of the data $\hat{\zeta}$.

(2.3) We use the paracontrolled structure of $u$ and the continuity results on correctors to write the term $\Pi(\Pi(u, u), \zeta)$ in multiplicative form. This gives

$$ \begin{aligned} \Pi(\Pi(u, u), \zeta) &= \Pi(\mathbf{P}_{u_i}\Pi(Z_i, u), \zeta) + \Pi(\mathbf{D}(u_i, Z_i, u), \zeta) + (5\alpha - 2) \\ &= u_i\Pi(\Pi(Z_i, u), \zeta) + \mathbf{C}^{\circ}(u_i, \Pi(Z_i, u), \zeta) \\ &\quad + u_{11}u_1\Pi(\mathbf{D}(Z_1, Z_1, Z_2), \zeta) + (5\alpha - 2) \\ &= u_iu_j\Pi(\Pi(Z_i, Z_j), \zeta) + u_{11}u_1(\star) + (5\alpha - 2) \end{aligned} $$

with

$$ (\star) := 2\mathbf{C}^{\circ}(Z_1, \Pi(Z_1, Z_1), \zeta) + 2\Pi(\mathbf{D}(Z_1, Z_1, Z_1), \zeta), $$

after elementary computations. The fact that $\Pi(\Pi(Z_1, Z_2), \zeta)$ and $(\star)$ are elements of $\mathcal{L}(C^{4\alpha})$ is part of the data $\hat{\zeta}$. Now,

$$ f^{(2)}(u)u_1^2 = \mathbf{P}_{2f^{(2)}(u)u_1u_{11}+f^{(3)}(u)u_1^3} Z_1 + (2\alpha), $$

so, recalling that an element in $C^{5\alpha-2}$ is of (b)-type, we have

$$ \begin{aligned} f^{(2)}(u)u_1^2 \Pi(\Pi(Z_1, Z_1), \zeta) &= \mathbf{P}_{f^{(2)}(u)u_1^2} \Pi(\Pi(Z_1, Z_1), \zeta) + (b) + \Pi(f^{(2)}(u)u_1^2, \Pi(\Pi(Z_1, Z_1), \zeta)) \\ &= \mathbf{P}_{f^{(2)}(u)u_1^2} \Pi(\Pi(Z_1, Z_1), \zeta) \\ &\quad + \mathbf{P}_{2f^{(2)}(u)u_1u_{11}+f^{(3)}(u)u_1^3} \Pi(Z_1, \Pi(\Pi(Z_1, Z_1), \zeta)) + (b). \end{aligned} $$

Here again, the fact that $\Pi(Z_1, \Pi(\Pi(Z_1, Z_1), \zeta))$ is of (b)-type is part of the data $\hat{\zeta}$, so the term $f^{(2)}(u)u_1^2\Pi(\Pi(Z_1, Z_1), \zeta)$ has indeed the right form. Similar computations show that $f^{(2)}(u)u_1u_2\Pi(\Pi(Z_1, Z_2), \zeta)$ and $u_{11}u_1(\star)$ also have the right form.
This gives the paracontrolled representation

$$ (\star)_{\ell} \Pi_{(1)}^{\ell}(u, \zeta) = \mathbf{P}_{(\star)_{\ell}u_1} \Pi_{(1)}^{\ell}(Z_1, \zeta) + (5\alpha - 2), \quad (3.26) $$

since $\alpha < 1/2$.

(2.2) Write {$u$} for the bracket term after $f^{(3)}(u)$. The first order paracontrolled structure of $u$ and the continuity properties of the correctors and commutators give

$$ f^{(3)}(u)\{u\} = \mathbf{P}_{f^{(3)}(u)u_1^3}\{Z_1\} + (5\alpha - 2), $$

for a remainder term $(5\alpha - 2) \in \mathcal{C}^{5\alpha-2}$. The fact that $\{Z_1\} \in \mathcal{L}(\mathcal{C}^{4\alpha})$ is part of the data $\hat{\zeta}$.

(2.3) We use the paracontrolled structure of $u$ and the continuity results on correctors to write the term $\Pi(\Pi(u, u), \zeta)$ in multiplicative form. This gives

$$ \begin{aligned} \Pi(\Pi(u, u), \zeta) &= \Pi(\mathbf{P}_{u_i}\Pi(Z_i, u), \zeta) + \Pi(\mathbf{D}(u_i, Z_i, u), \zeta) + (5\alpha - 2) \\ &= u_i\Pi(\Pi(Z_i, u), \zeta) + \mathbf{C}^{\circ}(u_i, \Pi(Z_i, u), \zeta) \\ &\quad + u_{11}u_1\Pi(\mathbf{D}(Z_1, Z_1, Z_2), \zeta) + (5\alpha - 2) \\ &= u_iu_j\Pi(\Pi(Z_i, Z_j), \zeta) + u_{11}u_1(\star) + (5\alpha - 2) \end{aligned} $$

with

$$ (\star) := 2\mathbf{C}^{\circ}(Z_1, \Pi(Z_1, Z_1), \zeta) + 2\Pi(\mathbf{D}(Z_1, Z_1, Z_1), \zeta), $$

after elementary computations. The fact that $\Pi(\Pi(Z_1, Z_2), \zeta)$ and $(\star)$ are elements of $\mathcal{L}(\mathcal{C}^{4\alpha})$ is part of the data $\hat{\zeta}$. Now,

$$ f^{(2)}(u)u_1^2 = \mathbf{P}_{2f^{(2)}(u)u_1u_{11}+f^{(3)}(u)u_1^3} Z_1 + (2\alpha), $$

so, recalling that an element in $\mathcal{C}^{5\alpha-2}$ is of (b)-type, we have

$$ \begin{aligned} f^{(2)}(u)u_1^2 \Pi(\Pi(Z_1, Z_1), \zeta) &= \mathbf{P}_{f^{(2)}(u)u_1^2} \Pi(\Pi(Z_1, Z_1), \zeta) + (b) + \Pi(f^{(2)}(u)u_1^2, \Pi(\Pi(Z_1, Z_1), \zeta)) \\ &= \mathbf{P}_{f^{(2)}(u)u_1^2} \Pi(\Pi(Z_1, Z_1), \zeta) \\ &\quad + \mathbf{P}_{2f^{(2)}(u)u_1u_{11}+f^{(3)}(u)u_1^3} \Pi(Z_1, \Pi(\Pi(Z_1, Z_1), \zeta)) + (b). \end{aligned} $$

Here again, the fact that $\Pi(Z_1, \Pi(\Pi(Z_1, Z_1), \zeta))$ is of (b)-type is part of the data $\hat{\zeta}$, so the term $f^{(2)}(u)u_1^2\Pi(\Pi(Z_1, Z_1), \zeta)$ has indeed the right form. Similar computations show that $f^{(2)}(u)u_1u_2\Pi(\Pi(Z_1, Z_2), \zeta)$ and $u_{11}u_1(\star)$ also have the right form.