Paracontrolled calculus for quasilinear singular PDEs

Abstract

We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. Continuity results for a number of operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser & Hairer and Otto, Sauer, Smith & Weber, or Furlan & Gubinelli’ study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we introduce here.

Appendices

A-Basics on high order paracontrolled calculus

We recall in this appendix a number of results from [3, 4] that we use in this work. This should help the reader understanding the computations of Appendix B and their mechanics.

We first describe some approximation operators ${\mathcal {P}}_t$ and ${\mathcal {Q}}_t$ that we use in place of the usual Littlewood-Paley projectors $\sum _{j\le n}\Delta _j$ and $\Delta _n$, in which the heat semigroup plays the role of Fourier theory. The parabolic Hölder space are defined from these operators. We also recall the form of the space-time paraproduct and resonant operators that we use and give a number of the continuity estimates on different correctors/commutators and their iterated versions.

Recall that we denote by M a 3-dimensional closed Riemannian manifold and set

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

for a finite positive time horizon T. We denote by $\rho (\cdot ,\cdot )$ the parabolic distance on ${\mathcal {M}}$ and by $e=(\tau ,x)$ a generic spacetime point. Denote by $\mu$ the Riemannian volume measure and define the parabolic measure

$$\begin{aligned} \nu := dt\otimes \mu . \end{aligned}$$

Recall the reformulation (2.7) of Eq. (1.1), where the operator $L=-\sum _{i=1}^\ell V_i^2$ is a second order differential operator in Hörmander form.

1.1 A.1 Approximation operators and parabolic Hölder spaces

In the flat setting of the torus, we can use Fourier theory to approximate Schwartz distributions by smooth functions. We have

$$\begin{aligned} f=\lim _{n\rightarrow \infty }S_n(f)=\sum _{i\ge -1}\Delta _i(f) \end{aligned}$$

with $\Delta _j$ the Paley-Littlewood projectors. Refer e.g. to [5] for basics on Littlewood-Paley theory. Using the heat semigroup, one has in a more general geometric framework

$$\begin{aligned} f=\lim _{t\rightarrow 0}P_t^{(b)}f=\int _0^1 Q_t^{(b)}f\frac{\mathrm dt}{t}+P_1^{(b)}f \end{aligned}$$

where

$$\begin{aligned} Q_t^{(b)}:=\frac{(tL)^be^{-tL}}{(b-1)!}\quad \text {and}\quad -t\partial _tP_t^{(b)}:=Q_t^{(b)} \end{aligned}$$

with $P_0=\text {Id}$. One can show that there exists a polynomial $p_b$ of degree $(b-1)$ such that $P_t^{(b)}=p_b(tL)e^{-tL}$ and $p_b(0)=1$. The operators $Q_t^{(b)}$ and $P_t^{(b)}$ play the role of Paley-Littlewood projector and Fourier series, respectively. Indeed, if one works on the torus, then

$$\begin{aligned} \widehat{Q_t^{(b)}}(\lambda )=\frac{\left( t|\lambda |^2\right) ^b}{(b-1)!}e^{-|\lambda |^2t}\quad \text {and}\quad \widehat{P_t^{(b)}}(\lambda )=p_b\left( t|\lambda |^2\right) e^{-|\lambda |^2t} \end{aligned}$$

so we see that $Q_t^{(b)}$ localize in frequency around the annulus $|\lambda |\sim t^{-\frac{1}{2}}$ and $P_t^{(b)}$ localize in frequency on the ball $|\lambda |\lesssim t^{-\frac{1}{2}}$. Since the measure dt/t gives unit mass to each interval $[2^{-(i+1)},2^{-i}]$, the operator $Q_t^{(b)}$ is a multiplier that is approximately localized at ‘frequencies’ of size $t^{-\frac{1}{2}}$. However, this decomposition using a continuous parameter does not satisfy the perfect cancellation property $\Delta _i\Delta _j=0$ for $|i-j|>1$, but the identity

$$\begin{aligned} Q_t^{(b)}Q_s^{(b)}=\left( \frac{ts}{(t+s)^2}\right) ^bQ_{t+s}^{(2b)} \end{aligned}$$

for any $s,t\in (0,1)$. The parameter b encodes a ‘degree’ of cancellation. In order to deal with time approximation, define for $m\in L^1({\mathbb {R}})$, with support in ${\mathbb {R}}_+$, the convolution operator

$$\begin{aligned} m^\star (f)(\tau ):=\int _0^\infty m(\tau -\sigma )f(\sigma )\mathrm d\sigma \quad \text {and}\quad m_t(\cdot ):=\frac{1}{t}m\left( \frac{\cdot }{t}\right) \end{aligned}$$

for $\tau \in {\mathbb {R}}$ and a positive scaling parameter t. Given $I=(i_1,\ldots ,i_n)\in \{1,\ldots ,\ell \}^n$, define the $n^\text {th}$-oder differential operator

$$\begin{aligned} V_I:=V_{i_n}\ldots V_{i_1}. \end{aligned}$$

We say that a family $({\mathcal {Q}}_t)_{t\in (0,1]}$ of operators is Gaussian if each the kernel of each ${\mathcal {Q}}_t$ is bounded pointwisely by the reference Gaussian kernel ${\mathcal {G}}_t$. (We do not recall its explicit expression here and refer the reader to Sect. 3.2 of [3]. It behaves as one expects.)

Definition

Let $a\in \llbracket 0,2b\rrbracket$. We define the standard collection $\mathsf {StGC}^a$ of operators with cancellation of order a as the family of operators

$$\begin{aligned} \left( \big (t^{\frac{|I|}{2}}V_I\big )\big (tL\big )^{\frac{j}{2}}P_t^{(c)}\otimes \varphi _t^\star \right) _{t\in (0,1]} \end{aligned}$$

where $a=|I|+j+2k$, $c\in \llbracket 1,b\rrbracket$ and $\varphi$ a smooth function supported in $[2^{-1},2]$ with bounded first derivative by 1 such that

$$\begin{aligned} \int \tau ^i\varphi (\tau )\mathrm d\tau =0\quad \text {for every }0\le i\le k-1. \end{aligned}$$

These operators are uniformly bounded in $L^p({\mathcal {M}})$ for every $p\in [1,\infty ]$, as functions of the parameter $t\in (0,1]$. We also set

$$\begin{aligned} \mathsf {StGC}^{[0,2b]}:=\bigcup _{0\le a\le 2b}\mathsf {StGC}^a. \end{aligned}$$

A standard family of operator ${\mathcal {Q}}\in \mathsf {StGC}^a$ can be seen as a bounded map $t\mapsto {\mathcal {Q}}_t$ from (0, 1] to the space of bounded linear operator on $L^p({\mathcal {M}})$. Since $V_iV_j\ne V_jV_i$, the operators $V_i$ do not commute with L so

$$\begin{aligned} V_IL^be^{-tL}\ne L^be^{-tL}V_I. \end{aligned}$$

We introduce for the needs of the next proposition the notation

$$\begin{aligned} \Big (V_I\psi (L)\Big )^\bullet :=\psi (L)V_I \end{aligned}$$

for any holomorphic function $\psi$. This notation is not related to any notion of duality.

Proposition 11

Consider ${\mathcal {Q}}^1\in \mathsf {StGC}^{a_1}$ and ${\mathcal {Q}}^2\in \mathsf {StGC}^{a_2}$ two standard collections with cancellation. Then for every $s,t\in (0,1]$, the composition ${\mathcal {Q}}_s^1\circ {\mathcal {Q}}_t^{2\bullet }$ has a kernel pointwisely bounded by

$$\begin{aligned} \begin{aligned} \Big |K_{{\mathcal {Q}}_s^1\circ {\mathcal {Q}}_t^{2\bullet }}(e,e')\Big |&\lesssim \left\{ \left( \frac{s}{t}\right) ^{\frac{a_1}{2}}{} \mathbf{1}_{s<t} + \left( \frac{t}{s}\right) ^{\frac{a_2}{2}}{} \mathbf{1}_{s\ge t}\right\} {\mathcal {G}}_{t+s}(e,e') \\&\lesssim \left( \frac{ts}{(s+t)^2}\right) ^{\frac{a}{2}}{\mathcal {G}}_{t+s}(e,e') \end{aligned} \end{aligned}$$

(A.1)

with $a:=\min (a_1,a_2)$.

Estimate (A.1) encodes a cancellation property that is in our setting the counterpart of the property $\Delta _i\Delta _j=0$ for $\vert i-j\vert > 1$. We also need operators that are not in the standard form but still have a useful cancellation property.

Definition

Let $a\in \llbracket 0,2b\rrbracket$. We define the collection ${{GC}^{a}}$ of operators with cancellation of order a as the set of families of Gaussian operators ${\mathcal {Q}}$ such as the following property holds. For every $s,t\in (0,1]$ and every $\mathsf S\in \mathsf {StGC}^{a'}$ with $a<a'\le 2b$, the composition ${\mathcal {Q}}_s\circ \mathsf S_t^\bullet$ has a kernel pointwisely bounded by

$$\begin{aligned} \left| K_{{\mathcal {Q}}_s\circ \mathsf S_t^\bullet }(e,e')\right| \lesssim \left( \frac{ts}{(t+s)^2}\right) ^{\frac{a}{2}}{\mathcal {G}}_{t+s}(e,e'). \end{aligned}$$

Definition

Given any $\alpha \in (-3,3)$, we define the parabolic Hölder spaces ${\mathcal {C}}^\alpha ({\mathcal {M}})$ as the set of distribution $f\in {\mathcal {D}}'({\mathcal {M}})$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal {C}}^\alpha }:=\left\| e^{-L}f\right\| _{L^\infty }+\sup _{\underset{|\alpha |<k\le 2b}{{\mathcal {Q}}\in \mathsf {StGC}^k}}\sup _{t\in (0,1]}t^{-\frac{\alpha }{2}}\Vert {\mathcal {Q}}_tf\Vert _{L^\infty }<\infty . \end{aligned}$$

1.2 A.2 Parabolic paraproducts, correctors and commutators

The Paley-Littlewood decomposition can be used to describe a product as

$$\begin{aligned} fg&=\lim _{n\rightarrow \infty }S_n(f)S_n(g) \\&=\sum _{i<j-2}\Delta _i(f)\Delta _j(g)+\sum _{|i-j|\le 1}\Delta _i(f)\Delta _j(g)+\sum _{i>j+1}\Delta _i(f)\Delta _j(g) \\&=\sum _i\Delta _{<i}(f)\Delta _i(g)+\sum _{|i-j|\le 1}\Delta _i(f)\Delta _j(g)+\sum _i\Delta _i(f)\Delta _{<i}(g) \\&=P_f^0g + \Pi ^0(f,g) + P_g^0f. \end{aligned}$$

The paraproducts $P_f^0g$ and $P_g^0f$ are always well-defined unlike the resonant term $\Pi ^0(f,g)$. We use here a slightly different identity

$$\begin{aligned} fg= & {} \lim _{t\rightarrow 0}{\mathcal {P}}_t^{(b)}\Big ({\mathcal {P}}_t^{(b)}f\cdot {\mathcal {P}}_t^{(b)}g\Big ) \nonumber \\= & {} \int _0^1\left\{ {\mathcal {Q}}_t^{(b)}\big ({\mathcal {P}}_t^{(b)}f\cdot {\mathcal {P}}_t^{(b)}g\big )+{\mathcal {P}}_t^{(b)}\big ({\mathcal {Q}}_t^{(b)}f\cdot {\mathcal {P}}_t^{(b)}g\big )+{\mathcal {P}}_t^{(b)}\big ({\mathcal {P}}_t^{(b)}f\cdot {\mathcal {Q}}_t^{(b)}g\big )\right\} \nonumber \\&\times \frac{dt}{t} +{\mathcal {P}}_1^{(b)}\Big ({\mathcal {P}}_1^{(b)}f\cdot {\mathcal {P}}_1^{(b)}g\Big ), \end{aligned}$$

(A.2)

which corresponds to writing

$$\begin{aligned} fg=\lim _{n\rightarrow \infty }S_n\big (S_n(f)S_n(g)\big ). \end{aligned}$$

Since ${\mathcal {P}}_t^{(b)}$ plays the role of $\Delta _{<i}$ and ${\mathcal {Q}}_t^{(b)}$ the role of $\Delta _i$ we want to manipulate this expression to get terms of the following forms

$$\begin{aligned}&\int _0^1{\mathcal {P}}_t^{1\bullet }\left( {\mathcal {Q}}_t^1f\cdot {\mathcal {Q}}_t^2g\right) \frac{dt}{t},\quad \text {or}\quad \int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {Q}}_t^2f\cdot {\mathcal {P}}_t^1g\right) \frac{dt}{t},\quad \text {and}\\&\int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {P}}_t^1f\cdot {\mathcal {Q}}_t^2g\right) \frac{dt}{t}, \end{aligned}$$

where ${\mathcal {Q}}_1,{\mathcal {Q}}_2\in \mathsf {StGC}^c$ encode some cancellation, so $c>0$, and ${\mathcal {P}}_1\in \mathsf {StGC}^{[0,d]}$ can encode no cancellation. This is done using repeatedly the Leibnitz rule $V_i(fg)=V_i(f)g+fV_i(g)$. We have for instance

$$\begin{aligned}&\int _0^1{\mathcal {P}}_t^{(b)}\bigg (b^{-1}(tL){\mathcal {Q}}_t^{(b-1)}f\cdot {\mathcal {P}}_t^{(b)}g\bigg )\frac{dt}{t} \\&\quad = b^{-1}\int _0^1{\mathcal {P}}_t^{(b)}(tL)\left( {\mathcal {Q}}_t^{(b-1)}f\cdot {\mathcal {P}}_t^{(b)}g\right) \frac{dt}{t}\\&\qquad - b^{-1}\int _0^1{\mathcal {P}}_t^{(b)}\left( {\mathcal {Q}}_t^{(b-1)}f\cdot (tL){\mathcal {P}}_t^{(b)}g\right) \frac{dt}{t} \\&\qquad - 2b^{-1}\sum _{i=1}^\ell \int _0^1{\mathcal {P}}_t^{(b)}(\sqrt{t}V_i)\left( {\mathcal {Q}}_t^{(b-1)}f\cdot (\sqrt{t}V_i){\mathcal {P}}_t^{(b)}g\right) \frac{dt}{t} \end{aligned}$$

where we ‘take’ some cancellation from ${\mathcal {Q}}_t^{(b)}$ to the other terms. Starting from identity (A.2) repeated use of this kind of decompositions allows to rewrite the product fg as

$$\begin{aligned} fg = \mathsf {P}_fg + \varvec{\Pi }(f,g) + \mathsf {P}_gf, \end{aligned}$$

where $\mathsf {P}_fg$ is a linear combination of terms of the form

$$\begin{aligned} \int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {P}}_t^1f\cdot {\mathcal {Q}}_t^2g\right) \frac{dt}{t}, \end{aligned}$$

and $\varvec{\Pi }(f,g)$ is a linear combination of terms of the form

$$\begin{aligned} \int _0^1{\mathcal {P}}_t^{1\bullet }\left( {\mathcal {Q}}_t^1f\cdot {\mathcal {Q}}_t^2g\right) \frac{dt}{t}, \end{aligned}$$

with ${\mathcal {Q}}^1,{\mathcal {Q}}^2\in \mathsf {StGC}^{\frac{b}{2}}$ and ${\mathcal {P}}^1\in \mathsf {StGC}^{[0,2b]}$, up to the smooth term ${\mathcal {P}}_1^{(b)}\left( {\mathcal {P}}_1^{(b)}f\cdot {\mathcal {P}}_1^{(b)}g\right)$. All the details on this construction and the classical estimates on the paraproduct $\mathsf {P}$ and the resonant $\varvec{\Pi }$ operators can be found in Sect. 4 of [3]. It is useful to introduce the conjugated paraproduct operator

$$\begin{aligned} {\widetilde{\mathsf {P}}}_fg = {\mathscr {L}}^{-1}\big (\mathsf {P}_f({\mathscr {L}}g)\big ) \end{aligned}$$

for any functions/distributions f and g. One can show that ${\widetilde{\mathsf {P}}}_fg$ is given as a linear combination of operators of the form

$$\begin{aligned} \int _0^1{{\widetilde{{\mathcal {Q}}}}}_t^{1\bullet }\left( {\mathcal {P}}_t^1f\cdot {\mathcal {Q}}_t^2\right) \frac{dt}{t} \end{aligned}$$

with ${{\widetilde{{\mathcal {Q}}}}}^1\in \mathsf {GC}^{\frac{b}{4}-2}$, ${\mathcal {Q}}^2\in \mathsf {StGC}^{\frac{b}{2}}$ and ${\mathcal {P}}^1\in \mathsf {StGC}^{[0,2b]}$. The only difference is that ${{\widetilde{{\mathcal {Q}}}}}^1$ is not given by a standard form but still encodes some cancellation. This is however sufficient for $\widetilde{\mathsf{P}}$ to enjoy the same continuity properties as $\mathsf P$. (See again Sect. 4 of [4].)

The study of semilinear singular SPDEs using paracontrolled calculus relies on a number of continuity estimate for different operators. We recall three of them here and refer the reader to [4] for a thorough account. Define the $\mathsf E$-type operator

$$\begin{aligned} \mathsf {C}(a,b,c):=\varvec{\Pi }\Big ({\widetilde{\mathsf {P}}}_ab,c\Big )-a\varvec{\Pi }\big (b,c\Big ) \end{aligned}$$

and its iterate

$$\begin{aligned} \mathsf {C}\Big ((a,b),c,d\Big ):=\mathsf {C}\Big ({\widetilde{\mathsf {P}}}_ab,c,d\Big )-a\mathsf {C}\Big (b,c,d\Big ). \end{aligned}$$

Proposition 12

The following two facts hold true.

• Let $\alpha \in (0,1)$ and $\beta ,\gamma \in (-3,3)$ such that

  $$\begin{aligned} \beta +\gamma<0\quad \text {and}\quad 0<\alpha +\beta +\gamma <1. \end{aligned}$$

  Then the corrector $\mathsf {C}$ has a unique extension as a continuous operator from ${\mathcal {C}}^\alpha \times {\mathcal {C}}^\beta \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\alpha +\beta +\gamma }$.

• Let $\alpha _1,\alpha _2\in (0,1)$ and $\beta ,\gamma \in (-3,3)$ such that

  $$\begin{aligned} \alpha _1+\beta +\gamma<0,\quad \alpha _2+\beta +\gamma<0\quad \text {and}\quad 0<\alpha _1+\alpha _2+\beta +\gamma <1. \end{aligned}$$

  Then the iterated corrector $\mathsf {C}$ has a unique extension as a continuous operator from ${\mathcal {C}}^{\alpha _1}\times {\mathcal {C}}^{\alpha _2}\times {\mathcal {C}}^\beta \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\alpha _1+\alpha _2+\beta +\gamma }$.

Note that the Hölder regularity exponent of the first argument in the corrector $\mathsf {C}$ has to be less than 1 in the above statement. In order to gain more information from a regularity exponent in the interval (1, 2) one needs to consider the refined corrector given for any $e\in {\mathcal {M}}$ by

$$\begin{aligned} \mathsf {C}_{(1)}\Big (a,b,c\Big )(e):=\mathsf {C}\Big (a,b,c\Big )(e)-\sum _{i=1}^\ell \gamma _i\big (V_ia\big )(e)\varvec{\Pi }\Big ({\widetilde{\mathsf {P}}}_{\delta _i(e,\cdot )}b,c\Big )(e) \end{aligned}$$

where the functions $\delta _i(\cdot )$ are given by

$$\begin{aligned} \delta _i(e,e') := \chi \big (d(x,x')\big )\langle V_i(x),\pi _{x,x'}\rangle _{T_xM}, \qquad e=(\tau ,x),\;e'=(\tau ',x'), \end{aligned}$$

with $\chi$ a smooth non-negative function on $[0,+\infty )$ equal to 1 in a neighbourhood of 0 with $\chi (r)=0$ for $r\ge r_m$, the injectivity radius of the compact Riemannian manifold M, and $\pi _{x,x'}$ a tangent vector of $T_xM$ of length d(x, y) whose associated geodesic reaches y at time 1. The functions $\gamma _i$ are defined from the identity

$$\begin{aligned} \nabla f = \sum _{i=1}^\ell \gamma _i(V_if)V_i, \end{aligned}$$

for all smooth real-valued functions f on M.

Proposition 13

Let $\alpha \in (1,2)$ and $\beta ,\gamma \in (-3,3)$ such that

$$\begin{aligned} \alpha +\beta +\gamma >0\quad \text {and}\quad \beta +\gamma <0. \end{aligned}$$

Then the operator $\mathsf {C}_{(1)}$ has a unique extension as a continuous operator from ${\mathcal {C}}^\alpha \times {\mathcal {C}}^\beta \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\alpha +\beta +\gamma }$.

Set

$$\begin{aligned} \mathsf {D}(a,b,c)&:= \varvec{\Pi }\big ({\widetilde{\mathsf {P}}}_ab,c\big ) - \mathsf {P}_a\big (\varvec{\Pi }(b,c)\big ), \\ \mathsf {R}(a,b,c)&:= \mathsf {P}_a({\widetilde{\mathsf {P}}}_bc) - \mathsf {P}_{ab}c, \\ \mathsf {R}^\circ (a,b,c)&:= \mathsf {P}_a(\mathsf {P}_bc) - \mathsf {P}_{ab}c. \end{aligned}$$

Proposition 14

The following two facts hold true.

• Let $\alpha ,\beta ,\gamma \in (0,3)$. Then the commutator $\mathsf {D}$ is continuous from ${\mathcal {C}}^\alpha \times {\mathcal {C}}^\beta \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\alpha +\beta +\gamma }$.

• Let $\beta \in (0,1)$ and $\gamma \in (-3,3)$ such that $\beta +\gamma \in (-3,3)$. Then the operators $\mathsf {R}$ and $\mathsf {R}^\circ$ are continuous from $L^\infty \times {\mathcal {C}}^\beta \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\beta +\gamma }$.

• Let $\alpha ,\beta \in (0,1/2)$ and $\gamma \in (-3,3)$. Then the operator $\mathsf {R}^\circ$ is continuous from ${\mathcal {C}}^\alpha \times {\mathcal {C}}^\beta \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\alpha +\beta +\gamma }$.

We also need continuity estimates on iterates of the operator $\mathsf {R}^\circ$. However in this case the expansion rule isdifferent depending on which argument we expand.

Proposition 15

The following two facts hold true.

• Let $\alpha _1,\alpha _2\in (0,1)$ and $\gamma \in (-3,3)$. Then the operator

  $$\begin{aligned} \mathsf {R}^\circ \big ((a_1,a_2),b,c\big ) := \mathsf {R}^\circ \big ({\widetilde{\mathsf {P}}}_{a_1}a_2,b,c\big ) - \mathsf {P}_{a_1}\mathsf {R}^\circ (a_2,b,c) \end{aligned}$$

  is continuous from ${\mathcal {C}}^{\alpha _1}\times {\mathcal {C}}^{\alpha _2}\times L^\infty \times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\alpha _1+\alpha _2+\gamma }$.

• Let $\beta _1,\beta _2\in (0,1)$ and $\gamma \in (-3,3)$. Then the operator

  $$\begin{aligned} \mathsf {R}^\circ \big (a,(b_1,b_2),c\big ) := \mathsf {R}^\circ \big (a,{\widetilde{\mathsf {P}}}_{b_1}b_2,c\big ) - \mathsf {R}^\circ (ab_1,b_2,c) \end{aligned}$$

  is continuous from $L^\infty \times {\mathcal {C}}^{\beta _1}\times {\mathcal {C}}^{\beta _2}\times {\mathcal {C}}^\gamma$ to ${\mathcal {C}}^{\beta _1+\beta _2+\gamma }$.

B-Correctors and commutators

In order to simplify the notation we write here $\Vert \cdot \Vert _\alpha$ for $\Vert \cdot \Vert _{{\mathcal {C}}^\alpha }$. The proofs of the corrector estimates follow the line of reasoning of similar estimates proved in [4]. Recall from Sect.  2.2.1 the definitions of the operators

$$\begin{aligned} \mathsf{C}_L^-\Big (a_1,a_2,b\Big )&= \mathsf {P}_{L{\widetilde{\mathsf {P}}}_{a_1}a_2}b-a_1\mathsf {P}_{La_2}b, \\ \mathsf{C}_L^+\Big (a,b_1,b_1\Big )&= \mathsf {P}_{La}\Big ({\widetilde{\mathsf {P}}}_{b_1}b_2\Big )-b_1\mathsf {P}_{La}b_2, \\ \mathsf{C}_L\Big (a_1,a_2,b\Big )&= \varvec{\Pi }\Big (L{\widetilde{\mathsf {P}}}_{a_1}a_2,b\Big )-a_1\varvec{\Pi }\Big (La_2,b\Big ). \end{aligned}$$

Proof of Theorem 3

We give here the details for the continuity estimate on $\mathsf {C}_L$ and explain how to adapt the proof for $\mathsf {C}_L^-,\mathsf {C}_L^+,\mathsf {C}_{V_i},\mathsf {C}_{V_i}^-$ and $\mathsf {C}_{V_i}^+$.

We want to compute the regularity of $\mathsf {C}_L(a_1,a_2,b)$ using a family ${\mathcal {Q}}$ of $\mathsf {StGC}^r$ with $r>|\alpha _1+\alpha _2+\beta -2|$. Recall that a term $\varvec{\Pi }(La,b)$ can be written as a linear combination of terms of the form

$$\begin{aligned} \int _0^1{\mathcal {P}}_t^{1\bullet }({\mathcal {Q}}_t^1(tL)a\cdot {\mathcal {Q}}_t^2b)\frac{dt}{t^2}, \end{aligned}$$

while ${\widetilde{\mathsf {P}}}_ba$ is a linear combination of terms of the form

$$\begin{aligned} \int _0^1{{\widetilde{{\mathcal {Q}}}}}_t^{3\bullet }\big ({{\widetilde{{\mathcal {Q}}}}}_t^4a\cdot {\mathcal {P}}_t^2b\big )\frac{dt}{t} \end{aligned}$$

with ${\mathcal {Q}}^1,{\mathcal {Q}}^2,{{\widetilde{{\mathcal {Q}}}}}^4\in \mathsf {StGC}^{\frac{3}{2}}$, ${{\widetilde{{\mathcal {Q}}}}}^3\in \mathsf {GC}^{\frac{3}{2}}$ and ${\mathcal {P}}^1,{\mathcal {P}}^2\in \mathsf {StGC}^{[0,3]}$. For the terms where ${\mathcal {P}}^2\in \mathsf {StGC}^{[1,3]}$, we already have the correct regularity since

$$\begin{aligned}&\int _0^1\int _0^1{\mathcal {Q}}_u{\mathcal {P}}_t^{1\bullet }\left( {\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( {\mathcal {P}}_s^2a_1\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b\right) \frac{ds}{s}\frac{dt}{t^2} \\&\quad \lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \int _0^1\int _0^1\left( \frac{ut}{(t+u)^2}\right) ^{\frac{r}{2}}\left( \frac{ts}{(s+t)^2}\right) ^{\frac{3}{2}}s^{\frac{\alpha _1+\alpha _2}{2}}t^{\frac{\beta }{2}}\frac{ds}{s}\frac{dt}{t^2} \\&\quad \lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \ u^{\frac{\alpha _1+\alpha _2+\beta -2}{2}} \end{aligned}$$

using that $\alpha _1\in (0,1)$. We only consider ${\mathcal {P}}^2\in \mathsf {StGC}^0$ for the remainder of the proof. For all $e\in {\mathcal {M}}$, we have

$$\begin{aligned}&\mathsf {C}_L(a_1,a_2,b)(e)=\varvec{\Pi }\left( L{\widetilde{\mathsf {P}}}_{a_1}a_2,b\right) (e)\\&-a_1(e)\cdot \varvec{\Pi }(La_2,b)(e)=\varvec{\Pi }\left( L{\widetilde{\mathsf {P}}}_{a_1}a_2-a_1(e)\cdot La_2,b\right) (e), \end{aligned}$$

since $\varvec{\Pi }$ is bilinear and $a_1(e)$ is a scalar. This yields that $\mathsf {C}_L(a_1,a_2,b)(e)$ is a linear combination of terms of the form

$$\begin{aligned} \int _0^1\int _0^1{\mathcal {P}}_t^{1\bullet }\bigg ({\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \left( {\mathcal {P}}_s^2a_1-a_1(e)\right) \cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b\bigg )(e)\,\frac{ds}{s}\frac{dt}{t^2} \end{aligned}$$

using that $\displaystyle \int _0^1L{{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }{{\widetilde{{\mathcal {Q}}}}}_s^4\frac{ds}{s}=L$ up to smooth terms. This gives $\big ({\mathcal {Q}}_u\mathsf {C}_L(a_1,a_2,b)\big )(e)$ as a linear combination of terms of the form

$$\begin{aligned}&\int K_{{\mathcal {Q}}_u}(e,e') {\mathcal {P}}_t^{1\bullet }\bigg ({\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \left( {\mathcal {P}}_s^2a_1-a_1(e')\right) \cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b\bigg )(e')\,\frac{ds}{s}\frac{dt}{t^2}\nu (de') \\&\quad = \int K_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\bigg ({\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \left( {\mathcal {P}}_s^2a_1-a_1(e'')\right) \cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b \bigg )\\&\quad \qquad \times \,(e'')\frac{ds}{s}\frac{dt}{t^2}\nu (de')\nu (de'') \\&\qquad +\int \int _0^u K_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\Big (a_1(e'')-a_1(e')\Big )\left( {\mathcal {Q}}_t^1(tL)a_2\cdot {\mathcal {Q}}_t^2b\right) \\ {}&\quad \qquad \times \,(e'')\frac{dt}{t^2}\nu (de')\nu (de'') \\&\qquad +\int \int _u^1 K_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\Big (a_1(e'')-a_1(e')\Big )\left( {\mathcal {Q}}_t^1(tL)a_2\cdot {\mathcal {Q}}_t^2b\right) \\ {}&\quad \qquad \times \,(e'')\frac{dt}{t^2}\nu (de')\nu (de'') \\&\quad =: A+B+C. \end{aligned}$$

The term A is bounded using cancellations properties. We have

$$\begin{aligned} |A|&= \int K_{{\mathcal {Q}}_u{\mathcal {P}}_t^{1\bullet }}(e,e')\bigg ({\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \left( {\mathcal {P}}_s^2a_1-a_1(e')\right) \cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b\bigg )\\ {}&\quad \times \,(e') \frac{ds}{s}\frac{dt}{t^2}\nu (de') \\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \left( \int _0^u\int _0^1\left( \frac{st}{(s+t)^2}\right) ^{\frac{3}{2}}(s+t)^{\frac{\alpha _1}{2}}s^{\frac{\alpha _2}{2}}t^{\frac{\beta }{2}}\frac{ds}{s}\frac{dt}{t^2}\right. \\&\quad \left. +\int _u^1\int _0^1\left( \frac{tu}{(t+u)^2}\right) ^{\frac{r}{2}}\left( \frac{st}{(s+t)^2}\right) ^{\frac{3}{2}}(s+t)^{\frac{\alpha _1}{2}}s^{\frac{\alpha _2}{2}}t^{\frac{\beta }{2}}\frac{ds}{s}\frac{dt}{t^2}\right) \\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \ u^{\frac{\alpha _1+\alpha _2+\beta -2}{2}}, \end{aligned}$$

using that $\alpha _1\in (0,1),{\mathcal {P}}^2\in \mathsf {StGC}^0$ and $(\alpha _1+\alpha _2+\beta -2)>0$.

For the term B, we have

$$\begin{aligned} |B|&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \int _{e',e''}\\ {}&\quad \times \,\int _0^uK_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\rho (e',e'')^{\alpha _1}t^{\frac{\alpha _2+\beta }{2}}\frac{dt}{t^2}\nu (de')\nu (de'')\\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \int _0^ut^{\frac{\alpha _1+\alpha _2+\beta -2}{2}}\frac{dt}{t}\\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \ u^{\frac{\alpha _1+\alpha _2+\beta -2}{2}}, \end{aligned}$$

using again that $\alpha _1\in (0,1)$ and $(\alpha _1+\alpha _2+\beta -2)>0$.

Finally for C, we also use cancellations properties to get

$$\begin{aligned} |C|&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \bigg \{\int _{e',e''}\int _u^1K_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\\&\quad \times \,\Big |a_1(e)-a_1(e')\Big |t^{\frac{\alpha _2+\beta }{2}}\frac{dt}{t^2}\nu (de')\nu (de'') \\&\quad + \int _{e',e''}\int _u^1K_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\Big |a_1(e')-a_1(e'')\Big |t^{\frac{\alpha _2+\beta }{2}}\frac{dt}{t^2}\nu (de')\nu (de'')\bigg \} \\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \bigg \{\int _{e',e''}\int _u^1K_{{\mathcal {Q}}_u}(e,e')K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\rho (e,e')^{\alpha _1}t^{\frac{\alpha _2+\beta }{2}}\\ {}&\quad \times \,\frac{dt}{t^2}\nu (de')\nu (de'') + \int _{e',e''}\int _u^1K_{{\mathcal {Q}}_u}(e,e')\\ {}&\quad \times \,K_{{\mathcal {P}}_t^{1\bullet }}(e',e'')\rho (e',e'')^{\alpha _1}t^{\frac{\alpha _2+\beta }{2}}\frac{dt}{t^2}\nu (de')\nu (de'')\bigg \}\\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \bigg \{u^{\frac{\alpha _1}{2}}\int _u^1t^{\frac{\alpha _2+\beta -2}{2}}\frac{dt}{t}+\int _u^1 \left( \frac{tu}{(t+u)^2}\right) ^{\frac{r}{2}}t^{\frac{\alpha _1+\alpha _2+\beta -2}{2}}\frac{dt}{t}\bigg \} \\&\lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \ u^{\frac{\alpha _1+\alpha _2+\beta -2}{2}}, \end{aligned}$$

using that $\alpha _1\in (0,1)$ and $(\alpha _2+\beta -2)<0$. In the end, we have

$$\begin{aligned} \Big \Vert {\mathcal {Q}}_u\mathsf {C}_L(a_1,a_2,b)\Big \Vert _\infty \lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \ u^{\frac{\alpha _1+\alpha _2+\beta -2}{2}} \end{aligned}$$

uniformly in $u\in (0,1]$, so the proof is complete for $\mathsf {C}_L$. The proofs for $\mathsf {C}_L^<$ and $\mathsf {C}_L^>$ are then easy to obtain since $\mathsf {P}_{La}b$ has the same form as $\varvec{\Pi }(La,b)$. Indeed, $\mathsf {P}_{La}b$ is a linear combination of

$$\begin{aligned} \int _0^1{\mathcal {Q}}_t^{1\bullet }\Big ({\mathcal {P}}_t^1(tL)a\cdot {\mathcal {Q}}_t^2b\Big )\,\frac{dt}{t^2} \end{aligned}$$

where ${\mathcal {Q}}^1,{\mathcal {Q}}^2\in \mathsf {StGC}^{\frac{3}{2}},{\mathcal {P}}^1\in \mathsf {StGC}^{[0,3]}$ and we have $\big ({\mathcal {P}}_t^1(tL)\big )_{0<t\le 1}\in \mathsf {StGC}^2$.

The proofs for $\mathsf {C}_{V_i},\mathsf {C}_{V_i}^-$ and $\mathsf {C}_{V_i}^+$ also follow from the same argument and using the Leibniz rule as for the corrector $\mathsf {C}_\partial$ used in Sect. 3.3 of [4] to solve the generalised (KPZ) equation. $\square$

Proof of Theorem 4

For the continuity estimate of $\mathsf {C}_{L,(1)}$, we also want to compute the regularity using a family ${\mathcal {Q}}$ of $\mathsf {StGC}^r$ with $r>|\alpha _1+\alpha _2+\beta -2|$. Again a term $\varvec{\Pi }(La,b)$ can be written as a linear combination of terms of the form

$$\begin{aligned} \int _0^1{\mathcal {P}}_t^{1\bullet }\big ({\mathcal {Q}}_t^1(tL)a\cdot {\mathcal {Q}}_t^2b\big )\frac{dt}{t^2}, \end{aligned}$$

while ${\widetilde{\mathsf {P}}}_ba$ is a linear combination of terms of the form

$$\begin{aligned} \int _0^1{{\widetilde{{\mathcal {Q}}}}}_t^{3\bullet }({{\widetilde{{\mathcal {Q}}}}}_t^4a\cdot {\mathcal {P}}_t^2b)\frac{dt}{t}, \end{aligned}$$

with ${\mathcal {Q}}^1,{\mathcal {Q}}^2,{{\widetilde{{\mathcal {Q}}}}}^3,{{\widetilde{{\mathcal {Q}}}}}^4\in \mathsf {StGC}^{\frac{3}{2}}$ and ${\mathcal {P}}^1,{\mathcal {P}}^2\in \mathsf {StGC}^{[0,3]}$. For the terms where ${\mathcal {P}}^2\in \mathsf {StGC}^{[2,3]}$, we already have the correct regularity since

$$\begin{aligned}&\int _0^1\int _0^1{\mathcal {Q}}_u{\mathcal {P}}_t^{1\bullet }\left( {\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( {\mathcal {P}}_s^2a_1\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b\right) \frac{ds}{s}\frac{dt}{t^2}\\&\quad \lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \int _0^1\int _0^1\left( \frac{ut}{(t+u)^2}\right) ^{\frac{r}{2}}\left( \frac{ts}{(s+t)^2}\right) ^{\frac{3}{2}}s^{\frac{\alpha _1+\alpha _2}{2}}t^{\frac{\beta }{2}}\frac{ds}{s}\frac{dt}{t^2}\\&\quad \lesssim \Vert a_1\Vert _{\alpha _1}\Vert a_2\Vert _{\alpha _2}\Vert b\Vert _\beta \ u^{\frac{\alpha _1+\alpha _2+\beta -2}{2}} \end{aligned}$$

using that $\alpha _1\in (1,2)$ so we only consider ${\mathcal {P}}^2\in \mathsf {StGC}^{[0,1]}$. For ${\mathcal {P}}^2\in \mathsf {StGC}^0$, we control it using the term $a_1\varvec{\Pi }(La_2,b)$ as in the proof of the continuity estimate of $\mathsf {C}$. We are left with

$$\begin{aligned}&\int {\mathcal {P}}_t^{1\bullet }\left( {\mathcal {Q}}_t^1(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \left( {\mathcal {P}}_s^2\Big (a_1-d\big ({\overline{u}}_0(e)\big )^{-1}\sum _{i=1}^\ell (V_ia_1)(e)\delta _i(\cdot ,e)\Big )\right) \cdot {{\widetilde{{\mathcal {Q}}}}}_s^4a_2\right) \cdot {\mathcal {Q}}_t^2b\right) \\&\quad \times \,(e)\frac{ds}{s}\frac{dt}{t^2} \end{aligned}$$

with ${\mathcal {P}}^2\in \mathsf {StGC}^1$. Then the result follows with the same proof using that ${\mathcal {P}}_s^21=0$ since it encodes some cancellation and the first order Taylor expansion

$$\begin{aligned} \left| a_1(e')-a_1(e)-d({\overline{u}}_0)^{-1}\sum _{i=1}^\ell (V_ia_1)(e)\delta _i(e',e)\right| \lesssim \rho (e,e')^\alpha . \end{aligned}$$

We let the reader prove the continuity resuls for $\mathsf {C}_{L,(1)}^-$ and $\mathsf {C}_{L,(1)}^+$; they can be proved by the same argument as above. $\square$

Proof of Theorems 5–8 / Elements

We give the proof for the continuity estimate on $\mathsf {L}$ and $\mathsf {L}_{(1)}$. We let the reader adapt the proof from [4] for the iterated operators of $\mathsf {L}$ since it relies on the same argument. The same holds for $\mathsf {V}_i(a,b)$ and its first iteration.

We want to compute the regularity of $\mathsf {L}(a,b)=L{\widetilde{\mathsf {P}}}_ab-\mathsf {P}_aLb$ using a family ${\mathcal {Q}}\in \mathsf {StGC}^r$ with $r>|\alpha +\beta -2|$. We write ${\widetilde{\mathsf {P}}}_ab$ and $\mathsf {P}_ab$ respectively as linear combination of

$$\begin{aligned} \int _0^1{{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( {\mathcal {P}}_s^2a\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\right) \frac{ds}{s}\quad \text {and}\quad \int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {P}}_t^1a\cdot {\mathcal {Q}}_t^2b\right) \frac{dt}{t} \end{aligned}$$

with ${\mathcal {Q}}^1,{\mathcal {Q}}^2,{{\widetilde{{\mathcal {Q}}}}}^4\in \mathsf {StGC}^{\frac{3}{2}},{{\widetilde{{\mathcal {Q}}}}}^3\in \mathsf {GC}^{\frac{3}{2}}$ and ${\mathcal {P}}^1,{\mathcal {P}}^2\in \mathsf {StGC}^{[0,3]}$. As done for $\mathsf {C}$, we only have to consider ${\mathcal {P}}^1,{\mathcal {P}}^2\in \mathsf {StGC}^0$ since the other terms already have the right regularity using that $\alpha \in (0,1)$. We consider a term

$$\begin{aligned} \int _0^1L{{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( {\mathcal {P}}_s^2a\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\right) \frac{ds}{s}-\int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {P}}_t^1a\cdot {\mathcal {Q}}_t^2(tL)b\right) \frac{dt}{t^2}. \end{aligned}$$

We use that $\displaystyle \int _0^1{\mathcal {Q}}_t^{1\bullet }{\mathcal {Q}}_t^2\frac{dt}{t}=\int _0^1{{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }{{\widetilde{{\mathcal {Q}}}}}_s^4\frac{ds}{s}=\text {Id}$, up to smooth term, to get

$$\begin{aligned}&\int _0^1\int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( {\mathcal {P}}_s^2a\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\right) -{\mathcal {P}}_t^1a\cdot {\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }{{\widetilde{{\mathcal {Q}}}}}_s^4b\right) \frac{dt}{t^2}\frac{ds}{s}\\&\quad =\int _0^1\int _0^1{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \big ({\mathcal {P}}_s^2a-{\mathcal {P}}_t^1a(\cdot )\big )\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\right) \right) \frac{dt}{t^2}\frac{ds}{s} \end{aligned}$$

where the variable of ${\mathcal {P}}_t^1a(\cdot )$ is frozen as before, in the sense that ${\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }$ does not act on it. Since $\alpha \in (0,1)$, we can use that for any $e,e'$

$$\begin{aligned} \left| {\mathcal {P}}_s^2a(e')-{\mathcal {P}}_t^1a(e)\right| \le \left| {\mathcal {P}}_s^2a(e')-a(e')\right| +|a(e')-a(e)|+\left| a(e)-{\mathcal {P}}_t^1a(e)\right| , \end{aligned}$$

to get

$$\begin{aligned}&\int _0^1\int _0^1{\mathcal {Q}}_u{\mathcal {Q}}_t^{1\bullet }\left( {\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\left( \big ({\mathcal {P}}_s^2a-{\mathcal {P}}_t^1a(\cdot )\big )\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\right) \right) \frac{dt}{t^2}\frac{ds}{s}\\&\quad \lesssim \Vert a\Vert _\alpha \Vert b\Vert _\beta \int _0^1\int _0^1\left( \frac{tu}{(t+u)^2}\right) ^{\frac{r}{2}}\left( \frac{st}{(s+t)^2}\right) ^{\frac{3}{2}}(t+s)^\alpha s^\beta \frac{dt}{t^2}\frac{ds}{s}\\&\quad \lesssim \Vert a\Vert _\alpha \Vert b\Vert _\beta \ u^{\frac{\alpha +\beta }{2}} \end{aligned}$$

which complete the proof for $\mathsf {L}(a,b)$.$\square$

We finally prove the estimate for the refined commutator $\mathsf {L}_{(1)}(a,b)$ that is given for any $e\in {\mathcal {M}}$ by

$$\begin{aligned} \mathsf {L}_{(1)}(a,b)(e)=\big (L{\widetilde{\mathsf {P}}}_ab\big )(e)-\big (\mathsf {P}_aLb\big )(e)-\sum _{i=1}^\ell \big (\mathsf {P}_{d({\overline{u}}_0)^{-1}V_ia)}^{(i)}b\big )(e). \end{aligned}$$

where

$$\begin{aligned} \displaystyle \big (\mathsf {P}_a^{(i)}b\big )(e)=\int _{e',e''}K(e;e',e'')a(e')\left( {\widetilde{\mathsf {P}}}_{\delta _i(\cdot ,e')}b\right) (e'')\nu (de')\nu (de''), \end{aligned}$$

with K the kernel of the bilinear operator $(a,b)\mapsto \mathsf {P}_ab$. As in the proof of $\mathsf {C}_{L,(1)}$, we are left with

$$\begin{aligned}&\int K_{{\mathcal {Q}}_t^{1\bullet }}(e,e') \bigg \{{\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\big ({\mathcal {P}}_s^2a\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\big ) \\&\qquad -\sum _{i=1}^\ell \big ({\mathcal {P}}_t^1(d(\overline{u_0})^{-1}V_ia)\big )(e')\cdot {\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet }\big ({\mathcal {P}}_s^2\delta _i(\cdot ,e')\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\big )\bigg \}(e') \, \frac{dt}{t^2}\frac{ds}{s}\nu (de') \\&\quad = \int K_{{\mathcal {Q}}_t^{1\bullet }}(e,e')\times \left( {\mathcal {Q}}_t^2(tL){{\widetilde{{\mathcal {Q}}}}}_s^{3\bullet } \left( {\mathcal {P}}_s^2\Big (a-\sum _{i=1}^\ell {\mathcal {P}}_t^1\big (d({\overline{u}}_0)a\big )(e')\big ) \delta _i(\cdot ,e')\Big )\cdot {{\widetilde{{\mathcal {Q}}}}}_s^4b\right) \right) \\&\qquad \times \,(e') \, \frac{dt}{t^2}\frac{ds}{s}\nu (de') \end{aligned}$$

with ${\mathcal {P}}^1,{\mathcal {P}}^2\in \mathsf {StGC}^1$. The result follows with the same proof using that ${\mathcal {P}}_s^21=0$ since it encodes some cancellation and the first order Taylor expansion for a.

C-Paracontrolled expansion

We use in the body of the text the following variation on the high order paracontrolled expansion formula from [4], Theorem 4 therein.

Theorem 16

Let $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a $C^4$ function and let u and v be respectively $C^\alpha$ and $C^{4\alpha }$ functions on $[0,T]\times {\mathbb {T}}^3$ with $\alpha \in (0,1)$. Then

$$\begin{aligned} f(u)v&=\mathsf {P}_{f'(u)v}u+\frac{1}{2}\Big \{\mathsf {P}_{f^{(2)}(u)v}u^2-2\mathsf {P}_{f^{(2)}(u)uv}u\Big \}\\&\quad +\frac{1}{3!}\Big \{\mathsf {P}_{f^{(3)}(u)v}u^3-3\mathsf {P}_{f^{(3)}(u)uv}u^2+3\mathsf {P}_{f^{(3)}(u)u^2v}u\Big \}+f_v(u)^\sharp \end{aligned}$$

for some remainder $f_v(u)^\sharp \in {\mathcal {C}}^{4\alpha }$.

Proof

We need to prove that

$$\begin{aligned} R&:=vf(u)-\mathsf {P}_{vf'(u)}u-\frac{1}{2}\Big \{\mathsf {P}_{vf^{(2)}(u)}u^2-2\mathsf {P}_{vf^{(2)}(u)u}u\Big \}\\&\quad -\frac{1}{3!}\Big \{\mathsf {P}_{vf^{(3)}(u)}u^3-3\mathsf {P}_{vf^{(3)}(u)u}u^2+3\mathsf {P}_{vf^{(3)}(u)u^2}u\Big \} \end{aligned}$$

is a $3\alpha$-Hölder function. Using that $\mathsf {P}_1vf(u)=vf(u)$ up to smooth term and that $\mathsf {P}_ab$ is the sum of terms of the form

$$\begin{aligned} \int _0^1{\mathcal {Q}}_t^{1\bullet }({\mathcal {Q}}_t^2a\cdot {\mathcal {P}}_t^1b)\frac{dt}{t} \end{aligned}$$

with ${\mathcal {Q}}^1,{\mathcal {Q}}^2\in \mathsf {StGC}^{\frac{3}{2}}$ and ${\mathcal {P}}^1\in \mathsf {StGC}^{[0,3]}$, R is a sum of terms of the form $\int _0^1{\mathcal {Q}}_t^{1\bullet }(r_t)\frac{dt}{t}$ with

$$\begin{aligned} r_t&:={\mathcal {Q}}_t^2\Big (vf(u)\Big )-{\mathcal {Q}}_t^2\Big (vf'(u)\Big ){\mathcal {P}}_t^1(u)-\frac{1}{2}{\mathcal {Q}}_t^2\Big (vf^{(2)}(u)\Big ){\mathcal {P}}_t^1(u^2)\\ {}&\quad +{\mathcal {Q}}_t^2\Big (vf^{(2)}(u)u\Big ){\mathcal {P}}_t^1(u )+\frac{1}{6}{\mathcal {Q}}_t^2\Big (vf^{(3)}(u)\Big ){\mathcal {P}}_t^1(u^3)\\&\quad +\frac{1}{2}{\mathcal {Q}}_t^2\Big (vf^{(3)}(u)u\Big ){\mathcal {P}}_t^1(u^2)-\frac{1}{2}{\mathcal {Q}}_t^2\Big (vf^{(3)}(u)u^2\Big ){\mathcal {P}}_t^1(u). \end{aligned}$$

We need to get a bound on $r_t$ in $L^\infty ({\mathcal {M}})$. We have for $e\in {\mathcal {M}}$

$$\begin{aligned} r_t(e)&=\int _{{\mathcal {M}}^2}K_{{\mathcal {Q}}_t^2}(e,e')K_{{\mathcal {P}}_t^1}(e,e'')\Big \{\Big (vf(u)\Big )(e')-\Big (vf'(u)\Big )(e')u(e'')\\&\quad -\frac{1}{2}\Big (vf^{(2)}(u)\Big )(e')u^2(e'')+\Big (vf^{(2)}(u)u\Big )(e')u(e'')+\frac{1}{6}\Big (vf^{(3)}(u)\Big )(e')u^3(e'')\\&\quad +\frac{1}{2}\Big (vf^{(3)}(u)u\Big )(e')u^2(e'')-\frac{1}{2}\Big (vf^{(3)}(u)u^2\Big )(e')u(e'')\Big \}\nu (de')\nu (de''). \end{aligned}$$

Using a Taylor expansion for f, we have

$$\begin{aligned} r_t(e)&=\int _{[0,1]^4}f^{(4)}\Big (u(e'')+s_4s_3s_2s_1\left( u(e')-u(e'')\right) \Big )\\ {}&\quad \times \,s_3s_2s_1\left( u(e')-u(e'')\right) ^4ds_4ds_3ds_2ds_1\\&\quad +v(e')\left( f(u(e''))+u(e'')f'(u(e''))+\frac{1}{2}u^2(e'')f^{(2)}(u(e''))\right. \\ {}&\quad \left. +\frac{1}{3!}u^3(e'')f^{(3)}(u(e''))\right) \\&=(1)+(2). \end{aligned}$$

For the first term, we have

$$\begin{aligned} (1)\le \Vert u\Vert _{\alpha }^4\ t^{\frac{4\alpha }{2}} \end{aligned}$$

and for the second term

$$\begin{aligned} (2)\le \Vert u\Vert _{L^\infty }\Vert v\Vert _{4\alpha }\ t^{\frac{4\alpha }{2}} \end{aligned}$$

which allows us to conclude. $\square$