Very similar computations give the right decomposition of the term $f^{(2)}(u) C^{\circ}(u, u, \zeta)$, once $C^{\circ}(u, u, \zeta)$ has been put in multiplicative form. We start from the identity

$$C^{\circ}(u, u, \zeta) = C^{\circ}(P_{u_1} Z_1, u, \zeta) + C^{\circ}(\mathbb{R}(1, u_1, Z_1), u, \zeta) + C^{\circ}(u^{(1)}, u, \zeta) + (5\alpha - 2)$$

with

$$u^{(1)} := u - \tilde{P}_{u_1} Z_1 - \tilde{P}_{u_2} Z_2$$

an element of $C^{3\alpha} \subset C^1$, since $3\alpha > 1$. Recall Definition 37 of the refined corrector, and observe that because of Theorem 10

$$C^{\circ}(u^{(1)}, u, \zeta) = u_1 C^{\circ}(u^{(1)}, Z_1, \zeta) + (5\alpha - 2),$$

since $3\alpha - 1 > 5\alpha - 2$. Then using the continuity result on the refined corrector $C_{(1)}$ from Theorem 6, and the fact that $\alpha > 5\alpha - 2$, one has

$$C^{\circ}(u^{(1)}, Z_1, \zeta) = \sum_{\ell=1}^{\ell_0} \gamma_{\ell} V_{\ell}(u^{(1)}) \Pi_{(1)}^{\ell}(Z_1, \zeta) + (5\alpha - 2). \quad (3.27)$$

The fact that $\Pi_{(1)}^{\ell}(Z_1, \zeta) \in \mathcal{L}(C^{2\alpha+1})$ is part of the data $\hat{\zeta}$, and it follows from the fact that $5\alpha - 2 > 0$, that the preceding sum is a (b)-term. This gives, from Proposition 14, the identity

$$C^{\circ}(u, u, \zeta) = u_i C^{\circ}(Z_i, u, \zeta) + C^{\circ}((u_i, Z_i), u, \zeta) + u_{111} C^{\circ}(\mathbb{R}(1, Z_1, Z_1), u, \zeta) + (b) + (5\alpha - 2).$$

The multiplicative decomposition of $C^{\circ}(u, u, \zeta)$ follows by an elementary computation.

(2.4) Last, we have the term $f'(u)\Pi(u, \zeta)$. We first put $\Pi(u, \zeta)$ in multiplicative form

$$\Pi(u, \zeta) = u_i \Pi(Z_i, \zeta) + u_{ij} C(Z_j, Z_i, \zeta) + u_{111} C((Z_1, Z_1), Z_1, \zeta) + C(u_1^\sharp, Z_1, \zeta) + (5\alpha - 2). \quad (3.28)$$

The term $C(u_1^\sharp, Z_1, \zeta)$ is treated using the refined corrector, as above, which gives a contribution

$$\sum_{\ell=1}^{\ell_0} f'(u) \gamma_{\ell} V_{\ell}(u_1^{\sharp}) \Pi_{(1)}^{\ell}(Z_1, \zeta) + (5\alpha - 2), \quad (3.29)$$
Very similar computations give the right decomposition of the term $f^{(2)}(u) C^{\circ}(u, u, \zeta)$, once $C^{\circ}(u, u, \zeta)$ has been put in multiplicative form. We start from the identity

$$C^{\circ}(u, u, \zeta) = C^{\circ}(P_{u_1} Z_1, u, \zeta) + C^{\circ}(\mathbb{R}(1, u_1, Z_1), u, \zeta) + C^{\circ}(u^{(1)}, u, \zeta) + (5\alpha - 2)$$

with

$$u^{(1)} := u - \tilde{P}_{u_1} Z_1 - \tilde{P}_{u_2} Z_2$$

an element of $C^{3\alpha} \subset C^1$, since $3\alpha > 1$. Recall Definition 37 of the refined corrector, and observe that because of Theorem 10

$$C^{\circ}(u^{(1)}, u, \zeta) = u_1 C^{\circ}(u^{(1)}, Z_1, \zeta) + (5\alpha - 2),$$

since $3\alpha - 1 > 5\alpha - 2$. Then using the continuity result on the refined corrector $C_{(1)}$ from Theorem 6, and the fact that $\alpha > 5\alpha - 2$, one has

$$C^{\circ}(u^{(1)}, Z_1, \zeta) = \sum_{\ell=1}^{\ell_0} \gamma_{\ell} V_{\ell}(u^{(1)}) \Pi_{(1)}^{\ell}(Z_1, \zeta) + (5\alpha - 2). \quad (3.27)$$

The fact that $\Pi_{(1)}^{\ell}(Z_1, \zeta) \in \mathcal{L}(C^{2\alpha+1})$ is part of the data $\hat{\zeta}$, and it follows from the fact that $5\alpha - 2 > 0$, that the preceding sum is a (b)-term. This gives, from Proposition 14, the identity

$$C^{\circ}(u, u, \zeta) = u_i C^{\circ}(Z_i, u, \zeta) + C^{\circ}((u_i, Z_i), u, \zeta) + u_{111} C^{\circ}(\mathbb{R}(1, Z_1, Z_1), u, \zeta) + (b) + (5\alpha - 2).$$

The multiplicative decomposition of $C^{\circ}(u, u, \zeta)$ follows by an elementary computation.

(2.4) Last, we have the term $f'(u)\Pi(u, \zeta)$. We first put $\Pi(u, \zeta)$ in multiplicative form

$$\Pi(u, \zeta) = u_i \Pi(Z_i, \zeta) + u_{ij} C(Z_j, Z_i, \zeta) + u_{111} C((Z_1, Z_1), Z_1, \zeta) + C(u_1^\sharp, Z_1, \zeta) + (5\alpha - 2). \quad (3.28)$$

The term $C(u_1^\sharp, Z_1, \zeta)$ is treated using the refined corrector, as above, which gives a contribution

$$\sum_{\ell=1}^{\ell_0} f'(u) \gamma_{\ell} V_{\ell}(u_1^{\sharp}) \Pi_{(1)}^{\ell}(Z_1, \zeta) + (5\alpha - 2), \quad (3.29)$$
for that term, in the analysis of $f'(u)\Pi(u, \zeta)$; this is a remainder term of (b)-type. Look at the term $f'(u)u_1\Pi(Z_1, \zeta)$; the other terms are dealt with similarly. One has

$$ 
\begin{aligned}
f'(u)u_1\Pi(Z_1, \zeta) &= P_{f'(u)u_1}\Pi(Z_1, \zeta) + P_{\Pi(Z_1, \zeta)}f'(u)u_1 + \Pi(f'(u)u_1, \Pi(Z_1, \zeta)) \\
&= P_{f'(u)u_1}\Pi(Z_1, \zeta) + P_{\Pi(Z_1, \zeta)}(P_{f'(u)}u_1 + P_{u_1}f'(u)) \\
&\quad + P_{\Pi(Z_1, \zeta)}\Pi(f'(u), u_1) \\
&\quad + f'(u)\Pi(u_1, \Pi(Z_1, \zeta)) + C^{\circ}(f'(u), u_1, \Pi(Z_1, \zeta)) \\
&\quad + u_1\Pi(f'(u), \Pi(Z_1, \zeta)) + C^{\circ}(u_1, f'(u), \Pi(Z_1, \zeta)) \\
&\quad + \Pi(\Pi(f'(u), u_1), \Pi(Z_1, \zeta)).
\end{aligned}
$$

The last six terms in the right hand side are of regularity $(4\alpha - 2)$; a first order expansion of $f'(u)$ and $u_1$ allows to put the in the right form (3.25). For the second and third terms in the right hand side, simply write

$$ P_{\Pi(Z_1, \zeta)}P_{f'(u)}u_1 = P_{f'(u)}P_{\Pi(Z_1, \zeta)}u_1 + S^{\circ}(\Pi(Z_1, \zeta), f'(u), u_1). $$

The last term has regularity $(4\alpha - 2)$, and a first order expansion of $f'(u)$ and $u_1$ allows to put the in the right form. Also we have

$$ 
\begin{aligned}
P_{f'(u)}P_{\Pi(Z_1, \zeta)}u_1 &= P_{f'(u)}P_{\Pi(Z_1, \zeta)}P_{u_1j}Z_j + (5\alpha - 2) \\
&= P_{f'(u)}P_{u_1j}P_{\Pi(Z_1, \zeta)}Z_j + P_{f'(u)}S^{\circ}(\Pi(Z_1, \zeta), u_1j, Z_j) + (5\alpha - 2) \\
&= P_{f'(u)u_1j}P_{\Pi(Z_1, \zeta)}Z_j + P_{f'(u)u_{111}}S^{\circ}(\Pi(Z_1, \zeta), Z_1, Z_1) + (b)
\end{aligned}
$$

from Proposition 14. Once again, the fact that $S^{\circ}(\Pi(Z_1, \zeta), Z_1, Z_1)$ is of (b)-type is part of the data on $\hat{\zeta}$; this eventually gives

$$ P_{f'(u)}P_{\Pi(Z_1, \zeta)}u_1 = P_{f'(u)u_1j}P_{\Pi(Z_1, \zeta)}Z_1 + (b). $$

The term $P_{\Pi(Z_1, \zeta)}P_{u_1}f'(u)$ is treated similarly. $\square$

<figure><img src="image_3.png" /></figure>

Remarks. (1) The proof of Proposition 18 gives the structure

$$ (\sharp) = (b) + P_{h_n(\hat{u})}Y_{4\alpha-2}^n + (5\alpha - 2) $$

for the remainder $(\sharp)$ in (3.25), with an implicit sum in $n$ in the right hand side, explicit functions $h_n(\hat{u}) \in C^{\alpha}$ depending continuously on $\hat{u}$, and reference distributions $Y_{4\alpha-2}^n$ assumed to be in $\mathcal{L}(C^{4\alpha})$, along the way; the latter are components of $\hat{\zeta}$ – see Section 4.1.

(2) The above computations have a simple structure, that can be summarized using the E and F notations from Section 1.2. Denote by $F^{\beta}(\cdot)$, or $F^{\beta}(\cdot, \cdot)$,
for that term, in the analysis of $f'(u)\Pi(u, \zeta)$; this is a remainder term of (b)-type. Look at the term $f'(u)u_1\Pi(Z_1, \zeta)$; the other terms are dealt with similarly. One has

$$ 
\begin{aligned}
f'(u)u_1\Pi(Z_1, \zeta) &= P_{f'(u)u_1}\Pi(Z_1, \zeta) + P_{\Pi(Z_1, \zeta)}f'(u)u_1 + \Pi(f'(u)u_1, \Pi(Z_1, \zeta)) \\
&= P_{f'(u)u_1}\Pi(Z_1, \zeta) + P_{\Pi(Z_1, \zeta)}(P_{f'(u)}u_1 + P_{u_1}f'(u)) \\
&\quad + P_{\Pi(Z_1, \zeta)}\Pi(f'(u), u_1) \\
&\quad + f'(u)\Pi(u_1, \Pi(Z_1, \zeta)) + C^{\circ}(f'(u), u_1, \Pi(Z_1, \zeta)) \\
&\quad + u_1\Pi(f'(u), \Pi(Z_1, \zeta)) + C^{\circ}(u_1, f'(u), \Pi(Z_1, \zeta)) \\
&\quad + \Pi(\Pi(f'(u), u_1), \Pi(Z_1, \zeta)).
\end{aligned}
$$

The last six terms in the right hand side are of regularity $(4\alpha - 2)$; a first order expansion of $f'(u)$ and $u_1$ allows to put the in the right form (3.25). For the second and third terms in the right hand side, simply write

$$ P_{\Pi(Z_1, \zeta)}P_{f'(u)}u_1 = P_{f'(u)}P_{\Pi(Z_1, \zeta)}u_1 + S^{\circ}(\Pi(Z_1, \zeta), f'(u), u_1). $$

The last term has regularity $(4\alpha - 2)$, and a first order expansion of $f'(u)$ and $u_1$ allows to put the in the right form. Also we have

$$ 
\begin{aligned}
P_{f'(u)}P_{\Pi(Z_1, \zeta)}u_1 &= P_{f'(u)}P_{\Pi(Z_1, \zeta)}P_{u_1j}Z_j + (5\alpha - 2) \\
&= P_{f'(u)}P_{u_1j}P_{\Pi(Z_1, \zeta)}Z_j + P_{f'(u)}S^{\circ}(\Pi(Z_1, \zeta), u_1j, Z_j) + (5\alpha - 2) \\
&= P_{f'(u)u_1j}P_{\Pi(Z_1, \zeta)}Z_j + P_{f'(u)u_{111}}S^{\circ}(\Pi(Z_1, \zeta), Z_1, Z_1) + (b)
\end{aligned}
$$

from Proposition 14. Once again, the fact that $S^{\circ}(\Pi(Z_1, \zeta), Z_1, Z_1)$ is of (b)-type is part of the data on $\hat{\zeta}$; this eventually gives

$$ P_{f'(u)}P_{\Pi(Z_1, \zeta)}u_1 = P_{f'(u)u_1j}P_{\Pi(Z_1, \zeta)}Z_1 + (b). $$

The term $P_{\Pi(Z_1, \zeta)}P_{u_1}f'(u)$ is treated similarly. $\square$

<figure><img src="image_3.png" /></figure>

Remarks. (1) The proof of Proposition 18 gives the structure

$$ (\sharp) = (b) + P_{h_n(\hat{u})}Y_{4\alpha-2}^n + (5\alpha - 2) $$

for the remainder $(\sharp)$ in (3.25), with an implicit sum in $n$ in the right hand side, explicit functions $h_n(\hat{u}) \in C^{\alpha}$ depending continuously on $\hat{u}$, and reference distributions $Y_{4\alpha-2}^n$ assumed to be in $\mathcal{L}(C^{4\alpha})$, along the way; the latter are components of $\hat{\zeta}$ – see Section 4.1.

(2) The above computations have a simple structure, that can be summarized using the E and F notations from Section 1.2. Denote by $F^{\beta}(\cdot)$, or $F^{\beta}(\cdot, \cdot)$,
a well-defined function on parabolic Hölder spaces that sends $C^{k\alpha}$ into $C^{k\alpha+\beta}$, respectively $C^{k\alpha} \times C^{\ell\alpha}$ into $C^{(k+\ell)\alpha+\beta}$, and enjoying the F-type expansion property. We allow ourselves to write identities like

$$F^{\beta}(P_{a_1} a_2) = P_{a_1} F^{\beta}(a_2) + F^{|a_2|+\beta}(a_1), \qquad (3.30)$$

for $a_2 \in C^{|a_2|}$. Also, denote by $E^{\beta}(\cdot, \cdot)$, or $E^{\beta}(\cdot, \cdot)$, an operator satisfying an E-type expansion formula, sending formally $C^{\alpha}$ into $C^{\alpha+\beta}$, respectively $C^{k\alpha} \times C^{\ell\alpha}$ into $C^{(k+\ell)\alpha+\beta}$. In those terms, we have

$$E^{\beta}(P_{a_1} a_2) = P_{a_1} E^{\beta}(a_2) + F^{\beta+|a_2|}(a_1) + E^{\beta+|a_2|}(a_1). \qquad (3.31)$$

The above proof of decomposition (3.25) starts from the identity

$$f(u)\zeta = P_{f(u)}\zeta + F^{\alpha-2}(f(u)) + E^{\alpha-2}(f(u)), $$

and proceeds by writing

$$ \begin{aligned} F^{\alpha-2}(f(u)) &= F^{\alpha-2}(u) + F^{\alpha-2}(f'(u), u) + F^{\alpha-2}(\Pi(u, u)) + \text{(b)} \\ &= F^{\alpha-2}(u) + F^{\alpha-2}(u, u) + \text{(b)}, \end{aligned} $$

and

$$ E^{\alpha-2}(f(u)) = f'(u) E^{\alpha-2}(u) + f^{(2)}(u) E^{\alpha-2}(u, u) \\ + f^{(3)}(u) E^{\alpha-2}(u, u, u) + \text{(b)}. $$

One then uses the paracontrolled structure of $u$ and the expansion rules (3.30) and (3.31) to run the computations.

(3) Note that Proposition 18 makes sense from a regularity structures point of view. Let us work in the regularity structure of the (gPAM) equation on the 3-dimensional torus, together with a model on it. Let $u$ be represented by the modelled distribution

$$ \underline{u} = u \mathbf{1} + u' X + u_{\tau} \underline{\tau} := \sum_{a \in \mathcal{A}} u_a \underline{Z}_a, $$

in $D^{\gamma}$, for $\gamma = (3/2)^{+}$. Denote by $\circ$ the noise symbol in the regularity structure. Then one would have

$$ f(\underline{u}) \circ = f(u) \circ + f'(u) u_a \underline{Z}_a \circ + \frac{1}{2!} f^{(2)}(u) u_a u_b \underline{Z}_a \underline{Z}_b \circ \\ + \frac{1}{3!} f^{(3)}(u) u_a u_b u_c \underline{Z}_a \underline{Z}_b \underline{Z}_c \circ, $$
a well-defined function on parabolic Hölder spaces that sends $C^{k\alpha}$ into $C^{k\alpha+\beta}$, respectively $C^{k\alpha} \times C^{\ell\alpha}$ into $C^{(k+\ell)\alpha+\beta}$, and enjoying the F-type expansion property. We allow ourselves to write identities like

$$F^{\beta}(P_{a_1} a_2) = P_{a_1} F^{\beta}(a_2) + F^{|a_2|+\beta}(a_1), \quad (3.30)$$

for $a_2 \in C^{|a_2|}$. Also, denote by $E^{\beta}(\cdot, \cdot)$, or $E^{\beta}(\cdot, \cdot)$, an operator satisfying an E-type expansion formula, sending formally $C^{\alpha}$ into $C^{\alpha+\beta}$, respectively $C^{k\alpha} \times C^{\ell\alpha}$ into $C^{(k+\ell)\alpha+\beta}$. In those terms, we have

$$E^{\beta}(P_{a_1} a_2) = P_{a_1} E^{\beta}(a_2) + F^{\beta+|a_2|}(a_1) + E^{\beta+|a_2|}(a_1). \quad (3.31)$$

The above proof of decomposition (3.25) starts from the identity

$$f(u)\zeta = P_{f(u)}\zeta + F^{\alpha-2}(f(u)) + E^{\alpha-2}(f(u)),$$

and proceeds by writing

$$\begin{aligned} F^{\alpha-2}(f(u)) &= F^{\alpha-2}(u) + F^{\alpha-2}(f'(u), u) + F^{\alpha-2}(\Pi(u, u)) + \text{(b)} \\ &= F^{\alpha-2}(u) + F^{\alpha-2}(u, u) + \text{(b)}, \end{aligned}$$

and

$$E^{\alpha-2}(f(u)) = f'(u) E^{\alpha-2}(u) + f^{(2)}(u) E^{\alpha-2}(u, u) \\ + f^{(3)}(u) E^{\alpha-2}(u, u, u) + \text{(b)}.$$

One then uses the paracontrolled structure of $u$ and the expansion rules (3.30) and (3.31) to run the computations.

(3) Note that Proposition 18 makes sense from a regularity structures point of view. Let us work in the regularity structure of the (gPAM) equation on the 3-dimensional torus, together with a model on it. Let $u$ be represented by the modelled distribution

$$\underline{u} = u \mathbf{1} + u' X + u_{\tau} \underline{\tau} := \sum_{a \in \mathcal{A}} u_a \underline{Z}_a,$$

in $D^{\gamma}$, for $\gamma = (3/2)^+$. Denote by $\circ$ the noise symbol in the regularity structure. Then one would have

$$f(\underline{u}) \circ = f(u) \circ + f'(u) u_a \underline{Z}_a \circ + \frac{1}{2!} f^{(2)}(u) u_a u_b \underline{Z}_a \underline{Z}_b \circ \\ + \frac{1}{3!} f^{(3)}(u) u_a u_b u_c \underline{Z}_a \underline{Z}_b \underline{Z}_c \circ,$$
with sums in $\mathcal{A}$ restricted to $|a|, |a| + |b|, |a| + |b| + |c| \le 3\alpha$. Using Theorem 1 in Bailleul and Hoshino's work [4], we would have for $f(u)\zeta$ the paracontrolled representation

$$ 
\begin{aligned}
& P_{f(u)}\zeta + P_{f'(u)u_a}[\underline{Z}_a\circ] + \frac{1}{2!}P_{f^{(2)}(u)u_a u_b}[\underline{Z}_a\underline{Z}_b\circ] \\
& \qquad + \frac{1}{3!}P_{f^{(3)}(u)u_a u_b u_c}[\underline{Z}_a\underline{Z}_b\underline{Z}_c\circ] + (5\alpha - 2) \\
& =: P_{f(u)}\zeta + P_{f'(u)u_a}[\underline{Z}_a\circ] + \frac{1}{2!}P_{f^{(2)}(u)u_a u_b}[\underline{Z}_a\underline{Z}_b\circ] + (b), \quad (3.32)
\end{aligned}
$$

for distributions $[\underline{Z}_a\circ], [\underline{Z}_a\underline{Z}_b\circ], [\underline{Z}_a\underline{Z}_b\underline{Z}_c\circ]$, built from the regularity structure and the model. This is the content of identity (3.25), and one reads off the functions $h_l(\hat{u})$ of point 1 in the preceding formula. The term $f'(u)u'X\circ$ from the regularity structures picture appears in the above paracontrolled analysis under the form $f'(u)u_3\Pi(Z_3, \zeta)$, in identity (3.28). This makes perfect sense if one considers that the piece of $u$ that is differentiable is in its paracontrolled representation is given by $\bar{\Pi}_{u_3}Z_3 + u_3^\sharp$, and we recall the pointwise first order expansion for paraproducts from Corollary 2. The term $f'(u)^2u'\circ\mathcal{I}(X\circ)$ from the regularity structures picture appears above under the form

$$ f'(u)\gamma_l V_l(u_1^\sharp) \Pi_{(1)}^l(Z_1, \zeta), $$

in identity (3.29), while the term $f^{(2)}(u)f(u)u'X\circ\mathcal{I}(\circ)$ from the regularity structures analysis appears in the form of identities (3.26) and (3.27). (Recall $\sum_{\ell=1}^{\ell_0} \gamma_l V_l(v) = v'$, on the flat torus.)

**3.3. Dealing with derivatives.** We work in this section in the one-dimensional torus $\mathbf{T}$, with $x$ as canonical coordinate and $L = \partial_x^2 =: \partial^2$. The study of the generalized KPZ equation requires the analysis of quantities of the form $P_{\partial\mathbf{P}/g}\partial h$, or similar quantities where $f, g$ or $h$ is itself given by a paraproduct. The following remark provides the key to the analysis of such terms.

Recall the notations $\mathcal{P}_t$ and $\mathcal{Q}_t$ for the parabolic projectors from the standard collection of operators with cancellation – Definition 27 in Appendix A.1, and the notation $v = dt \otimes dx$ for the parabolic volume measure.

**LEMMA 20.** Both $P_{\partial f} \partial g$ and $\Pi(\partial f, \partial g)$ can be written under the form

$$ \int_0^1 \frac{1}{t} \mathcal{P}_t^*(\tilde{\mathcal{Q}}_t^1 f \tilde{\mathcal{Q}}_t^2 g) \frac{dt}{t}, $$

with $\tilde{\mathcal{Q}}^1, \tilde{\mathcal{Q}}^2 \in \mathrm{GC}^1$, and $\mathcal{P}_t \in \mathrm{StGC}^{[0,b]}$.
with sums in $\mathcal{A}$ restricted to $|a|, |a| + |b|, |a| + |b| + |c| \le 3\alpha$. Using Theorem 1 in Bailleul and Hoshino's work [4], we would have for $f(u)\zeta$ the paracontrolled representation

$$ \begin{aligned} & P_{f(u)\zeta} + P_{f'(u)u_a}[\underline{Z}_a\circ] + \frac{1}{2!}P_{f^{(2)}(u)u_a u_b}[\underline{Z}_a\underline{Z}_b\circ] \\ & \qquad + \frac{1}{3!}P_{f^{(3)}(u)u_a u_b u_c}[\underline{Z}_a\underline{Z}_b\underline{Z}_c\circ] + (5\alpha - 2) \\ & =: P_{f(u)\zeta} + P_{f'(u)u_a}[\underline{Z}_a\circ] + \frac{1}{2!}P_{f^{(2)}(u)u_a u_b}[\underline{Z}_a\underline{Z}_b\circ] + (b), \end{aligned} \quad (3.32) $$

for distributions $[\underline{Z}_a\circ], [\underline{Z}_a\underline{Z}_b\circ], [\underline{Z}_a\underline{Z}_b\underline{Z}_c\circ]$, built from the regularity structure and the model. This is the content of identity (3.25), and one reads off the functions $h_l(\hat{u})$ of point 1 in the preceding formula. The term $f'(u)u'X\circ$ from the regularity structures picture appears in the above paracontrolled analysis under the form $f'(u)u_3\Pi(Z_3, \zeta)$, in identity (3.28). This makes perfect sense if one considers that the piece of $u$ that is differentiable is in its paracontrolled representation is given by $\bar{\Pi}_{u_3}Z_3 + u_3^\sharp$, and we recall the pointwise first order expansion for paraproducts from Corollary 2. The term $f'(u)^2u'\circ\mathcal{I}(X\circ)$ from the regularity structures picture appears above under the form

$$ f'(u)\gamma_l V_l(u_1^\sharp) \Pi_{(1)}^l(Z_1, \zeta), $$

in identity (3.29), while the term $f^{(2)}(u)f(u)u'X\circ\mathcal{I}(\circ)$ from the regularity structures analysis appears in the form of identities (3.26) and (3.27). (Recall $\sum_{\ell=1}^{\ell_0} \gamma_l V_l(v) = v'$, on the flat torus.)

**3.3. Dealing with derivatives.** We work in this section in the one-dimensional torus $\mathbf{T}$, with $x$ as canonical coordinate and $L = \partial_x^2 =: \partial^2$. The study of the generalized KPZ equation requires the analysis of quantities of the form $P_{\partial P/g} \partial h$, or similar quantities where $f, g$ or $h$ is itself given by a paraproduct. The following remark provides the key to the analysis of such terms.

Recall the notations $\mathcal{P}_t$ and $\mathcal{Q}_t$ for the parabolic projectors from the standard collection of operators with cancellation – Definition 27 in Appendix A.1, and the notation $v = dt \otimes dx$ for the parabolic volume measure.

**LEMMA 20.** Both $P_{\partial f} \partial g$ and $\Pi(\partial f, \partial g)$ can be written under the form

$$ \int_0^1 \frac{1}{t} \mathcal{P}_t^*(\tilde{\mathcal{Q}}_t^1 f \tilde{\mathcal{Q}}_t^2 g) \frac{dt}{t}, $$

with $\tilde{\mathcal{Q}}^1, \tilde{\mathcal{Q}}^2 \in \text{GC}^1$, and $\mathcal{P}_t \in \text{StGC}^{[0,b]}$.
*Proof.* Consider $\mathcal{P}_t$, one of our localization operator at the parabolic scale $t^{1/2}$, then by integrating by parts in space, we see that for $e := (\tau, x)$

$$ 
\begin{align*}
(\mathcal{P}_t f)(e) &= \int_{\mathcal{M}} K_{\mathcal{P}_t}(e, e') \partial f(e') \nu(de') \\
&= -\int_{\mathcal{M}} \partial_{x'} K_{\mathcal{P}_t}(e, e') f(e') \nu(de') \\
&= -t^{-1/2} \int_{\mathcal{M}} t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e') f(e') \nu(de').
\end{align*}
$$

Then since we assume regularity estimates on the heat kernel, it follows that $t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e')$ satisfies the same kind of pointwise estimates as $K_{\mathcal{P}_t}$. Moreover, its first momentum is null, which is a cancellation property of order 1

$$ \int_{\mathcal{M}} t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e') \nu(de') = \int_{\mathcal{M}} t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e') \nu(de) = 0. $$

In terms of the notation introduced and described in Appendix A, the collection $(t^{1/2} \partial \mathcal{P}_t)_{t>0}$ belongs to the class $\mathcal{GC}^1$. That legitimates to use the notation

$$ \tilde{\mathcal{Q}}_t := t^{1/2} \partial \mathcal{P}_t, $$

from which the representation of the statement follows. A similar observation holds for the operator $\Pi(\partial f, \partial g)$. $\square$

*Remark.* In the model setting of spatial paraproducts on the one-dimensional torus

$$ P_{\partial f} \partial g = \sum_k S_{k-2}(\partial f) \Delta_k(\partial g), $$

so an integration by parts shows that

$$ S_{k-2}(\partial f) = 2^{k-2} \tilde{\Delta}_{k-2} f, $$

for some Fourier multiplier $\tilde{\Delta}_{k-2}$ acting on a distribution $f(\cdot) = \sum c_n e^{inx}$, as

$$ (\tilde{\Delta}_{k-2} f)(x) = 2^{-(k-2)} \sum_{|n| \le 2^{k-2}} c_n i n e^{inx} = \sum_{|n| \le 2^{k-2}} \frac{i n}{2^{k-2}} c_n e^{inx}, $$

with symbol $(in/2^{k-2})1_{|n| \le 2^{k-2}}$. This symbol is not exactly supported on the annulus at scale $2^{k-2}$, as it is the case for the Fourier projector $\Delta_k$, but it satisfies some decay property at 0 and at infinity, so it still encodes some cancellation property. We have

$$ P_{\partial f} \partial g = \sum_k 2^{2k-2} (\tilde{\Delta}_k f)(\Delta'_k g), $$
*Proof.* Consider $\mathcal{P}_t$, one of our localization operator at the parabolic scale $t^{1/2}$, then by integrating by parts in space, we see that for $e := (\tau, x)$

$$ 
\begin{align*}
(\mathcal{P}_t f)(e) &= \int_{\mathcal{M}} K_{\mathcal{P}_t}(e, e') \partial f(e') \nu(de') \\
&= -\int_{\mathcal{M}} \partial_{x'} K_{\mathcal{P}_t}(e, e') f(e') \nu(de') \\
&= -t^{-1/2} \int_{\mathcal{M}} t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e') f(e') \nu(de').
\end{align*}
$$

Then since we assume regularity estimates on the heat kernel, it follows that $t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e')$ satisfies the same kind of pointwise estimates as $K_{\mathcal{P}_t}$. Moreover, its first momentum is null, which is a cancellation property of order 1

$$ \int_{\mathcal{M}} t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e') \nu(de') = \int_{\mathcal{M}} t^{1/2} \partial_{x'} K_{\mathcal{P}_t}(e, e') \nu(de) = 0. $$

In terms of the notation introduced and described in Appendix A, the collection $(t^{1/2} \partial \mathcal{P}_t)_{t>0}$ belongs to the class $\mathcal{G}^1$. That legitimates to use the notation

$$ \tilde{\mathcal{Q}}_t := t^{1/2} \partial \mathcal{P}_t, $$

from which the representation of the statement follows. A similar observation holds for the operator $\Pi(\partial f, \partial g)$. $\square$

*Remark.* In the model setting of spatial paraproducts on the one-dimensional torus

$$ P_{\partial f} \partial g = \sum_k S_{k-2}(\partial f) \Delta_k(\partial g), $$

so an integration by parts shows that

$$ S_{k-2}(\partial f) = 2^{k-2} \tilde{\Delta}_{k-2} f, $$

for some Fourier multiplier $\tilde{\Delta}_{k-2}$ acting on a distribution $f(\cdot) = \sum c_n e^{inx}$, as

$$ (\tilde{\Delta}_{k-2} f)(x) = 2^{-(k-2)} \sum_{|n| \le 2^{k-2}} c_n i n e^{inx} = \sum_{|n| \le 2^{k-2}} \frac{i n}{2^{k-2}} c_n e^{inx}, $$

with symbol $(in/2^{k-2})1_{|n| \le 2^{k-2}}$. This symbol is not exactly supported on the annulus at scale $2^{k-2}$, as it is the case for the Fourier projector $\Delta_k$, but it satisfies some decay property at 0 and at infinity, so it still encodes some cancellation property. We have

$$ P_{\partial f} \partial g = \sum_k 2^{2k-2} (\tilde{\Delta}_k f)(\Delta'_k g), $$
for operators $\Delta_k'$ perfectly localized at frequencies of scale $2^k$. The resonant operator $\Pi(\partial f, \partial g)$ has the same structure

$$ \Pi(\partial f, \partial g) = \sum_k 2^{2k} \hat{\Delta}_k(f) \Delta_k'(g), $$

for operators $\hat{\Delta}_k$ perfectly localized at frequencies of scale $2^k$.

It follows from that lemma that all the continuity results of Section 3.1 on the corrector $\mathcal{C}$ and its iterates have direct counterparts in terms of the operator $(f, g) \mapsto \mathcal{P}_{\partial f} g$. We single out three of them here to make that point clear. Define on the space of bounded measurable functions on the parabolic space $\mathcal{M}$ the correctors

$$ \mathcal{C}_{\partial}^{<}((f_1, f_2), g) := \mathcal{P}_{\partial \tilde{P}_{f_1 f_2}} \partial g - f_1 \mathcal{P}_{\partial f_2} \partial g, $$

$$ \mathcal{C}_{\partial}^{<}(f, (g_1, g_2)) := \mathcal{P}_{\partial f}(\tilde{\mathcal{P}}_{g_1} g_2) - g_1 \mathcal{P}_{\partial f} \partial g_2, $$

$$ \mathcal{C}_{\partial}^{=}((f_1, f_2), g) := \Pi(\partial \tilde{\mathcal{P}}_{f_1 f_2}, \partial g) - f_1 \Pi(\partial f_2, \partial g). $$

We use the exponent $<$ in the notation to remind the reader from the fact that paraproducts $\mathcal{P}_f g$ are defined in a Fourier setting with frequencies of $f$ strictly smaller than those of $g$, while the resonant operator involves frequencies that are essentially equal.

**THEOREM 21.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha \in (0, 1)$ and $\alpha + \beta \le 1$. If

$$ \beta + \gamma - 2 < 0 < \alpha + \beta + \gamma - 2, $$

then the maps

$$ \mathcal{C}^{\alpha} \times \mathcal{C}^{\beta} \times \mathcal{C}^{\gamma} \rightarrow \mathcal{C}^{\alpha+\beta+\gamma-2} \\ (f, g, h) \mapsto \mathcal{C}_{\partial}^{<}((f, g), h), \mathcal{C}_{\partial}^{<}((f, g, h)) $$

are continuous.

Let $\alpha, \beta, \gamma_1, \gamma_2$ be regularity exponents, with $\alpha, \gamma_1 \in (0, 1)$ and $\alpha + \beta \le 1$. If

$$ \beta + \gamma_1 + \gamma_2 - 2 < 0 < \alpha + \beta + \gamma_1 + \gamma_2 - 2, $$

then the map

$$ \mathcal{C}^{\alpha} \times \mathcal{C}^{\beta} \times \mathcal{C}^{\gamma_1} \times \mathcal{C}^{\gamma_2} \rightarrow \mathcal{C}^{\alpha+\beta+\gamma_1+\gamma_2-2} \\ ((f, g), (h_1, h_2)) \mapsto \mathcal{C}_{\partial}^{<}((f, g), \tilde{\mathcal{P}}_{h_1 h_2}) - h_1 \mathcal{C}_{\partial}((f, g), h_2) $$

is continuous.
for operators $\Delta'_k$ perfectly localized at frequencies of scale $2^k$. The resonant operator $\Pi(\partial f, \partial g)$ has the same structure

$$ \Pi(\partial f, \partial g) = \sum_k 2^{2k} \hat{\Delta}_k(f) \Delta'_k(g), $$

for operators $\hat{\Delta}_k$ perfectly localized at frequencies of scale $2^k$.

It follows from that lemma that all the continuity results of Section 3.1 on the corrector $\mathcal{C}$ and its iterates have direct counterparts in terms of the operator $(f, g) \mapsto \mathcal{P}_{\partial f} g$. We single out three of them here to make that point clear. Define on the space of bounded measurable functions on the parabolic space $\mathcal{M}$ the correctors

$$ \begin{aligned} \mathcal{C}_{\partial}^{<}((f_1, f_2), g) &:= \mathcal{P}_{\partial \tilde{P}_{f_1} f_2} \partial g - f_1 \mathcal{P}_{\partial f_2} \partial g, \\ \mathcal{C}_{\partial}^{<}(f, (g_1, g_2)) &:= \mathcal{P}_{\partial f}(\tilde{\mathcal{P}}_{g_1} g_2) - g_1 \mathcal{P}_{\partial f} \partial g_2, \\ \mathcal{C}_{\partial}^{=}((f_1, f_2), g) &:= \Pi(\partial \tilde{\mathcal{P}}_{f_1} f_2, \partial g) - f_1 \Pi(\partial f_2, \partial g). \end{aligned} $$

We use the exponent $<$ in the notation to remind the reader from the fact that paraproducts $\mathcal{P}_f g$ are defined in a Fourier setting with frequencies of $f$ strictly smaller than those of $g$, while the resonant operator involves frequencies that are essentially equal.

**THEOREM 21.** Let $\alpha, \beta, \gamma$ be regularity exponents, with $\alpha \in (0, 1)$ and $\alpha + \beta \le 1$. If

$$ \beta + \gamma - 2 < 0 < \alpha + \beta + \gamma - 2, $$

then the maps

$$ \begin{aligned} \mathcal{C}^{\alpha} \times \mathcal{C}^{\beta} \times \mathcal{C}^{\gamma} &\rightarrow \mathcal{C}^{\alpha+\beta+\gamma-2} \\ (f, g, h) &\mapsto \mathcal{C}_{\partial}^{<}((f, g), h), \mathcal{C}_{\partial}^{<}(f, (g, h)) \end{aligned} $$

are continuous.

Let $\alpha, \beta, \gamma_1, \gamma_2$ be regularity exponents, with $\alpha, \gamma_1 \in (0, 1)$ and $\alpha + \beta \le 1$. If

$$ \beta + \gamma_1 + \gamma_2 - 2 < 0 < \alpha + \beta + \gamma_1 + \gamma_2 - 2, $$

then the map

$$ \begin{aligned} \mathcal{C}^{\alpha} \times \mathcal{C}^{\beta} \times \mathcal{C}^{\gamma_1} \times \mathcal{C}^{\gamma_2} &\rightarrow \mathcal{C}^{\alpha+\beta+\gamma_1+\gamma_2-2} \\ ((f, g), (h_1, h_2)) &\mapsto \mathcal{C}_{\partial}^{<}((f, g), \tilde{\mathcal{P}}_{h_1} h_2) - h_1 \mathcal{C}_{\partial}((f, g), h_2) \end{aligned} $$

is continuous.
*Proof.* Let us concentrate first on the first statement, in the model case of the flat torus, where

$$C_{\partial}^{<}((f,g),h) = \sum 2^{2k-2}(\tilde{\Delta}_{k-2}(\Pi_{f}^{0}g) - f(\tilde{\Delta}_{k-2}g))\Delta_{k}'(h).$$

Note that since

$$\tilde{\Delta}_{k-2}(\Pi_{f}^{0}g) - f\tilde{\Delta}_{k-2}g = \sum_{\ell \le k-2} 2^{-(k-2-\ell)}(S_{\ell}f - f)\Delta_{\ell}g,$$

we have the estimate

$$\|\tilde{\Delta}_{k-2}(\Pi_{f}^{0}g) - f(\tilde{\Delta}_{k-2}g)\|_{\infty} \lesssim \sum_{\ell \le k-2} 2^{-(k-2-\ell)} 2^{-\ell(\alpha+\beta)} \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \\ \lesssim 2^{-k(\alpha+\beta)} \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}},$$

since $\alpha + \beta \le 1$. We see here the importance of the decay of the symbol of the operator $\tilde{\Delta}_{k-2}$, encoded in the factor $2^{-(k-2-\ell)}$. The conclusion follows then from the estimate

$$\|\Delta_n(C_{\partial}^{<}((f,g),h))\|_{\infty} \\ \lesssim \left( \sum_{k \ge n-1} 2^{-k(\alpha+\beta+\gamma-2)} + \sum_{k \le n-1} 2^{-n\alpha} 2^{-k(\beta+\gamma-2)} \right) \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \|h\|_{C^{\gamma}} \\ \lesssim 2^{-n(\alpha+\beta+\gamma-2)} \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \|h\|_{C^{\gamma}}.$$

If now $h = \Pi_{h_1}^0 h_2$, then $\Delta_k(h) \simeq (S_{k-2}h_1)(\Delta_k h_2)$, and we have

$$C_{\partial}^{<}((f,g), P_{h_1}h_2) - h_1 C_{\partial}((f,g), h_2) \\ = \sum 2^{k-2}(\tilde{\Delta}_{k-2}(P_f g) - f\tilde{\Delta}_{k-2}g)\Delta_k(h_2)(h_1 - S_{k-2}h_1)$$

and we may conclude by the same reasoning as above, with the extra exponents coming from the positive regularity of $h_1$, since

$$\|h_1 - S_{k-2}h_1\|_{\infty} \lesssim 2^{-k\gamma_1} \|h_1\|_{C^{\gamma_1}}. \quad \square$$

We let the reader state and prove the other continuity results for the iterated versions of $C_{\partial}^{<}$ and $C_{\partial}^{=}$. Set

$$\mathcal{A} := \{i, ij, ijk\}_{i,(i+j),(i+j+k)\le 3}, \qquad (3.33)$$

and set

$$|i| := i, \quad |ij| := i + j, \quad |ijk| := i + j + k.$$
*Proof.* Let us concentrate first on the first statement, in the model case of the flat torus, where

$$C_{\partial}^{<}((f,g),h) = \sum 2^{2k-2}(\tilde{\Delta}_{k-2}(\Pi_{f}^{0}g) - f(\tilde{\Delta}_{k-2}g))\Delta_{k}'(h).$$

Note that since

$$\tilde{\Delta}_{k-2}(\Pi_{f}^{0}g) - f\tilde{\Delta}_{k-2}g = \sum_{\ell \le k-2} 2^{-(k-2-\ell)}(S_{\ell}f - f)\Delta_{\ell}g,$$

we have the estimate

$$\|\tilde{\Delta}_{k-2}(\Pi_{f}^{0}g) - f(\tilde{\Delta}_{k-2}g)\|_{\infty} \lesssim \sum_{\ell \le k-2} 2^{-(k-2-\ell)} 2^{-\ell(\alpha+\beta)} \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \\ \lesssim 2^{-k(\alpha+\beta)} \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}},$$

since $\alpha + \beta \le 1$. We see here the importance of the decay of the symbol of the operator $\tilde{\Delta}_{k-2}$, encoded in the factor $2^{-(k-2-\ell)}$. The conclusion follows then from the estimate

$$\|\Delta_n(C_{\partial}^{<}((f,g),h))\|_{\infty} \\ \lesssim \left( \sum_{k \ge n-1} 2^{-k(\alpha+\beta+\gamma-2)} + \sum_{k \le n-1} 2^{-n\alpha} 2^{-k(\beta+\gamma-2)} \right) \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \\ \lesssim 2^{-n(\alpha+\beta+\gamma-2)} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}.$$

If now $h = \Pi_{h_1}^0 h_2$, then $\Delta_k(h) \simeq (S_{k-2}h_1)(\Delta_k h_2)$, and we have

$$C_{\partial}^{<}((f,g), P_{h_1}h_2) - h_1 C_{\partial}((f,g), h_2) \\ = \sum 2^{k-2}(\tilde{\Delta}_{k-2}(P_f g) - f\tilde{\Delta}_{k-2}g)\Delta_k(h_2)(h_1 - S_{k-2}h_1)$$

and we may conclude by the same reasoning as above, with the extra exponents coming from the positive regularity of $h_1$, since

$$\|h_1 - S_{k-2}h_1\|_{\infty} \lesssim 2^{-k\gamma_1} \|h_1\|_{C^{\gamma_1}}. \quad \square$$

We let the reader state and prove the other continuity results for the iterated versions of $C_{\partial}^{<}$ and $C_{\partial}^{=}$. Set

$$\mathcal{A} := \{i, ij, ijk\}_{i,(i+j),(i+j+k) \le 3}, \qquad (3.33)$$

and set

$$|i| := i, \quad |ij| := i + j, \quad |ijk| := i + j + k.$$
**PROPOSITION 22.** Let $u$ be given by a paracontrolled system with reference functions $\mathbf{Z}$, together with a function $g \in C_b^4$. Then one can write the product $g(u)(\partial u)^2$ under the form

$$ g(u)(\partial u)^2 = \sum_{\substack{c_1, c_2 \in \mathcal{A} \\ |c_1| + |c_2| \le 4}} P_{g(u)u_{c_1}u_{c_2}} X_{c_1c_2} + \sum_{\substack{a, c_1, c_2 \in \mathcal{A} \\ |a| + |c_1| + |c_2| \le 4}} P_{g'(u)u_a u_{c_1} u_{c_2}} X_{ac_1c_2} \\ + \frac{1}{2!} P_{g^{(2)}(u)u_1^4} X_{1111} + (5\alpha - 2) \quad (3.34) $$

for some remainder term of regularity $5\alpha - 2$, and some explicit reference distributions $X$'s that depend only on the noise $\zeta$ and $\mathbf{Z}$.

As in Corollary 18 on the paracontrolled representation of $f(u)\zeta$, the proper interpretation of the above statement is that the remainder is of parabolic regularity $5\alpha - 2$, provided the $X$'s are well defined as elements of their natural spaces. In the present work, these $X$'s are given by the enhancement $\tilde{\zeta}$ of the noise.

*Proof.* If one is not interested in the precise form of the $X_\alpha$ in (3.34), one can proceed very efficiently making only computations with $\mathbf{E}$ and $\mathbf{F}$ notations, as in the proof of Corollary 18. We first provide a multiplicative decomposition for

$$ (\partial u)^2 = (\partial u)^2 = 2P_{\partial u}\partial u + \Pi(\partial u, \partial u) := \mathbf{E}^{-2}(u, u) $$

as follows. The function $\mathbf{E}^{-2}$ is a function of $\mathbf{E}$-type with respect to its two arguments. Recall that two $\mathbf{E}$ operators in the same identity may mean different $\mathbf{E}$-type operators. Distributions of regularity $\beta$ that do not depend on $u$ are denoted by $X^\beta$. Write $\simeq$ to mean equality up to a remainder term of regularity $5\alpha - 2$. One has

$$ \begin{aligned} (\partial u)^2 &= \mathbf{E}^{-2}(u, u) \\ &= u_i \mathbf{E}^{i\alpha-2}(u) + \mathbf{E}^{i\alpha-2}(u_i, u) \\ &= u_i u_j X^{(i+j)\alpha-2} + \mathbf{E}^{(i+j)\alpha-2}(u_j) + u_{ij} \mathbf{E}^{(i+j)\alpha-2}(u) \\ &= u_i u_j X^{(i+j)\alpha-2} + u_i u_{jk} X^{(i+j+k)\alpha-2} + u_i \mathbf{E}^{(i+j+k)\alpha-2}(u_{jk}) \\ &\quad + u_{ij} u_k X^{(i+j+k)\alpha-2} + u_{ij} \mathbf{E}^{(i+j+k)\alpha-2}(u_k) \\ &\simeq u_i u_j X^{(i+j)\alpha-2} + u_i u_{jk} X^{(i+j+k)\alpha-2} + u_i u_{jk\ell} X^{(i+j+k+\ell)\alpha-2} \\ &\quad + u_{ij} u_k X^{(i+j+k)\alpha-2} + u_{ij} u_{k\ell} \mathbf{E}^{(i+j+k+\ell)\alpha-2} \\ &\simeq u_i u_j X^{(i+j)\alpha-2} + u_i u_{jk} X^{(i+j+k)\alpha-2} + (u_1 u_{111} + u_{11}^2) X^{4\alpha-2}. \end{aligned} $$

The above implicit sums are restricted to $(i+j)$, $(i+j+k)$, $(i+j+k+\ell) \le 4$. Look now at the term $g(u)u_1^2 X^{2\alpha-2}$ and show that it can be written in the
**PROPOSITION 22.** Let $u$ be given by a paracontrolled system with reference functions $\mathbf{Z}$, together with a function $g \in C_b^4$. Then one can write the product $g(u)(\partial u)^2$ under the form

$$ g(u)(\partial u)^2 = \sum_{\substack{c_1, c_2 \in \mathcal{A} \\ |c_1| + |c_2| \le 4}} \mathbf{P}_{g(u)u_{c_1}u_{c_2}} X_{c_1c_2} + \sum_{\substack{a, c_1, c_2 \in \mathcal{A} \\ |a| + |c_1| + |c_2| \le 4}} \mathbf{P}_{g'(u)u_a u_{c_1} u_{c_2}} X_{ac_1c_2} \\ + \frac{1}{2!} \mathbf{P}_{g^{(2)}(u)u_1^4} X_{1111} + (5\alpha - 2) \quad (3.34) $$

for some remainder term of regularity $5\alpha - 2$, and some explicit reference distributions $X$'s that depend only on the noise $\zeta$ and $\mathbf{Z}$.

As in Corollary 18 on the paracontrolled representation of $f(u)\zeta$, the proper interpretation of the above statement is that the remainder is of parabolic regularity $5\alpha - 2$, provided the $X$'s are well defined as elements of their natural spaces. In the present work, these $X$'s are given by the enhancement $\tilde{\zeta}$ of the noise.

*Proof.* If one is not interested in the precise form of the $X_\alpha$ in (3.34), one can proceed very efficiently making only computations with $\mathbf{E}$ and $\mathbf{F}$ notations, as in the proof of Corollary 18. We first provide a multiplicative decomposition for

$$ (\partial u)^2 = (\partial u)^2 = 2\mathbf{P}_{\partial u}\partial u + \Pi(\partial u, \partial u) := \mathbf{E}^{-2}(u, u) $$

as follows. The function $\mathbf{E}^{-2}$ is a function of $\mathbf{E}$-type with respect to its two arguments. Recall that two $\mathbf{E}$ operators in the same identity may mean different $\mathbf{E}$-type operators. Distributions of regularity $\beta$ that do not depend on $u$ are denoted by $X^\beta$. Write $\simeq$ to mean equality up to a remainder term of regularity $5\alpha - 2$. One has

$$ \begin{aligned} (\partial u)^2 &= \mathbf{E}^{-2}(u, u) \\ &= u_i \mathbf{E}^{i\alpha-2}(u) + \mathbf{E}^{i\alpha-2}(u_i, u) \\ &= u_i u_j X^{(i+j)\alpha-2} + \mathbf{E}^{(i+j)\alpha-2}(u_j) + u_{ij} \mathbf{E}^{(i+j)\alpha-2}(u) \\ &= u_i u_j X^{(i+j)\alpha-2} + u_i u_{jk} X^{(i+j+k)\alpha-2} + u_i \mathbf{E}^{(i+j+k)\alpha-2}(u_{jk}) \\ &\quad + u_{ij} u_k X^{(i+j+k)\alpha-2} + u_{ij} \mathbf{E}^{(i+j+k)\alpha-2}(u_k) \\ &\simeq u_i u_j X^{(i+j)\alpha-2} + u_i u_{jk} X^{(i+j+k)\alpha-2} + u_i u_{jk\ell} X^{(i+j+k+\ell)\alpha-2} \\ &\quad + u_{ij} u_k X^{(i+j+k)\alpha-2} + u_{ij} u_{k\ell} \mathbf{E}^{(i+j+k+\ell)\alpha-2} \\ &\simeq u_i u_j X^{(i+j)\alpha-2} + u_i u_{jk} X^{(i+j+k)\alpha-2} + (u_1 u_{111} + u_{11}^2) X^{4\alpha-2}. \end{aligned} $$

The above implicit sums are restricted to $(i+j)$, $(i+j+k)$, $(i+j+k+\ell) \le 4$. Look now at the term $g(u)u_1^2 X^{2\alpha-2}$ and show that it can be written in the
form (3.34); the other terms are easier to deal with. We have

$$g(u)u_1^2 X^{2\alpha-2} = P_{g(u)u_1^2} X^{2\alpha-2} + F^{2\alpha-2}(g(u)u_1^2) + E^{2\alpha-2}(g(u)u_1^2), \quad (3.35)$$

with

$$\begin{aligned} F^{2\alpha-2}(g(u)u_1^2) &= P_{g(u)} F^{2\alpha-2}(u_1^2) + P_{u_1^2} F^{2\alpha-2}(g(u)) + F^{2\alpha-2}(g(u), u_1^2) \\ &= P_{g(u)} P_{u_1} F^{2\alpha-2}(u_1) + P_{g(u)} F^{2\alpha-2}(u_1, u_1) + P_{u_1^2} F^{2\alpha-2}(g(u)) \\ &\quad + P_{g'(u)u_1^2 u_{11}} X^{4\alpha-2} + (5\alpha-2) \\ &= P_{g(u)u_1} F^{2\alpha-2}(u_1) + P_{g(u)u_{11}^2} X^{4\alpha-2} + P_{u_1^2} F^{2\alpha-2}(g(u)) \\ &\quad + P_{g'(u)u_1^2 u_{11}} X^{4\alpha-2} + (5\alpha-2). \end{aligned}$$

One writes the first term in the right hand side in the good form

$$\begin{aligned} P_{g(u)u_1} F^{2\alpha-2}(u_1) &= P_{g(u)u_1} P_{u_{1j}} X^{(2+j)\alpha-2} + F^{3\alpha-2}(u_{11}) \\ &= P_{g(u)u_1 u_{1j}} X^{(2+j)\alpha-2} + P_{u_{111}} X^{4\alpha-2} + (5\alpha-2). \end{aligned}$$

We use the paracontrolled expansion

$$g(u) = P_{g'(u)u_i} Z_i + P_{g'(u)u_{11}} \mathbb{R}(1, Z_1, Z_1) + \frac{1}{2} P_{g^{(2)}(u)u_1^2} \Pi(Z_1, Z_1) + (3\alpha),$$

of $g(u)$ to order 2 to get

$$\begin{aligned} P_{u_1^2} F^{2\alpha-2}(g(u)) &= P_{g'(u)u_1^3} X^{3\alpha-2} + P_{g^{(2)}(u)u_1^3 u_{11}} X^{4\alpha-2} + P_{g'(u)u_2 u_1^2} X^{4\alpha-2} \\ &\quad + P_{g'(u)u_1^2 u_{11}} X^{4\alpha-2} + P_{g^{(2)}(u)u_1^4} X^{4\alpha-2} + (5\alpha-2). \end{aligned}$$

The term $E^{2\alpha-2}(g(u)u_1^2)$ in (3.35) is dealt with similarly, using the E-type expansion rule

$$E^{2\alpha-2}(g(u)u_1^2) = g(u)E^{2\alpha-2}(u_1^2) + u_1^2 E^{2\alpha-2}(g(u)) \\ + E^{2\alpha-2}(g(u), u_1^2) + E^{2\alpha-2}(\Pi(g(u), u_1^2))$$

and the above second order paracontrolled expansion for $g(u)$, to put $E^{2\alpha-2}(g(u)u_1^2)$ in multiplicative form first, and then use the F-type expansion to put it in the form of Equation (3.34). Details are left to the reader. □

*Remark.* Similarly to Proposition 18, Proposition 22 makes perfect sense from a regularity structures point of view. Note also that the above proof implicitly uses a refined version of the $C_{\beta}^{<}$ corrector to deal with a first argument $u_1^\beta$ of regularity strictly bigger than 1. This term corresponds to the term $2g(u)f(u)f'(u)u'\partial\mathcal{I}(X\circ)\partial\mathcal{I}(\circ)$, that appears in the regularity structures analysis.
form (3.34); the other terms are easier to deal with. We have

$$g(u)u_1^2 X^{2\alpha-2} = P_{g(u)u_1^2} X^{2\alpha-2} + F^{2\alpha-2}(g(u)u_1^2) + E^{2\alpha-2}(g(u)u_1^2), \quad (3.35)$$

with

$$\begin{aligned} F^{2\alpha-2}(g(u)u_1^2) &= P_{g(u)} F^{2\alpha-2}(u_1^2) + P_{u_1^2} F^{2\alpha-2}(g(u)) + F^{2\alpha-2}(g(u), u_1^2) \\ &= P_{g(u)} P_{u_1} F^{2\alpha-2}(u_1) + P_{g(u)} F^{2\alpha-2}(u_1, u_1) + P_{u_1^2} F^{2\alpha-2}(g(u)) \\ &\quad + P_{g'(u)u_1^2 u_{11}} X^{4\alpha-2} + (5\alpha-2) \\ &= P_{g(u)u_1} F^{2\alpha-2}(u_1) + P_{g(u)u_{11}^2} X^{4\alpha-2} + P_{u_1^2} F^{2\alpha-2}(g(u)) \\ &\quad + P_{g'(u)u_1^2 u_{11}} X^{4\alpha-2} + (5\alpha-2). \end{aligned}$$

One writes the first term in the right hand side in the good form

$$\begin{aligned} P_{g(u)u_1} F^{2\alpha-2}(u_1) &= P_{g(u)u_1} P_{u_1j} X^{(2+j)\alpha-2} + F^{3\alpha-2}(u_{11}) \\ &= P_{g(u)u_1 u_{1j}} X^{(2+j)\alpha-2} + P_{u_{111}} X^{4\alpha-2} + (5\alpha-2). \end{aligned}$$

We use the paracontrolled expansion

$$g(u) = P_{g'(u)u_i} Z_i + P_{g'(u)u_{11}} \mathbb{R}(1, Z_1, Z_1) + \frac{1}{2} P_{g^{(2)}(u)u_1^2} \Pi(Z_1, Z_1) + (3\alpha),$$

of $g(u)$ to order 2 to get

$$\begin{aligned} P_{u_1^2} F^{2\alpha-2}(g(u)) &= P_{g'(u)u_1^3} X^{3\alpha-2} + P_{g^{(2)}(u)u_1^3 u_{11}} X^{4\alpha-2} + P_{g'(u)u_2 u_1^2} X^{4\alpha-2} \\ &\quad + P_{g'(u)u_1^2 u_{11}} X^{4\alpha-2} + P_{g^{(2)}(u)u_1^4} X^{4\alpha-2} + (5\alpha-2). \end{aligned}$$

The term $E^{2\alpha-2}(g(u)u_1^2)$ in (3.35) is dealt with similarly, using the E-type expansion rule

$$\begin{aligned} E^{2\alpha-2}(g(u)u_1^2) &= g(u)E^{2\alpha-2}(u_1^2) + u_1^2 E^{2\alpha-2}(g(u)) \\ &\quad + E^{2\alpha-2}(g(u), u_1^2) + E^{2\alpha-2}(\Pi(g(u), u_1^2)) \end{aligned}$$

and the above second order paracontrolled expansion for $g(u)$, to put $E^{2\alpha-2}(g(u)u_1^2)$ in multiplicative form first, and then use the F-type expansion to put it in the form of Equation (3.34). Details are left to the reader. □

**Remark.** Similarly to Proposition 18, Proposition 22 makes perfect sense from a regularity structures point of view. Note also that the above proof implicitly uses a refined version of the $C_{\delta}^{<}$ corrector to deal with a first argument $u_1^\#^\#$ of regularity strictly bigger than 1. This term corresponds to the term $2g(u)f(u)f'(u)u'\partial\mathcal{I}(X\circ)\partial\mathcal{I}(\circ)$, that appears in the regularity structures analysis.
We summarize here the notations introduced in this section.

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

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

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

$$R(f, g, h) = P_f(\tilde{P}_g h) - P_f g h.$$

We use the same letters for the iterates of these operators, such as they were defined above. The ° operators are defined by the corresponding formulae where $\tilde{P}$ is replaced by $P$. To deal specifically with the derivatives, we have the operators

$$C_{\partial}^{<}(f_1, f_2), g) := P_{\partial \tilde{P}_{f_1} f_2} \partial g - f_1 P_{\partial f_2} \partial g,$$

$$C_{\partial}^{<}(f, (g_1, g_2)) := P_{\partial f}(\partial \tilde{P}_{g_1} g_2) - g_1 P_{\partial f} \partial g_2,$$

$$C_{\partial}^{=}(f_1, f_2, g) := \Pi(\partial P_{f_1} f_2, \partial g) - f_1 \Pi(\partial f_2, \partial g),$$

and their iterates. The only thing that matters from a computational point of view is to identify which operators are of E-type, and which operators are of F-type.

## 4. Nonlinear singular PDEs

We choose to illustrate the use of paracontrolled calculus for the study of singular partial differential equations on the examples of the 3-dimensional generalized parabolic Anderson model equation

$$\mathcal{L}u = f(u)\zeta,$$

and the generalized (KPZ) equation

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

on the one-dimensional torus. We introduce the notion of enhanced noise, and consistent enhancement, in Section 4.1. We define in Section 4.2 the maps that are used to give a fixed point formulation of the (gPAM) and (gKPZ) equations in Section 4.3.

**4.1. Enhanced noise.** The prototype of singular equation is given by the controlled ordinary differential equation

$$dx_t = V(x_t) dh_t, \quad (4.1)$$
We summarize here the notations introduced in this section.

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

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

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

$$R(f, g, h) = P_f(\tilde{P}_g h) - P_{fg}h.$$

We use the same letters for the iterates of these operators, such as they were defined above. The ° operators are defined by the corresponding formulae where $\tilde{P}$ is replaced by $P$. To deal specifically with the derivatives, we have the operators

$$C_{\partial}^{<}(f_1, f_2), g) := P_{\partial \tilde{P}_{f_1} f_2} \partial g - f_1 P_{\partial f_2} \partial g,$$

$$C_{\partial}^{<}(f, (g_1, g_2)) := P_{\partial f}(\partial \tilde{P}_{g_1} g_2) - g_1 P_{\partial f} \partial g_2,$$

$$C_{\partial}^{=}(f_1, f_2, g) := \Pi(\partial P_{f_1} f_2, \partial g) - f_1 \Pi(\partial f_2, \partial g),$$

and their iterates. The only thing that matters from a computational point of view is to identify which operators are of E-type, and which operators are of F-type.

## 4. Nonlinear singular PDEs

We choose to illustrate the use of paracontrolled calculus for the study of singular partial differential equations on the examples of the 3-dimensional generalized parabolic Anderson model equation

$$\mathcal{L}u = f(u)\zeta,$$

and the generalized (KPZ) equation

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

on the one-dimensional torus. We introduce the notion of enhanced noise, and consistent enhancement, in Section 4.1. We define in Section 4.2 the maps that are used to give a fixed point formulation of the (gPAM) and (gKPZ) equations in Section 4.3.

### 4.1. Enhanced noise.
The prototype of singular equation is given by the controlled ordinary differential equation

$$dx_t = V(x_t) dh_t, \quad (4.1)$$
where $h$ is a nondifferentiable $\mathbf{R}^\ell$-valued control and $V$ an $L(\mathbf{R}^\ell, \mathbf{R}^d)$-valued one form on $\mathbf{R}^d$, say. Think of a Brownian path for the control $h$. One of the deepest insights of Lyons in his theory of rough paths [22] was to understand that one needs to change the notion of control to make sense of such an equation, and that this enhanced control takes values in a very specific universal algebraic structure. In simple terms, the enhanced control consists of $h$ and the collection of a number of objects playing the role of the nonexisting iterated integrals $\int_{s \le s_1 \le \dots \le s_k \le t} dh_{s_1} \otimes \dots \otimes dh_{s_k}$ – such iterated integrals cannot be defined as continuous functions of their integrands, here $(h, \dots, h)$, if $h$ is not sufficiently regular; see Proposition 1.29 in [8]. Once given these extra data, one can make sense of, and solve uniquely, the controlled ordinary differential equation (4.1) under some appropriate regularity conditions on the one form $V$, and the solution path happens to be a continuous function of the enhanced control, in some appropriate topology. The enhancement of the control cannot be made on a purely analytic basis and requires some extra input, typically the use of probabilistic methods when the control $h$ is random.

Hairer's theory of regularity structures provides a conceptually close framework for the study of a large class of singular partial differential equations containing the generalized parabolic Anderson model equation

$$ \mathcal{L}u = f(u)\zeta \quad (4.2) $$

as a particular case. To make sense of Equation (4.2), one needs to enhance the distribution $\zeta$ with the *a priori* datum of a number of other distributions. Contrary to the case of the controlled ordinary differential equation (4.1), this enhanced ‘control’ takes values in an equation-dependent algebraic structure. The resolution process is also different, as the equation is first recast in some abstract space of jets of solutions, where it can be solved under appropriate conditions. This corresponds to looking for a solution in a specific space of distributions where one can actually make sense of all the terms in the equation, especially some *a priori* ill-defined products. A fundamental tool, the reconstruction operator, allows then to associate to this abstract solution a classical distribution. The equation-dependent algebraic structure in which the enhanced distribution lives also allows to give sense to this solution distribution as a limit of solutions to some family of classically well-posed equations in which the distribution $\zeta$ has been smoothened. The latter point is related to renormalization matters.

Recall now Proposition 22, and Remark 1 following its proof, giving the paracontrolled expansion (3.34) of $f(u)\zeta$, under the assumption that a number of quantities $(Y_{k\alpha-2})_{2 \le k \le 4}$ are given *a priori* as elements of the $\mathcal{L}(C^{k\alpha})$-spaces, as measurable functions of $\zeta$ and $\mathbf{Z}$; write $Y := Y(\zeta, \mathbf{Z})$. A choice of $\mathfrak{Z}_k \in C^{k\alpha}$,
where $h$ is a nondifferentiable $\mathbf{R}^\ell$-valued control and $V$ an $L(\mathbf{R}^\ell, \mathbf{R}^d)$-valued one form on $\mathbf{R}^d$, say. Think of a Brownian path for the control $h$. One of the deepest insights of Lyons in his theory of rough paths [22] was to understand that one needs to change the notion of control to make sense of such an equation, and that this enhanced control takes values in a very specific universal algebraic structure. In simple terms, the enhanced control consists of $h$ and the collection of a number of objects playing the role of the nonexisting iterated integrals $\int_{s \le s_1 \le \dots \le s_k \le t} dh_{s_1} \otimes \dots \otimes dh_{s_k}$ – such iterated integrals cannot be defined as continuous functions of their integrands, here $(h, \dots, h)$, if $h$ is not sufficiently regular; see Proposition 1.29 in [8]. Once given these extra data, one can make sense of, and solve uniquely, the controlled ordinary differential equation (4.1) under some appropriate regularity conditions on the one form $V$, and the solution path happens to be a continuous function of the enhanced control, in some appropriate topology. The enhancement of the control cannot be made on a purely analytic basis and requires some extra input, typically the use of probabilistic methods when the control $h$ is random.

Hairer's theory of regularity structures provides a conceptually close framework for the study of a large class of singular partial differential equations containing the generalized parabolic Anderson model equation

$$ \mathcal{L}u = f(u)\zeta \quad (4.2) $$

as a particular case. To make sense of Equation (4.2), one needs to enhance the distribution $\zeta$ with the *a priori* datum of a number of other distributions. Contrary to the case of the controlled ordinary differential equation (4.1), this enhanced ‘control’ takes values in an equation-dependent algebraic structure. The resolution process is also different, as the equation is first recast in some abstract space of jets of solutions, where it can be solved under appropriate conditions. This corresponds to looking for a solution in a specific space of distributions where one can actually make sense of all the terms in the equation, especially some *a priori* ill-defined products. A fundamental tool, the reconstruction operator, allows then to associate to this abstract solution a classical distribution. The equation-dependent algebraic structure in which the enhanced distribution lives also allows to give sense to this solution distribution as a limit of solutions to some family of classically well-posed equations in which the distribution $\zeta$ has been smoothened. The latter point is related to renormalization matters.

Recall now Proposition 22, and Remark 1 following its proof, giving the paracontrolled expansion (3.34) of $f(u)\zeta$, under the assumption that a number of quantities $(Y_{k\alpha-2})_{2 \le k \le 4}$ are given *a priori* as elements of the $\mathcal{L}(C^{k\alpha})$-spaces, as measurable functions of $\zeta$ and $\mathbf{Z}$; write $Y := Y(\zeta, \mathbf{Z})$. A choice of $\mathfrak{Z}_k \in C^{k\alpha}$,
with $2 \le k \le 4$, and

$$ \mathcal{L}\mathfrak{Z}_k = Y_{k\alpha-2}, $$

defines an **enhancement** $\hat{\zeta} = (\zeta, \mathfrak{Z})$ **of the noise** $\zeta$ for the generalized parabolic Anderson model equation

$$ \mathcal{L}u = f(u)\zeta. $$

Enlarge the finite collection $\mathfrak{Z}_2$ by adding $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ into it. This defines the collection $\bar{\mathfrak{Z}}_2$. Set $\bar{\mathfrak{Z}}_k := \mathfrak{Z}_k$, for $3 \le k \le 4$. We define this way the finite collection $\bar{\mathfrak{Z}}$. An enhancement $\hat{\zeta}$ is said to be **coherent** if

$$ \mathcal{L}\mathfrak{Z}_j = Y_{j\alpha-2}(\zeta, (\bar{\mathfrak{Z}}_\ell)_{2\le\ell\le 3}), \quad 2 \le j \le 3. $$

A coherent enhancement of the noise $\zeta$ for the (gPAM) equation can be used to work with $\mathbf{Z} = \bar{\mathfrak{Z}}$ in a paracontrolled setting. The introduction of $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ in the system is necessary not only to give a paracontrolled representation of the solution $u$ of the equation but also a lower order paracontrolled representation of its derivatives, amongst which $f(u)$. This is reminiscent of the use of two spaces of trees in the regularity structures setting.

The study of the generalized (KPZ) equation

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

requires the introduction of further quantities ($X_{c_1c_2}, X_{ac_1c_2}, X_{1111}, \dots$) that appear in (3.34) and the proof of this decomposition. They are assumed to be elements of $\mathcal{L}(C^{k\alpha})$-spaces, and measurable functions of $\zeta$ and $\mathbf{Z}$. Write

$$ X_{k\alpha-2} = X_{k\alpha-2}(\zeta, \mathbf{Z}), $$

and recall that $Z_2, Z_3$ may stand for tuples $(Z_2^{n_2}), (Z_3^{n_3})$. The joint choice of $\bar{\mathfrak{Z}}$ and elements $\mathfrak{X}^k \in C^{k\alpha}$ such that

$$ \mathcal{L}\mathfrak{X}_k = X_{k\alpha-2}(\zeta, \mathbf{Z}), $$

defines an **enhancement** $\hat{\zeta} = (\zeta, \mathfrak{Z}, \mathfrak{X})$ **of the noise** $\zeta$ for the generalized (KPZ) equation. This enhancement is said to be coherent if

$$ \mathcal{L}\mathfrak{X}_k = X_{k\alpha-2}(\zeta, (\bar{\mathfrak{Z}}^\ell)_{2\le\ell\le 3}, (\mathfrak{X}^\ell)_{2\le\ell\le 3})). $$

In accordance with the regularity structures picture, the different terms that form the enhanced noise correspond to the different pieces of a paracontrolled/resonant expansion of all the formal products that appear in the tree of negative homogeneity in the regularity structures expansions of the
with $2 \le k \le 4$, and

$$ \mathcal{L}\mathfrak{Z}_k = Y_{k\alpha-2}, $$

defines an **enhancement** $\hat{\zeta} = (\zeta, \mathfrak{Z})$ **of the noise** $\zeta$ for the generalized parabolic Anderson model equation

$$ \mathcal{L}u = f(u)\zeta. $$

Enlarge the finite collection $\mathfrak{Z}_2$ by adding $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ into it. This defines the collection $\bar{\mathfrak{Z}}_2$. Set $\bar{\mathfrak{Z}}_k := \mathfrak{Z}_k$, for $3 \le k \le 4$. We define this way the finite collection $\bar{\mathfrak{Z}}$. An enhancement $\hat{\zeta}$ is said to be **coherent** if

$$ \mathcal{L}\mathfrak{Z}_j = Y_{j\alpha-2}(\zeta, (\bar{\mathfrak{Z}}_\ell)_{2\le\ell\le 3}), \quad 2 \le j \le 3. $$

A coherent enhancement of the noise $\zeta$ for the (gPAM) equation can be used to work with $\mathbf{Z} = \bar{\mathfrak{Z}}$ in a paracontrolled setting. The introduction of $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ in the system is necessary not only to give a paracontrolled representation of the solution $u$ of the equation but also a lower order paracontrolled representation of its derivatives, amongst which $f(u)$. This is reminiscent of the use of two spaces of trees in the regularity structures setting.

The study of the generalized (KPZ) equation

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

requires the introduction of further quantities ($X_{c_1c_2}, X_{ac_1c_2}, X_{1111}, \dots$) that appear in (3.34) and the proof of this decomposition. They are assumed to be elements of $\mathcal{L}(C^{k\alpha})$-spaces, and measurable functions of $\zeta$ and $\mathbf{Z}$. Write

$$ X_{k\alpha-2} = X_{k\alpha-2}(\zeta, \mathbf{Z}), $$

and recall that $Z_2, Z_3$ may stand for tuples $(Z_2^{n_2}), (Z_3^{n_3})$. The joint choice of $\bar{\mathfrak{Z}}$ and elements $\mathfrak{X}^k \in C^{k\alpha}$ such that

$$ \mathcal{L}\mathfrak{X}_k = X_{k\alpha-2}(\zeta, \mathbf{Z}), $$

defines an **enhancement** $\hat{\zeta} = (\zeta, \mathfrak{Z}, \mathfrak{X})$ **of the noise** $\zeta$ for the generalized (KPZ) equation. This enhancement is said to be coherent if

$$ \mathcal{L}\mathfrak{X}_k = X_{k\alpha-2}(\zeta, (\bar{\mathfrak{Z}}^\ell)_{2\le\ell\le 3}, (\mathfrak{X}^\ell)_{2\le\ell\le 3}). $$

In accordance with the regularity structures picture, the different terms that form the enhanced noise correspond to the different pieces of a paracontrolled/resonant expansion of all the formal products that appear in the tree of negative homogeneity in the regularity structures expansions of the
right hand sides of the generalized (PAM) and (KPZ) equations. Constructing coherent enhancements is the task of renormalization of stochastic singular partial differential equations. This has been implemented in the setting of regularity structures in the groundbreaking works [7, 10] of Hairer and co-authors. This work has not been done yet in the paracontrolled setting, but the use of Bailleul and Hoshino's results in [4] allows to construct coherent enhancements from the renormalized models built from [10].

**4.2. Fixed point equation.** We give in this section the fixed point formulations of the generalized (PAM) and (KPZ) equations. In both cases, we assume a coherent enhancement $\hat{\zeta}$ of the noise $\zeta$ is given, and work with the associated reference system $\mathbf{Z}$. Recall $\frac{2}{5} < \alpha \le \frac{1}{2}$, so $5\alpha - 2 > 0$. To avoid working with time-weighted Hölder spaces with exploding weights, we assume the initial conditions $u_0 \in C^{4\alpha}$, to have $Pu_0 \in C^{4\alpha}$, and treat this term in the integral formulation of the equation as a remainder term – recall $P$ stands for the heat propagator. See [3, 17] or [23] for a sample of works where $u_0 \in C^\alpha$, in a first order paracontrolled setting.

**4.2.1. Generalized (PAM) equation.** Recall from (3.33) the definition of the index set $\mathcal{A}$. We obtained in Proposition 18 the paracontrolled decomposition

$$f(u)\zeta = P_{f(u)}\zeta + P_{f'(u)u_a}Y_a + \frac{1}{2!}P_{f^{(2)}(u)u_1^2}Y_{11} + (b) + P_{h_l(\hat{u})}Y_{4\alpha-2}^{(\ell)} + (5\alpha - 2)(\hat{u}),$$

with implicit sums restricted to $|a| \le 2\alpha$, and $(b) \in \mathcal{L}(C^{4\alpha})$. The function $(5\alpha - 2)(\hat{u}) \in C^{5\alpha-2}$, is a locally Lipschitz function of $\hat{u}$. Define $v \in C^\alpha$, setting, with obvious notations,

$$\begin{aligned} v &:= Pu_0 + \mathcal{L}^{-1}(f(u)\zeta) \\ &= \tilde{P}_{f(u)}Z_1 + \tilde{P}_{f'(u)u_a}Z_{|a|+1} + \frac{1}{2!}\tilde{P}_{f^{(2)}(u)u_1^2}Z_3^{(1)} + \mathcal{L}^{-1}(b) \\ &\quad + \tilde{P}_{h_l(\hat{u})}Z_4^{(\ell)} + \mathcal{L}^{-1}((5\alpha - 2)(\hat{u})) + Pu_0. \end{aligned} \quad (4.3)$$

Note that $f(u)$ has a second order paracontrolled expansion

$$\begin{aligned} f(u) &= \tilde{P}_{f'(u)u_1}Z_1 + \tilde{P}_{f'(u)u_2}Z_2 + \tilde{P}_{f^{(2)}(u)u_1^2+2f'(u)u_{11}}R(1, Z_1, Z_1) \\ &\quad + \frac{1}{2!}\tilde{P}_{f^{(2)}(u)u_1^2}\Pi(Z_1, Z_1) + f(u)^{\sharp}(\hat{u}), \end{aligned} \quad (4.4)$$

$$f'(u)u_1 = \tilde{P}_{f^{(2)}(u)u_1^2+f'(u)u_{11}}Z_1 + (f'(u)u_1)^{\sharp}(\hat{u}).$$

Equations (4.3) and (4.4) show that $v$ is the first component of a paracontrolled system $\hat{v}$. Write $\hat{u}^{\sharp}$ for the collection of all the remainders that define $\hat{u}$, and $\hat{v}^{\sharp}$ for the collection of all the remainders that define $\hat{v}$. Set

$$\Phi(\hat{u}^{\sharp}) := \hat{v}^{\sharp}. \quad (4.5)$$
right hand sides of the generalized (PAM) and (KPZ) equations. Constructing coherent enhancements is the task of renormalization of stochastic singular partial differential equations. This has been implemented in the setting of regularity structures in the groundbreaking works [7, 10] of Hairer and co-authors. This work has not been done yet in the paracontrolled setting, but the use of Bailleul and Hoshino's results in [4] allows to construct coherent enhancements from the renormalized models built from [10].

**4.2. Fixed point equation.** We give in this section the fixed point formulations of the generalized (PAM) and (KPZ) equations. In both cases, we assume a coherent enhancement $\hat{\zeta}$ of the noise $\zeta$ is given, and work with the associated reference system $\mathbf{Z}$. Recall $\frac{2}{5} < \alpha \le \frac{1}{2}$, so $5\alpha - 2 > 0$. To avoid working with time-weighted Hölder spaces with exploding weights, we assume the initial conditions $u_0 \in C^{4\alpha}$, to have $Pu_0 \in C^{4\alpha}$, and treat this term in the integral formulation of the equation as a remainder term – recall $P$ stands for the heat propagator. See [3, 17] or [23] for a sample of works where $u_0 \in C^\alpha$, in a first order paracontrolled setting.

**4.2.1. Generalized (PAM) equation.** Recall from (3.33) the definition of the index set $\mathcal{A}$. We obtained in Proposition 18 the paracontrolled decomposition

$$f(u)\zeta = P_{f(u)}\zeta + P_{f'(u)u_a}Y_a + \frac{1}{2!}P_{f^{(2)}(u)u_1^2}Y_{11} + (b) + P_{h_l(\hat{u})}Y_{4\alpha-2}^{(\ell)} + (5\alpha - 2)(\hat{u}),$$

with implicit sums restricted to $|a| \le 2\alpha$, and $(b) \in \mathcal{L}(C^{4\alpha})$. The function $(5\alpha - 2)(\hat{u}) \in C^{5\alpha-2}$, is a locally Lipschitz function of $\hat{u}$. Define $v \in C^\alpha$, setting, with obvious notations,

$$\begin{aligned} v &:= Pu_0 + \mathcal{L}^{-1}(f(u)\zeta) \\ &= \tilde{P}_{f(u)}Z_1 + \tilde{P}_{f'(u)u_a}Z_{|a|+1} + \frac{1}{2!}\tilde{P}_{f^{(2)}(u)u_1^2}Z_3^{(1)} + \mathcal{L}^{-1}(b) \\ &\quad + \tilde{P}_{h_l(\hat{u})}Z_4^{(\ell)} + \mathcal{L}^{-1}((5\alpha - 2)(\hat{u})) + Pu_0. \end{aligned} \quad (4.3)$$

Note that $f(u)$ has a second order paracontrolled expansion

$$\begin{aligned} f(u) &= \tilde{P}_{f'(u)u_1}Z_1 + \tilde{P}_{f'(u)u_2}Z_2 + \tilde{P}_{f^{(2)}(u)u_1^2+2f'(u)u_{11}}R(1, Z_1, Z_1) \\ &\quad + \frac{1}{2!}\tilde{P}_{f^{(2)}(u)u_1^2}\Pi(Z_1, Z_1) + f(u)^{\sharp}(\hat{u}), \end{aligned} \quad (4.4)$$

$$f'(u)u_1 = \tilde{P}_{f^{(2)}(u)u_1^2+f'(u)u_{11}}Z_1 + (f'(u)u_1)^{\sharp}(\hat{u}).$$

Equations (4.3) and (4.4) show that $v$ is the first component of a paracontrolled system $\hat{v}$. Write $\hat{u}^{\sharp}$ for the collection of all the remainders that define $\hat{u}$, and $\hat{v}^{\sharp}$ for the collection of all the remainders that define $\hat{v}$. Set

$$\Phi(\hat{u}^{\sharp}) := \hat{v}^{\sharp}. \quad (4.5)$$
4.2.2. Generalized (KPZ) equation. Set

$$
\mathcal{B} := \{(c_1c_2), (c_1c_2c_3), (c_1c_2c_3c_4); c_i \in \mathcal{A}, |c_1| + |c_2|, |c_1| + |c_2| + |c_3|, \\
|c_1| + |c_2| + |c_3| + |c_4| \le 4\alpha\} \tag{4.6}
$$

Propositions 18 and 22 together give the decomposition

$$
\begin{align}
f(u)\zeta + g(u)(\partial u)^2 & \\
&= P_{f(u)}\zeta + \sum_{|a|\le 2\alpha} P_{f'(u)u_a} Y_a + \frac{1}{2!} P_{f^{(2)}(u)u_1^2} Y_{11} + (b) + P_{h_\ell(\hat{u})} Y_\ell^{4\alpha-2} \nonumber \\
&\quad + \sum_{b \in \mathcal{B}} P_{g_b(\hat{u})} X_b + (5\alpha-2)(\hat{u}), \tag{4.7}
\end{align}
$$

for the explicit functions $g_b$ of $\hat{u}$ that appear in formula (3.34). We have for instance $h_{11}(\hat{u}) = g(u)u_1^2$, and one checks that $h_{11}(\hat{u})$ has a second order paracontrolled expansion, using the paracontrolled expansion of $g(u)$. One has

$$
g(u)u_1^2 = \tilde{P}_{(\star)_1} Z_1 + \tilde{P}_{g'(u)u_1^2 u_2} Z_2 + \tilde{P}_{(\star)_2} \Pi(Z_1, Z_1) + \tilde{P}_{(\star)_3} \mathbb{R}(1, Z_1, Z_1) + (3\alpha)(\hat{u}), \quad (4.8)
$$

with

$$
(\star)_1 := 2g(u)u_1 u_{11} + g'(u)u_1^3,
$$

$$
(\star)_2 := g(u)u_{11}^2 + \frac{1}{2} g^{(2)}(u)u_1^4 + 2g'(u)u_1^2 u_{11}
$$

$$
(\star)_3 := f^{(2)}(u)u_1^4 + f'(u)u_1^2 u_{11} + g^{(2)}(u)u_1^4 + g'(u)u_1^2 u_{11} \\
\qquad + 2g'(u)u_1^2 u_{11} + 2g(u)u_{11}^2 + 2g(u)u_1 u_{111} + g^{(2)}(u)u_1^4 + 3g'(u)u_1^2 u_{11},
$$

and a remainder in $C^{3\alpha}$ that is a locally Lipschitz function of $\hat{u}$. (Note that the two
reference functions $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ are already in the $Z_2$-collection,
since we are working with a coherent enhancement of the noise. They appear
separately in (4.8) as a result of computations.) The other functions $g_b(\hat{u})$ that
appear in formula (4.7) all have a first order paracontrolled expansion, obtained
by elementary means. Setting

$$
v := P u_0 + \mathcal{L}^{-1} (f(u) \zeta + g(u) (\partial u)^2)
$$

defines the first component of a paracontrolled system. (Recall we assume
$u_0 \in C^{4\alpha}$, so $P u_0 \in C^{4\alpha}$, and one can treat it as a remainder term.) Write $\hat{v}^{\sharp}$ for
the collection of all the remainders that define $\hat{v}$, and set

$$
\Phi(\hat{u}^{\sharp}) := \hat{v}^{\sharp}. \tag{4.9}
$$
4.2.2. Generalized (KPZ) equation. Set

$$
\mathcal{B} := \{(c_1c_2), (c_1c_2c_3), (c_1c_2c_3c_4); c_i \in \mathcal{A}, |c_1| + |c_2|, |c_1| + |c_2| + |c_3|, \\
|c_1| + |c_2| + |c_3| + |c_4| \le 4\alpha\}. \tag{4.6}
$$

Propositions 18 and 22 together give the decomposition

$$
\begin{align}
f(u)\zeta + g(u)(\partial u)^2 & = P_{f(u)}\zeta + \sum_{|a|\le 2\alpha} P_{f'(u)u_a}Y_a + \frac{1}{2!}P_{f^{(2)}(u)u_1^2}Y_{11} + (b) + P_{h_\ell(\hat{u})}Y_\ell^{4\alpha-2} \nonumber \\
& \phantom{=} + \sum_{b\in\mathcal{B}} P_{g_b(\hat{u})}X_b + (5\alpha-2)(\hat{u}), \tag{4.7}
\end{align}
$$

for the explicit functions $g_b$ of $\hat{u}$ that appear in formula (3.34). We have for instance $h_{11}(\hat{u}) = g(u)u_1^2$, and one checks that $h_{11}(\hat{u})$ has a second order paracontrolled expansion, using the paracontrolled expansion of $g(u)$. One has

$$
g(u)u_1^2 = \tilde{P}_{(\star)_1} Z_1 + \tilde{P}_{g'(u)u_1^2 u_2} Z_2 + \tilde{P}_{(\star)_2} \Pi(Z_1, Z_1) + \tilde{P}_{(\star)_3} \mathbb{R}(1, Z_1, Z_1) + (3\alpha)(\hat{u}), \quad (4.8)
$$

with

$$
(\star)_1 := 2g(u)u_1 u_{11} + g'(u)u_1^3,
$$

$$
(\star)_2 := g(u)u_{11}^2 + \frac{1}{2}g^{(2)}(u)u_1^4 + 2g'(u)u_1^2 u_{11}
$$

$$
(\star)_3 := f^{(2)}(u)u_1^4 + f'(u)u_1^2 u_{11} + g^{(2)}(u)u_1^4 + g'(u)u_1^2 u_{11} \\
\qquad + 2g'(u)u_1^2 u_{11} + 2g(u)u_{11}^2 + 2g(u)u_1 u_{111} + g^{(2)}(u)u_1^4 + 3g'(u)u_1^2 u_{11},
$$

and a remainder in $C^{3\alpha}$ that is a locally Lipschitz function of $\hat{u}$. (Note that the two
reference functions $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ are already in the $Z_2$-collection,
since we are working with a coherent enhancement of the noise. They appear
separately in (4.8) as a result of computations.) The other functions $g_b(\hat{u})$ that
appear in formula (4.7) all have a first order paracontrolled expansion, obtained
by elementary means. Setting

$$
v := P u_0 + \mathcal{L}^{-1} (f(u) \zeta + g(u) (\partial u)^2)
$$

defines the first component of a paracontrolled system. (Recall we assume
$u_0 \in C^{4\alpha}$, so $P u_0 \in C^{4\alpha}$, and one can treat it as a remainder term.) Write $\hat{v}^{\sharp}$ for
the collection of all the remainders that define $\hat{v}$, and set

$$
\Phi(\hat{u}^{\sharp}) := \hat{v}^{\sharp}. \tag{4.9}
$$
**4.3. Solving the equation.** Recall we work on the parabolic space $[0, T) \times M$, for some possibly infinite positive time horizon $T$.

**4.3.1. Generalized (PAM) equation.** Let an initial condition $u_0 \in C^{4\alpha}(M)$ for (gPAM) equation be given. Assume a coherent enhancement $\hat{\zeta}$ of the noise $\zeta$ is given. We work with the associated reference system $Z = (Z_1, Z_2, Z_3) \in \prod_{k=1}^3 C^{k\alpha}$. Pick regularity exponents $(\beta_a)_{a \in \mathcal{A}}$ such that

$$ \frac{2}{5} < \beta_i < \beta_{ij} < \beta_{ijk} < \alpha, \quad (i), (ij), (ijk) \in \mathcal{A}, \qquad (4.10) $$

and set $\beta := \min_{1 \le i \le 3} \beta_i$. Define a map

$$ \prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta_a} \times C^{3\alpha + \beta} \to C^{\alpha} \times \prod_{a \in \mathcal{A}} C^{\alpha} \\ ((u_a^\sharp)_{a \in \mathcal{A}}, u^\sharp) \mapsto \hat{u} := (u, (u_a)_{a \in \mathcal{A}}), $$

given by the paracontrolled system

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

Write $\hat{u}^\sharp$ for $((u_a^\sharp)_{a \in \mathcal{A}}, u^\sharp)$. Recall the synthetic form (3.32) obtained in Section 3.2.3 for $f(u)\zeta$, and the second order paracontrolled expansion (4.4) for $f(u)$. One reads on these formulae the fact that a 'solution' of the equation needs to satisfy the constraint

$$ u_a = h_a(u), $$

for explicit functions $h_a$. (A proper definition of a solution to the (gPAM) equation is given in Definition 4.1 below.) One has for instance

$$ \begin{aligned} u_1 &= f(u), \quad u_2^{(1)} = (f'f)(u), \quad u_3^{(1)} = f'(u)u_2 = ((f')^2f)(u), \\ u_3^{(2)} &= \frac{1}{2} f^{(2)}(u)u_1^2 = \frac{1}{2} (f^{(2)}(f')^2)(u), \end{aligned} $$

and so on. Note that a 'solution' $u$ has null derivatives in the $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ directions. Define

$$ S_T^{\text{pam}}(u_0) := \{\hat{u}^\sharp; u_{a|\tau=0}^\sharp = h_a(u_0), u_{|\tau=0}^\sharp = u_0\}; $$
**4.3. Solving the equation.** Recall we work on the parabolic space $[0, T) \times M$, for some possibly infinite positive time horizon $T$.

**4.3.1. Generalized (PAM) equation.** Let an initial condition $u_0 \in C^{4\alpha}(M)$ for (gPAM) equation be given. Assume a coherent enhancement $\hat{\zeta}$ of the noise $\zeta$ is given. We work with the associated reference system $Z = (Z_1, Z_2, Z_3) \in \prod_{k=1}^3 C^{k\alpha}$. Pick regularity exponents $(\beta_a)_{a \in \mathcal{A}}$ such that

$$ \frac{2}{5} < \beta_i < \beta_{ij} < \beta_{ijk} < \alpha, \quad (i), (ij), (ijk) \in \mathcal{A}, \quad (4.10) $$

and set $\beta := \min_{1 \le i \le 3} \beta_i$. Define a map

$$ \prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta_a} \times C^{3\alpha + \beta} \to C^{\alpha} \times \prod_{a \in \mathcal{A}} C^{\alpha} \\ ((u_a^\sharp)_{a \in \mathcal{A}}, u^\sharp) \mapsto \hat{u} := (u, (u_a)_{a \in \mathcal{A}}), $$

given by the paracontrolled system

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

Write $\hat{u}^\sharp$ for $((u_a^\sharp)_{a \in \mathcal{A}}, u^\sharp)$. Recall the synthetic form (3.32) obtained in Section 3.2.3 for $f(u)\zeta$, and the second order paracontrolled expansion (4.4) for $f(u)$. One reads on these formulae the fact that a 'solution' of the equation needs to satisfy the constraint

$$ u_a = h_a(u), $$

for explicit functions $h_a$. (A proper definition of a solution to the (gPAM) equation is given in Definition 4.1 below.) One has for instance

$$ \begin{aligned} u_1 &= f(u), & u_2^{(1)} &= (f'f)(u), & u_3^{(1)} &= f'(u)u_2 = ((f')^2f)(u), \\ u_3^{(2)} &= \frac{1}{2}f^{(2)}(u)u_1^2 &= \frac{1}{2}(f^{(2)}(f')^2)(u), \end{aligned} $$

and so on. Note that a 'solution' $u$ has null derivatives in the $\Pi(Z_1, Z_1)$ and $\mathbb{R}(1, Z_1, Z_1)$ directions. Define

$$ S_T^{\text{pam}}(u_0) := \{\hat{u}^\sharp; u_{a|\tau=0}^\sharp = h_a(u_0), u_{|\tau=0}^\sharp = u_0\}; $$
equipped with the natural norm induced from $\prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta_a} \times C^{3\alpha + \beta}$, the space $S_T^{\text{pam}}(u_0)$ is complete. Recall the definition of the map $\Phi$ defined in (4.5).

**PROPOSITION 23.** A choice of time horizon $T$ sufficiently small ensures that $\Phi$ is a contraction of $S_T^{\text{pam}}(u_0)$.

*Proof.* The formulae defining the family of remainders $\hat{v}^{\sharp}$ are actually explicit – see the end of the proof for a sample, and one can read on them that

$$ \Phi(S_T^{\text{pam}}(u_0)) \subset \prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta'_a} \times C^{3\alpha + \beta'} $$

with $\beta'_a > \beta_a$ and $\beta' > \beta$, as a consequence of the choice of exponents (4.10) – we use the regularity assumptions on the components of the enhanced noise $\hat{\zeta}$ and the classical Schauder estimate on Hölder space of positive regularity. So not only do we have that $\Phi$ sends $S_T^{\text{pam}}(u_0)$ into itself, but we also have that $\Phi$ is locally Lipschitz from $S_T^{\text{pam}}(u_0)$ into $\prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta'_a} \times C^{3\alpha + \beta'}$, as a consequence of the locally Lipschitz character of the corrector and commutator and their iterates, and the refined corrector. The contraction property of $\Phi: S_T^{\text{pam}}(u_0) \to S_T^{\text{pam}}(u_0)$, for $T$ sufficiently small, follows then from the elementary estimate

$$ \|w\|_{C^{\delta_1}} \lesssim T^{\frac{\delta_2 - \delta_1}{2}} \|w\|_{C^{\delta_2}}, \quad w|_{\tau=0} = 0 \qquad (4.11) $$

that holds for any $\delta_2 > \delta_1 > 0$.

Here are explicit formulae for the components $v_1^{\sharp}$ and $v_2^{\sharp} = v_{11}^{\sharp}$ of $\hat{v}^{\sharp}$. The term $v_1^{\sharp}$ is the remainder in the second order paracontrolled expansion of $f(u)$. Denote by $(3\alpha)_f(u)$ the $3\alpha$-remainder in the second order paracontrolled expansion for $f(u)$, defined by

$$ f(u) = \mathbf{P}_{f'(u)}u + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)}\Pi(u, u) + (3\alpha)_f(u). $$

The function $(3\alpha)_f$ sends continuously $C^\alpha$ into $C^{3\alpha}$. One has

$$ \begin{aligned} v_1^{\sharp} = & \sum_{i=1}^{3} \mathbf{R}^o(f'(u), u_i, Z_i) + \sum_{j=2}^{3} \mathbf{P}_{f'(u)}\mathbf{R}(1, u_j, Z_j) + \mathbf{P}_{f'(u)}\mathbf{R}(1, \tilde{P}_{u_{12}}Z_2 + u_1^{\sharp}, Z_1) \\ & + \mathbf{R}(1, (u_{11}, Z_1), Z_1) + \mathbf{P}_{f'(u)}u^{\sharp} + \frac{1}{2}\mathbf{P}_{f^{(2)}(u)}\mathbf{D}(u_i, Z_i, u) + (3\alpha)_f(u) \\ & + \frac{1}{2}\mathbf{R}(f^{(2)}(u), u_i, \Pi(Z_i, u)) + \mathbf{R}(f'(u), u_{11}, \mathbf{R}(1, Z_1, Z_1)) \\ & + \frac{1}{2}\sum_{j=2}^{3} \mathbf{P}_{f^{(2)}(u)u_j}\Pi(Z_j, u) \end{aligned} $$
equipped with the natural norm induced from $\prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta_a} \times C^{3\alpha + \beta}$, the space $S_T^{\text{pam}}(u_0)$ is complete. Recall the definition of the map $\Phi$ defined in (4.5).

**PROPOSITION 23.** A choice of time horizon $T$ sufficiently small ensures that $\Phi$ is a contraction of $S_T^{\text{pam}}(u_0)$.

*Proof.* The formulae defining the family of remainders $\hat{v}^{\sharp}$ are actually explicit – see the end of the proof for a sample, and one can read on them that

$$ \Phi(S_T^{\text{pam}}(u_0)) \subset \prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta'_a} \times C^{3\alpha + \beta'} $$

with $\beta'_a > \beta_a$ and $\beta' > \beta$, as a consequence of the choice of exponents (4.10) – we use the regularity assumptions on the components of the enhanced noise $\hat{\zeta}$ and the classical Schauder estimate on Hölder space of positive regularity. So not only do we have that $\Phi$ sends $S_T^{\text{pam}}(u_0)$ into itself, but we also have that $\Phi$ is locally Lipschitz from $S_T^{\text{pam}}(u_0)$ into $\prod_{a \in \mathcal{A}} C^{(3-|a|)\alpha + \beta'_a} \times C^{3\alpha + \beta'}$, as a consequence of the locally Lipschitz character of the corrector and commutator and their iterates, and the refined corrector. The contraction property of $\Phi : S_T^{\text{pam}}(u_0) \to S_T^{\text{pam}}(u_0)$, for $T$ sufficiently small, follows then from the elementary estimate

$$ \|w\|_{C^{\delta_1}} \lesssim T^{\frac{\delta_2 - \delta_1}{2}} \|w\|_{C^{\delta_2}}, \quad w|_{\tau=0} = 0 \qquad (4.11) $$

that holds for any $\delta_2 > \delta_1 > 0$.

Here are explicit formulae for the components $v_1^{\sharp}$ and $v_2^{\sharp} = v_{11}^{\sharp}$ of $\hat{v}^{\sharp}$. The term $v_1^{\sharp}$ is the remainder in the second order paracontrolled expansion of $f(u)$. Denote by $(3\alpha)_f(u)$ the $3\alpha$-remainder in the second order paracontrolled expansion for $f(u)$, defined by

$$ f(u) = \mathbf{P}_{f'(u)}u + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)}\Pi(u, u) + (3\alpha)_f(u). $$

The function $(3\alpha)_f$ sends continuously $C^\alpha$ into $C^{3\alpha}$. One has

$$ \begin{aligned} v_1^{\sharp} = & \sum_{i=1}^{3} \mathbf{R}^o(f'(u), u_i, Z_i) + \sum_{j=2}^{3} \mathbf{P}_{f'(u)}\mathbf{R}(1, u_j, Z_j) + \mathbf{P}_{f'(u)}\mathbf{R}(1, \tilde{P}_{u_{12}}Z_2 + u_1^{\sharp}, Z_1) \\ & + \mathbf{R}(1, (u_{11}, Z_1), Z_1) + \mathbf{P}_{f'(u)}u^{\sharp} + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)}\mathbf{D}(u_i, Z_i, u) + (3\alpha)_f(u) \\ & + \frac{1}{2} \mathbf{R}(f^{(2)}(u), u_i, \Pi(Z_i, u)) + \mathbf{R}(f'(u), u_{11}, \mathbf{R}(1, Z_1, Z_1)) \\ & + \frac{1}{2} \sum_{j=2}^{3} \mathbf{P}_{f^{(2)}(u)u_j}\Pi(Z_j, u) \end{aligned} $$
$$ 
\begin{aligned}
&+ \frac{1}{2} \sum_{j=2}^{3} \mathbf{P}_{f^{(2)}(u)u_1} \Pi(Z_1, \tilde{\mathbf{P}}_{u_k} Z_k + u^{\sharp}) + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)u_1} \mathbf{D}(u_1, Z_1, Z_1) \\
&+ \frac{1}{2} \mathbf{R}(f^{(2)}(u)u_1, u_1, \Pi(Z_1, Z_1)) \\
&+ \mathbf{R}(1, f'(u)u_2, Z_2) + \mathbf{R}(1, f'(u)u_{11}, \mathbf{R}(1, Z_1, Z_1)) \\
&+ \frac{1}{2} \mathbf{R}(1, f^{(2)}(u)u_1^2, \Pi(Z_1, Z_1)) + \mathbf{R}(1, v_{11}^{\sharp}, Z_1) \\
&+ \mathbf{R}(1, f^{(2)}(u)u_1^2 + f'(u)u_{11}, Z_1, Z_1) \\
&+ \mathbf{R}(1, f^{(2)}(u)u_1^2, \Pi(Z_1, Z_1)), Z_1).
\end{aligned}
$$

The term $v_2^\sharp = v_{11}^\sharp$ is the $2\alpha$-remainder in the first order paracontrolled expansion of $f'(u)u_1$. Denote by $(2\alpha)_{f'}(u)$ the $2\alpha$-remainder in the paralinearization formula for $f'(u)$, defined by

$$ f'(u) = \mathbf{P}_{f^{(2)}(u)} u + (2\alpha)_{f'}(u). $$

One has

$$ 
\begin{aligned}
v_2^\sharp = v_{11}^\sharp = \Pi(f'(u), u_1) + \mathbf{P}_{f'(u)}(\tilde{\mathbf{P}}_{u_{12}} Z_1 + u_1^\sharp) + (2\alpha)_{f'}(u) + \mathbf{R}(f'(u), u_{11}, Z_1) \\
+ \mathbf{R}^o(u_1, f^{(2)}(u), u) + \mathbf{R}(1, f'(u)u_{11}, Z_1) + \sum_{k=2}^{3} \mathbf{P}_{f^{(2)}(u)}(\tilde{\mathbf{P}}_{u_k} Z_k + u^\sharp) \\
+ \mathbf{R}^o(f^{(2)}(u)u_1, u_1, Z_1) + \mathbf{R}(1, f^{(2)}(u)u_1^2, Z_1).
\end{aligned}
$$

The explicit expressions for the other terms of $\hat{v}^\sharp$ are similar or easier to obtain. These expression are simpler in the case where $f(u) = u$, as $f'(u) = 1$ and $f^{(2)}(u) = 0$, and $\mathbf{R}^o(1, \dots) = 0$ and $\mathbf{P}_1(*) = (*)$. $\square$

**DEFINITION 4.1.** A **solution to the generalized (PAM) equation** is a fixed point of $\Phi$ in $S_T^{\mathrm{pam}}(u_0)$.

We obtain the following well-posedness result as a direct consequence of Proposition 23.

**THEOREM 24.** *The generalized (PAM) equation has a unique solution in $S_T^{\mathrm{pam}}(u_0)$; it depends continuously on the coherent enhancement $\xi$ of the noise $\zeta$.*

*Remarks.* • So far, the theory of regularity structures has not been developed in a manifold setting. The recent work [15] of Dahlqvist–Diehl–Driver shows how this can be done in the simplest case where the noise is not too rough,
$$ 
\begin{aligned}
&+ \frac{1}{2} \sum_{j=2}^{3} \mathbf{P}_{f^{(2)}(u)u_1} \Pi(Z_1, \tilde{\mathbf{P}}_{u_k} Z_k + u^{\sharp}) + \frac{1}{2} \mathbf{P}_{f^{(2)}(u)u_1} \mathbf{D}(u_1, Z_1, Z_1) \\
&+ \frac{1}{2} \mathbf{R}(f^{(2)}(u)u_1, u_1, \Pi(Z_1, Z_1)) \\
&+ \mathbf{R}(1, f'(u)u_2, Z_2) + \mathbf{R}(1, f'(u)u_{11}, \mathbf{R}(1, Z_1, Z_1)) \\
&+ \frac{1}{2} \mathbf{R}(1, f^{(2)}(u)u_1^2, \Pi(Z_1, Z_1)) + \mathbf{R}(1, v_{11}^{\sharp}, Z_1) \\
&+ \mathbf{R}(1, f^{(2)}(u)u_1^2 + f'(u)u_{11}, Z_1, Z_1) \\
&+ \mathbf{R}(1, f^{(2)}(u)u_1^2, \Pi(Z_1, Z_1)), Z_1).
\end{aligned}
$$

The term $v_2^\sharp = v_{11}^\sharp$ is the $2\alpha$-remainder in the first order paracontrolled expansion of $f'(u)u_1$. Denote by $(2\alpha)_{f'}(u)$ the $2\alpha$-remainder in the paralinearization formula for $f'(u)$, defined by

$$ f'(u) = \mathbf{P}_{f^{(2)}(u)} u + (2\alpha)_{f'}(u). $$

One has

$$ 
\begin{aligned}
v_2^\sharp = v_{11}^\sharp = \Pi(f'(u), u_1) + \mathbf{P}_{f'(u)}(\tilde{\mathbf{P}}_{u_{12}} Z_1 + u_1^\sharp) + (2\alpha)_{f'}(u) + \mathbf{R}(f'(u), u_{11}, Z_1) \\
+ \mathbf{R}^o(u_1, f^{(2)}(u), u) + \mathbf{R}(1, f'(u)u_{11}, Z_1) + \sum_{k=2}^{3} \mathbf{P}_{f^{(2)}(u)}(\tilde{\mathbf{P}}_{u_k} Z_k + u^\sharp) \\
+ \mathbf{R}^o(f^{(2)}(u)u_1, u_1, Z_1) + \mathbf{R}(1, f^{(2)}(u)u_1^2, Z_1).
\end{aligned}
$$

The explicit expressions for the other terms of $\hat{v}^\sharp$ are similar or easier to obtain. These expression are simpler in the case where $f(u) = u$, as $f'(u) = 1$ and $f^{(2)}(u) = 0$, and $\mathbf{R}^o(1, \dots) = 0$ and $\mathbf{P}_1(*) = (*)$. $\square$

**DEFINITION 4.1.** A **solution to the generalized (PAM) equation** is a fixed point of $\Phi$ in $S_T^{\mathrm{pam}}(u_0)$.

We obtain the following well-posedness result as a direct consequence of Proposition 23.

**THEOREM 24.** *The generalized (PAM) equation has a unique solution in $S_T^{\mathrm{pam}}(u_0)$; it depends continuously on the coherent enhancement $\xi$ of the noise $\zeta$.*

*Remarks.* • So far, the theory of regularity structures has not been developed in a manifold setting. The recent work [15] of Dahlqvist–Diehl–Driver shows how this can be done in the simplest case where the noise is not too rough,
corresponding in our setting to a regularity exponent $\alpha > \frac{2}{3}$. A first order description of the objects is sufficient in that setting, as was the case in our previous work [1], whose content covers partly their results.

* We assume here that the initial condition is in $C^{4\alpha}$. We use that fact to put the term $P(u_0)$ in the remainder. One can improve upon this constraint on $u_0$ and only require that $u_0 \in C^\alpha$, at the price of working with weighted Hölder spaces with a temporal weight, explosive at $\tau = 0$, for example a space equipped with the norm

$$ \sup_{0<\tau\le T} \tau^\gamma \|u(\tau)\|_{C^\alpha} $$

for some $\gamma > 0$. See [3, 17] or [23] for a sample of works where $u_0 \in C^\alpha$, in a first order paracontrolled setting.

The next statement is about the linear (PAM) equation

$$ \mathcal{L}u = u\zeta. \qquad (4.12) $$

**THEOREM 25.** Assume the noise $\zeta$ in the (PAM) equation is a space white noise on $M$, and the components $\mathfrak{Z}_k$, $\mathfrak{X}_k$ of the enhanced noise $\hat{\zeta}$ are such that both $\mathcal{L}\mathfrak{Z}_k$ and $\mathcal{L}\mathfrak{X}_k$ take values in $L^\infty_\tau C^{k\alpha-2}_x$. Then Equation (4.12) has a unique, global in time, solution in $S^{\text{pam}}_\infty(u_0)$. Its restriction to any finite time interval depends continuously on the coherent enhancement $\hat{\zeta}$ of the noise $\zeta$.

*Proof.* Define a temporal weight

$$ w(\tau) := e^{\kappa\tau}, \qquad (4.13) $$

for some nonnegative constant $\kappa$ to be fixed later, and work in weighted parabolic Hölder space $C_w^\gamma$, described in Appendix A. We let the reader to check that all the proofs of continuity for the paraproduct, resonant, correctors and commutators, still hold in the setting of weighted parabolic Hölder spaces, with estimates that are uniform in $\kappa \ge 0$. This uniform character of the continuity estimates comes from the important fact that all our approximation operators $P_t, Q_t, \dots$ are supported only on ‘past time’-half spaces. Indeed, if $T$ is an operator acting on the time variable with a kernel $K(\tau, \sigma)$ supported on the past time half space $\{(\tau, \sigma), 0 \le \sigma \le \tau\}$, then

$$ \begin{aligned} w(\tau)^{-1}T(f)(\tau) &= e^{-\kappa\tau}T(f)(\tau) = \int K(\tau, \sigma)e^{-\kappa\tau}f(\sigma)d\sigma \\ &= \int_0^\tau K(\tau, \sigma)e^{-\kappa(\tau-\sigma)}e^{-\kappa\sigma}f(\sigma)d\sigma, \end{aligned} $$
corresponding in our setting to a regularity exponent $\alpha > \frac{2}{3}$. A first order description of the objects is sufficient in that setting, as was the case in our previous work [1], whose content covers partly their results.

* We assume here that the initial condition is in $C^{4\alpha}$. We use that fact to put the term $P(u_0)$ in the remainder. One can improve upon this constraint on $u_0$ and only require that $u_0 \in C^\alpha$, at the price of working with weighted Hölder spaces with a temporal weight, explosive at $\tau = 0$, for example a space equipped with the norm

$$ \sup_{0<\tau\le T} \tau^\gamma \|u(\tau)\|_{C^\alpha} $$

for some $\gamma > 0$. See [3, 17] or [23] for a sample of works where $u_0 \in C^\alpha$, in a first order paracontrolled setting.

The next statement is about the linear (PAM) equation

$$ \mathcal{L}u = u\zeta. \qquad (4.12) $$

**THEOREM 25.** Assume the noise $\zeta$ in the (PAM) equation is a space white noise on $M$, and the components $\mathfrak{Z}_k$, $\mathfrak{X}_k$ of the enhanced noise $\hat{\zeta}$ are such that both $\mathcal{L}\mathfrak{Z}_k$ and $\mathcal{L}\mathfrak{X}_k$ take values in $L^\infty_\tau C^k_{x,\alpha-2}$. Then Equation (4.12) has a unique, global in time, solution in $S^\text{pam}_\infty(u_0)$. Its restriction to any finite time interval depends continuously on the coherent enhancement $\hat{\zeta}$ of the noise $\zeta$.

*Proof.* Define a temporal weight

$$ w(\tau) := e^{\kappa\tau}, \qquad (4.13) $$

for some nonnegative constant $\kappa$ to be fixed later, and work in weighted parabolic Hölder space $C_w^\gamma$, described in Appendix A. We let the reader to check that all the proofs of continuity for the paraproduct, resonant, correctors and commutators, still hold in the setting of weighted parabolic Hölder spaces, with estimates that are uniform in $\kappa \ge 0$. This uniform character of the continuity estimates comes from the important fact that all our approximation operators $P_t, Q_t, \dots$ are supported only on ‘past time’-half spaces. Indeed, if $T$ is an operator acting on the time variable with a kernel $K(\tau, \sigma)$ supported on the past time half space $\{(\tau, \sigma), 0 \le \sigma \le \tau\}$, then

$$ \begin{aligned} w(\tau)^{-1}T(f)(\tau) &= e^{-\kappa\tau}T(f)(\tau) = \int K(\tau, \sigma)e^{-\kappa\tau}f(\sigma)d\sigma \\ &= \int_0^\tau K(\tau, \sigma)e^{-\kappa(\tau-\sigma)}e^{-\kappa\sigma}f(\sigma)d\sigma, \end{aligned} $$
so we are reduced to the case of $T(w^{-1}f)$, up to an extra coefficient $e^{-\kappa(\tau-\sigma)}$ which is $\kappa$-uniformly bounded by 1, because of the time support of the kernel $K$.

With this point in mind, we repeat the previous computations in the weighted Hölder spaces. The assumptions on the components of the enhanced noise allow us to use Schauder estimates, Theorem 30, and gain an extra factor $\kappa^{-(\delta_2-\delta_1)/2}$. This is the analogue of (4.11) in the above unweighted proof. Taking $\kappa$ large enough allows to compensate any ‘implicit’ constants in the different estimates and get the contraction property of the map $\Phi$. We then conclude the proof of the existence and uniqueness of the fixed point, for an arbitrary horizon time. This approach using weighted spaces only works for a linear application $f$. $\square$

### 4.3.2. Generalized (KPZ) equation.
We proceed exactly as in the preceding section, working with an initial condition $u_0 \in C^{4\alpha}(\mathbf{T})$. We assume a coherent enhancement $\tilde{\zeta}$ of the (gKPZ) noise $\zeta$ is given, and work with the associated reference system $\mathbf{Z}$. Recall from (4.6) that the components of $\mathbf{Z}$ are naturally indexed by the index sets $\mathcal{A}$ and $\mathcal{B}_{\le 3\alpha} := \{b \in \mathcal{B}; |b| \le 3\alpha\}$. Here again, a ‘formal’ solution of the equation needs to satisfy the constraint

$$u_b = h_b(u),$$

for $b \in \mathcal{B}_{\le 3\alpha}$ and explicit functions $h_b$, in addition to the constraints $u_a = h_a(u)$, $a \in \mathcal{A}$, on the components that are common with the (gPAM) equation. One has for instance

$$h_{(1)(1)}(u) = (gf^2)(u), \quad h_{(1)(2)}(u) = (gf^2 f')(u), \quad h_{(1)((1)(1))}(u) = (g^2 f^2)(u).$$

(Our index notation becomes a bit messy on concrete examples, and the tree-indexed notation of regularity structures seems more appropriate to get concise notations.) Set

$$S_T^{\text{kpz}}(u_0) := \{\hat{u}^\sharp; u_{a|\tau=0}^\sharp = h_a(u_0), u_{|\tau=0}^\sharp = u_0\}.$$

Equipped with the natural norm induced from $\prod_{a \in \mathcal{A} \cup \mathcal{B}_{\le 3\alpha}} C^{(3-|a|)\alpha + \beta_a} \times C^{3\alpha + \beta}$, the space $S_T^{\text{kpz}}(u_0)$ is complete. Recall the definition of the map $\Psi$ defined in (4.5).

**DEFINITION.** A solution of the (gKPZ) equation is a fixed point of $\Psi$ in $S_T^{\text{kpz}}(u_0)$.

The very same reasoning as above provides the local in time well-posed character of the (gKPZ) equation; details are left to the reader.

**THEOREM 26.** Given $u_0 \in C^{4\alpha}(\mathbf{T})$, one can choose the time horizon $T$ sufficiently small for the generalized (KPZ) equation to have a unique solution in $S_T^{\text{kpz}}(u_0)$. This solution depends continuously on the coherent enhancement $\tilde{\zeta}$ of the noise $\zeta$.
so we are reduced to the case of $T(w^{-1}f)$, up to an extra coefficient $e^{-\kappa(\tau-\sigma)}$ which is $\kappa$-uniformly bounded by 1, because of the time support of the kernel $K$.

With this point in mind, we repeat the previous computations in the weighted Hölder spaces. The assumptions on the components of the enhanced noise allow us to use Schauder estimates, Theorem 30, and gain an extra factor $\kappa^{-(\delta_2-\delta_1)/2}$. This is the analogue of (4.11) in the above unweighted proof. Taking $\kappa$ large enough allows to compensate any ‘implicit’ constants in the different estimates and get the contraction property of the map $\Phi$. We then conclude the proof of the existence and uniqueness of the fixed point, for an arbitrary horizon time. This approach using weighted spaces only works for a linear application $f$. $\square$

### 4.3.2. Generalized (KPZ) equation.
We proceed exactly as in the preceding section, working with an initial condition $u_0 \in C^{4\alpha}(\mathbf{T})$. We assume a coherent enhancement $\tilde{\zeta}$ of the (gKPZ) noise $\zeta$ is given, and work with the associated reference system $\mathbf{Z}$. Recall from (4.6) that the components of $\mathbf{Z}$ are naturally indexed by the index sets $\mathcal{A}$ and $\mathcal{B}_{\le 3\alpha} := \{b \in \mathcal{B}; |b| \le 3\alpha\}$. Here again, a ‘formal’ solution of the equation needs to satisfy the constraint

$$u_b = h_b(u),$$

for $b \in \mathcal{B}_{\le 3\alpha}$ and explicit functions $h_b$, in addition to the constraints $u_a = h_a(u)$, $a \in \mathcal{A}$, on the components that are common with the (gPAM) equation. One has for instance

$$h_{(1)(1)}(u) = (gf^2)(u), \quad h_{(1)(2)}(u) = (gf^2 f')(u), \quad h_{(1)((1)(1))}(u) = (g^2 f^2)(u).$$

(Our index notation becomes a bit messy on concrete examples, and the tree-indexed notation of regularity structures seems more appropriate to get concise notations.) Set

$$S_T^{\text{kpz}}(u_0) := \{\hat{u}^{\sharp}; u_{a|\tau=0}^{\sharp} = h_a(u_0), u_{|\tau=0}^{\sharp} = u_0\}.$$

Equipped with the natural norm induced from $\prod_{a \in \mathcal{A} \cup \mathcal{B}_{\le 3\alpha}} C^{(3-|a|)\alpha + \beta_a} \times C^{3\alpha + \beta}$, the space $S_T^{\text{kpz}}(u_0)$ is complete. Recall the definition of the map $\Psi$ defined in (4.5).

**DEFINITION.** A solution of the (gKPZ) equation is a fixed point of $\Psi$ in $S_T^{\text{kpz}}(u_0)$.

The very same reasoning as above provides the local in time well-posed character of the (gKPZ) equation; details are left to the reader.

**THEOREM 26.** Given $u_0 \in C^{4\alpha}(\mathbf{T})$, one can choose the time horizon $T$ sufficiently small for the generalized (KPZ) equation to have a unique solution in $S_T^{\text{kpz}}(u_0)$. This solution depends continuously on the coherent enhancement $\tilde{\zeta}$ of the noise $\zeta$.
## Appendix A. Details on the parabolic setting

For the reader's convenience, we recall in this appendix a number of notions/facts introduced and studied in detail in our previous work [2], with the hope that this will make the reading of the present work self-contained. We refer the reader to [2] for the proofs of the different statements given here. We describe in Section A.1 a class of operators with some cancellation property; they play here the role played by the Fourier projectors $\Delta_i$ on dyadic blocks in Littlewood-Paley theory. Parabolic Hölder spaces are described in Section A.2, together with Schauder estimates in this scale of spaces. We introduce the pair $(\mathbb{P}, \tilde{\mathbb{P}})$ of paraproducts in Section A.3. The statements given here are explicitly used in the proofs of the continuity results of Section 3, to be found in Appendix C.

We use the notations introduced in Section 1.3 and assume the operator $L$ satisfies the assumption stated there. Recall we denote by $e$ a generic element of the parabolic space $\mathcal{M}$.

**A.1. Approximation operators.** The use of paraproducts and other kind of singular operators involve the fundamental notion of approximation operators, some aspects of which we discuss in this section. Recall typical space/time points are denoted by $(\tau, x)$ and $(\sigma, y)$.

The following parabolic Gaussian-like kernels $(\mathcal{G}_t)_{0<t\le 1}$ will be used as reference kernels. For $0 < t \le 1$ and $\sigma \le \tau$, set

$$ \mathcal{G}_t((\tau, x), (\sigma, y)) := \nu(B_{\mathcal{M}}((\tau, x), \sqrt{t}))^{-1} \left(1 + c \frac{\rho((\tau, x), (\sigma, y))^2}{t}\right)^{-\ell_1} $$

and set $\mathcal{G}_t \equiv 0$ if $\tau \le \sigma$. We do not emphasize the dependence of $\mathcal{G}$ on the positive constant $c$ in the above definition, and we shall allow ourselves to abuse notations and write $\mathcal{G}_t$ for two functions corresponding to two different values of that constant. So we have for instance, for $s, t \in (0, 1)$, the estimate

$$ \int_{\mathcal{M}} \mathcal{G}_t((\tau, x), (\sigma, y)) \mathcal{G}_s((\sigma, y), (\lambda, z)) \nu(d\sigma dy) \lesssim \mathcal{G}_{t+s}((\tau, x), (\lambda, z)). \quad (A.1) $$

Presently, note that a choice of large enough constant $\ell_1$ in the definition of $\mathcal{G}_t$ ensures that we have

$$ \sup_{t \in (0,1]} \sup_{(\tau,x) \in \mathcal{M}} \int_{\mathcal{M}} \mathcal{G}_t((\tau, x), (\sigma, y)) \nu(d\sigma dy) < \infty, $$

so any linear operator on $\mathcal{M}$, with a kernel pointwisely bounded by some $\mathcal{G}_t$ is bounded in $L^p(\nu)$ for every $p \in [1, \infty]$.
## Appendix A. Details on the parabolic setting

For the reader's convenience, we recall in this appendix a number of notions/facts introduced and studied in detail in our previous work [2], with the hope that this will make the reading of the present work self-contained. We refer the reader to [2] for the proofs of the different statements given here. We describe in Section A.1 a class of operators with some cancellation property; they play here the role played by the Fourier projectors $\Delta_i$ on dyadic blocks in Littlewood-Paley theory. Parabolic Hölder spaces are described in Section A.2, together with Schauder estimates in this scale of spaces. We introduce the pair $(\mathbb{P}, \tilde{\mathbb{P}})$ of paraproducts in Section A.3. The statements given here are explicitly used in the proofs of the continuity results of Section 3, to be found in Appendix C.

We use the notations introduced in Section 1.3 and assume the operator $L$ satisfies the assumption stated there. Recall we denote by $e$ a generic element of the parabolic space $\mathcal{M}$.

**A.1. Approximation operators.** The use of paraproducts and other kind of singular operators involve the fundamental notion of approximation operators, some aspects of which we discuss in this section. Recall typical space/time points are denoted by $(\tau, x)$ and $(\sigma, y)$.

The following parabolic Gaussian-like kernels $(\mathcal{G}_t)_{0<t\le 1}$ will be used as reference kernels. For $0 < t \le 1$ and $\sigma \le \tau$, set

$$ \mathcal{G}_t((\tau, x), (\sigma, y)) := \nu(B_{\mathcal{M}}((\tau, x), \sqrt{t}))^{-1} \left( 1 + c \frac{\rho((\tau, x), (\sigma, y))^2}{t} \right)^{-\ell_1} $$

and set $\mathcal{G}_t \equiv 0$ if $\tau \le \sigma$. We do not emphasize the dependence of $\mathcal{G}$ on the positive constant $c$ in the above definition, and we shall allow ourselves to abuse notations and write $\mathcal{G}_t$ for two functions corresponding to two different values of that constant. So we have for instance, for $s, t \in (0, 1)$, the estimate

$$ \int_{\mathcal{M}} \mathcal{G}_t((\tau, x), (\sigma, y)) \mathcal{G}_s((\sigma, y), (\lambda, z)) \nu(d\sigma dy) \lesssim \mathcal{G}_{t+s}((\tau, x), (\lambda, z)). \quad (A.1) $$

Presently, note that a choice of large enough constant $\ell_1$ in the definition of $\mathcal{G}_t$ ensures that we have

$$ \sup_{t \in (0,1]} \sup_{(\tau,x) \in \mathcal{M}} \int_{\mathcal{M}} \mathcal{G}_t((\tau, x), (\sigma, y)) \nu(d\sigma dy) < \infty, $$

so any linear operator on $\mathcal{M}$, with a kernel pointwisely bounded by some $\mathcal{G}_t$ is bounded in $L^p(\nu)$ for every $p \in [1, \infty]$.
DEFINITION. We shall denote throughout by $\mathcal{G}$ the set of families $(P_t)_{0<t\le 1}$ of linear operators on $\mathcal{M}$, with kernels pointwisely bounded by

$$|K_{P_t}(e, e')| \lesssim \mathcal{G}_t(e, e').$$

Given a real-valued integrable function $\phi$ on $\mathbb{R}$, set

$$\phi_t(\cdot) := \frac{1}{t} \phi\left(\frac{\cdot}{t}\right);$$

the family $(\phi_t)_{0<t\le 1}$ is uniformly bounded in $L^1(\mathbb{R})$. We also define the 'convolution' operator $\phi^*$ associated with $\phi$ via the formula

$$\phi^*(f)(\tau) := \int_0^\infty \phi(\tau - \sigma) f(\sigma) d\sigma.$$

Note that if $\phi$ has support in $\mathbb{R}_+$, then the operator $\phi^*$ has a kernel supported on the same set $\{(\sigma, \tau); \sigma \le \tau\}$ as our Gaussian-like kernel. Moreover, we let the reader check that if $\phi_1, \phi_2$ are two functions in $L^1$, with $\phi_2$ supported on $[0, \infty)$ then

$$(\phi_1 * \phi_2)^* = \phi_1^* \circ \phi_2^*,$$

where $\phi_1 * \phi_2$ stands for the usual convolution of $\phi_1$ and $\phi_2$.

Given an integer $b \ge 1$, we define a family of operators on $L^2(M)$ setting

$$Q_t^{(b)} := \gamma_b^{-1}(tL)^b e^{-tL} \quad \text{and} \quad -t \partial_t P_t^{(b)} = Q_t^{(b)},$$

with $\gamma_b := (b-1)!$; so $P_t^{(b)}$ is an operator of the form $p_b(tL)e^{-tL}$, for some polynomial $p_b$ of degree $b-1$, with value 1 in 0. Recall from Section 1.3 the definition of the differential operators $V_I$, for $I$ a multiindex. Under the assumptions on $L$ stated in Section 1.3, the operators $P_t^{(b)}$ and $Q_t^{(b)}$ both satisfy, for any multiindex $I$, the Gaussian regularity estimates

$$|K_{t^{|I|/2}V_I R}(x, y)| \vee |K_{t^{|I|/2}RV_I}(x, y)| \lesssim \frac{1}{\mu(B(x, \sqrt{t}))} e^{-c(d(x, y)^2/t)},$$

with $R$ standing here for $P_t^{(b)}$ or $Q_t^{(b)}$, as well as the following pointwise regularity estimates. For $d(x, z) \le \sqrt{t}$, we have

$$|K(x, y) - K(z, y)| \lesssim \frac{d(x, z)}{\sqrt{t}} \frac{1}{V(x, \sqrt{t})} e^{-c(d(x, y)^2/\sqrt{t})},$$

where $K$ is the kernel of either $t^{|I|/2}V_I R$ or $t^{|I|/2}RV_I$.
DEFINITION. We shall denote throughout by $\mathcal{G}$ the set of families $(P_t)_{0<t\le 1}$ of linear operators on $\mathcal{M}$, with kernels pointwisely bounded by

$$|K_{P_t}(e, e')| \lesssim G_t(e, e').$$

Given a real-valued integrable function $\phi$ on $\mathbb{R}$, set

$$\phi_t(\cdot) := \frac{1}{t} \phi\left(\frac{\cdot}{t}\right);$$

the family $(\phi_t)_{0<t\le 1}$ is uniformly bounded in $L^1(\mathbb{R})$. We also define the 'convolution' operator $\phi^*$ associated with $\phi$ via the formula

$$\phi^*(f)(\tau) := \int_0^\infty \phi(\tau - \sigma) f(\sigma) d\sigma.$$

Note that if $\phi$ has support in $\mathbb{R}_+$, then the operator $\phi^*$ has a kernel supported on the same set $\{(\sigma, \tau); \sigma \le \tau\}$ as our Gaussian-like kernel. Moreover, we let the reader check that if $\phi_1, \phi_2$ are two functions in $L^1$, with $\phi_2$ supported on $[0, \infty)$ then

$$(\phi_1 * \phi_2)^* = \phi_1^* \circ \phi_2^*,$$

where $\phi_1 * \phi_2$ stands for the usual convolution of $\phi_1$ and $\phi_2$.

Given an integer $b \ge 1$, we define a family of operators on $L^2(M)$ setting

$$Q_t^{(b)} := \gamma_b^{-1}(tL)^b e^{-tL} \quad \text{and} \quad -t \partial_t P_t^{(b)} = Q_t^{(b)},$$

with $\gamma_b := (b-1)!$; so $P_t^{(b)}$ is an operator of the form $p_b(tL)e^{-tL}$, for some polynomial $p_b$ of degree $b-1$, with value 1 in 0. Recall from Section 1.3 the definition of the differential operators $V_I$, for $I$ a multiindex. Under the assumptions on $L$ stated in Section 1.3, the operators $P_t^{(b)}$ and $Q_t^{(b)}$ both satisfy, for any multiindex $I$, the Gaussian regularity estimates

$$|K_{t^{|I|/2}V_I R}(x, y)| \vee |K_{t^{|I|/2}RV_I}(x, y)| \lesssim \frac{1}{\mu(B(x, \sqrt{t}))} e^{-c(d(x, y)^2/t)},$$

with $R$ standing here for $P_t^{(b)}$ or $Q_t^{(b)}$, as well as the following pointwise regularity estimates. For $d(x, z) \le \sqrt{t}$, we have

$$|K(x, y) - K(z, y)| \lesssim \frac{d(x, z)}{\sqrt{t}} \frac{1}{V(x, \sqrt{t})} e^{-c(d(x, y)^2/2t)},$$

where $K$ is the kernel of either $t^{|I|/2}V_I R$ or $t^{|I|/2}RV_I$.
The parameters $b$ and $l_1$ are chosen large enough and fixed once and for all – see [2] to see how to choose these parameters. The reader should simply keep in mind that the higher $b$ and $l_1$ are, the higher order of regularity we can deal with. In our applications, we need all the objects to have a regularity order in the range $(-3, 3)$, so $b$ and $l_1$ are chosen big enough to allow for this range in all the following continuity results.

**DEFINITION 27.** Let an integer $a \in [0, 2b]$ be given. The following collection of families of operators is called the **standard collection of operators with cancellation of order a**, denoted by $\text{StGC}^a$. It is made up of all the space-time operators

$$((t^{|J|/2}V_J)(tL)^{(a-|J|-2k)/2}P_t^{(c)} \otimes m_t^*)_{0<t\le 1}$$

where $k$ is an integer with $2k + |J| \le a$, and $c \in [1, b]$, and $m$ is any smooth function supported on $[\frac{1}{2}, 2]$ such that

$$\int \tau^i m(\tau) d\tau = 0, \qquad (A.2)$$

for all $0 \le i \le k - 1$, with its first $b$ derivatives bounded by 1. We also set

$$\text{StGC}^{[0,2b]} := \bigcup_{0 \le a \le 2b} \text{StGC}^a.$$

The above mentioned cancellation effect is quantified by the property (A.3) stated in Proposition 28 below. Note here that it makes sense at an intuitive level to say that $L^{(a-|J|-2k)/2}$ encodes cancellation in the space variable of order $a - |J| - 2k$, that $V_J$ encodes a cancellation in space of order $|J|$ and that the moment condition (A.2) encodes a cancellation property in the time variable of order $k$ for the convolution operator $m_t^*$. Since we are in the parabolic scaling, a cancellation of order $k$ in time corresponds to a cancellation of order $2k$ in space, so the operator $V_J L^{(a-|J|-2k)/2} P_t^{(c)} \otimes m_t^*$ is expected to have a space-time cancellation property of order $a$.

**DEFINITION.** Given an operator $Q := V_I \phi(L)$, with $|I| \ge 1$, defined by functional calculus from some appropriate function $\phi$, we write $Q^*$ for the **formal dual operator**

$$Q^* := \phi(L)V_I.$$

For $I = \emptyset$, and $Q = \phi(L)$, we set $Q^* := Q$. For an operator $Q$ as above we set

$$(Q \otimes m^*)^* := Q^* \otimes m^*.$$
The parameters $b$ and $l_1$ are chosen large enough and fixed once and for all – see [2] to see how to choose these parameters. The reader should simply keep in mind that the higher $b$ and $l_1$ are, the higher order of regularity we can deal with. In our applications, we need all the objects to have a regularity order in the range $(-3, 3)$, so $b$ and $l_1$ are chosen big enough to allow for this range in all the following continuity results.

**DEFINITION 27.** Let an integer $a \in [0, 2b]$ be given. The following collection of families of operators is called the **standard collection of operators with cancellation of order a**, denoted by $\text{StGC}^a$. It is made up of all the space-time operators

$$((t^{|J|/2}V_J)(tL)^{(a-|J|-2k)/2}P_t^{(c)} \otimes m_t^*)_{0<t\le 1}$$

where $k$ is an integer with $2k + |J| \le a$, and $c \in [1, b]$, and $m$ is any smooth function supported on $[\frac{1}{2}, 2]$ such that

$$\int \tau^i m(\tau) d\tau = 0, \qquad (A.2)$$

for all $0 \le i \le k - 1$, with its first $b$ derivatives bounded by 1. We also set

$$\text{StGC}^{[0,2b]} := \bigcup_{0 \le a \le 2b} \text{StGC}^a.$$

The above mentioned cancellation effect is quantified by the property (A.3) stated in Proposition 28 below. Note here that it makes sense at an intuitive level to say that $L^{(a-|J|-2k)/2}$ encodes cancellation in the space variable of order $a - |J| - 2k$, that $V_J$ encodes a cancellation in space of order $|J|$ and that the moment condition (A.2) encodes a cancellation property in the time variable of order $k$ for the convolution operator $m_t^*$. Since we are in the parabolic scaling, a cancellation of order $k$ in time corresponds to a cancellation of order $2k$ in space, so the operator $V_J L^{(a-|J|-2k)/2} P_t^{(c)} \otimes m_t^*$ is expected to have a space-time cancellation property of order $a$.

**DEFINITION.** Given an operator $Q := V_I \phi(L)$, with $|I| \ge 1$, defined by functional calculus from some appropriate function $\phi$, we write $Q^*$ for the **formal dual operator**

$$Q^* := \phi(L)V_I.$$

For $I = \emptyset$, and $Q = \phi(L)$, we set $Q^* := Q$. For an operator $Q$ as above we set

$$(Q \otimes m^*)^* := Q^* \otimes m^*.$$
Note that the above definition is not related to any classical notion of duality, and emphasize that we do not assume that $L$ is self-adjoint in $L^2(\mu)$. This notation is only used to indicate that a $Q_t$ operator, respectively a $Q_t^*$ operator, can be composed on the right, respectively on the left, by another operator $\psi(L)$, for a suitable function $\psi$, due to the functional calculus for $L$. $L$ is supposed to be sectorial and to have a holomorphic functional calculus, so for example it is known to give a sense to $\phi(L)$ for every holomorphic function $\phi$ (or function which are holomorphic and bounded in a small sector of the complex plane around $(0, \infty)$, as $\phi(z) = z^k e^{-z}$ for example).

**PROPOSITION 28.** Consider $Q^1 \in \text{StGC}^{a_1}$ and $Q^2 \in \text{StGC}^{a_2}$ two standard collections with cancellation, and set $a := \min(a_1, a_2)$. Then for every $s$, $t \in (0, 1]$, the operator $Q_s^1 \circ Q_s^2$ has a kernel pointwisely bounded by

$$ |K_{Q_s^1 \circ Q_s^2}(e, e')| \lesssim \left( \frac{ts}{(s+t)^2} \right)^{a/2} \mathcal{G}_{t+s}(e, e'). \quad (A.3) $$

The above mentioned orthogonality property of standard operators with cancellation is encoded in the factor $(ts/(s+t)^2)^{a/2}$ that appears in the above estimate. This factor is small as soon as $s$ or $t$ is small compared to the other.

**DEFINITION.** Let $a$ be an integer in $[0, 2b]$. We define the subset $\text{GC}^a$ of $\mathcal{G}$ of families of operators with the cancellation property of order $a$ as the set of elements $Q$ of $\mathcal{G}$ with the following cancellation property. For every $0 < s, t \le 1$ and every standard family $S \in \text{StGC}^{a'}$, with $a' \in [a, 2b]$, the operator $Q_t \circ S^*$ has a kernel pointwisely bounded by

$$ |K_{Q_t \circ S^*}(e, e')| \lesssim \left( \frac{st}{(s+t)^2} \right)^{a/2} \mathcal{G}_{t+s}(e, e'). \quad (A.4) $$

We introduced above the operators $Q_t^{(b)}$ and $P_t^{(b)}$ acting on functions/distributions on $M$; we now introduce their parabolic counterpart. Choose arbitrarily a smooth real-valued function $\varphi$ on $\mathbb{R}$, with support in $[\frac{1}{2}, 2]$, unit integral and such that for every integer $k = 1, \dots, b$

$$ \int \tau^k \varphi(\tau) d\tau = 0. $$

Set

$$ \mathcal{P}_t^{(b)} := P_t^{(b)} \otimes \varphi_t^* \quad \text{and} \quad Q_t^{(b)} := -t \partial_t \mathcal{P}_t^{(b)}. $$
Note that the above definition is *not* related to any classical notion of duality, and emphasize that we do *not* assume that $L$ is self-adjoint in $L^2(\mu)$. This notation is only used to indicate that a $Q_t$ operator, respectively a $Q_t^\bullet$ operator, can be composed on the right, respectively on the left, by another operator $\psi(L)$, for a suitable function $\psi$, due to the functional calculus for $L$. $L$ is supposed to be sectorial and to have a holomorphic functional calculus, so for example it is known to give a sense to $\phi(L)$ for every holomorphic function $\phi$ (or function which are holomorphic and bounded in a small sector of the complex plane around $(0, \infty)$, as $\phi(z) = z^k e^{-z}$ for example).

**PROPOSITION 28.** Consider $Q^1 \in \text{StGC}^{a_1}$ and $Q^2 \in \text{StGC}^{a_2}$ two standard collections with cancellation, and set $a := \min(a_1, a_2)$. Then for every $s$, $t \in (0, 1]$, the operator $Q_s^1 \circ Q_s^2$ has a kernel pointwisely bounded by

$$|K_{Q_s^1 \circ Q_s^2}(e, e')| \lesssim \left( \frac{ts}{(s+t)^2} \right)^{a/2} \mathcal{G}_{t+s}(e, e'). \quad (A.3)$$

The above mentioned *orthogonality* property of standard operators with cancellation is encoded in the factor $(ts/(s+t)^2)^{a/2}$ that appears in the above estimate. This factor is small as soon as $s$ or $t$ is small compared to the other.

**DEFINITION.** Let $a$ be an integer in $[0, 2b]$. We define the subset $\mathcal{GC}^a$ of $\mathcal{G}$ of families of operators with the cancellation property of order $a$ as the set of elements $Q$ of $\mathcal{G}$ with the following cancellation property. For every $0 < s, t \le 1$ and every standard family $S \in \text{StGC}^{a'}$, with $a' \in [a, 2b]$, the operator $Q_t \circ S^\bullet$ has a kernel pointwisely bounded by

$$|K_{Q_t \circ S^\bullet}(e, e')| \lesssim \left( \frac{st}{(s+t)^2} \right)^{a/2} \mathcal{G}_{t+s}(e, e'). \quad (A.4)$$

We introduced above the operators $Q_t^{(b)}$ and $P_t^{(b)}$ acting on functions/distributions on $M$; we now introduce their parabolic counterpart. Choose arbitrarily a smooth real-valued function $\varphi$ on $\mathbb{R}$, with support in $[\frac{1}{2}, 2]$, unit integral and such that for every integer $k = 1, \dots, b$

$$\int \tau^k \varphi(\tau) d\tau = 0.$$

Set

$$\mathcal{P}_t^{(b)} := P_t^{(b)} \otimes \varphi_t^\star \quad \text{and} \quad \mathcal{Q}_t^{(b)} := -t \partial_t \mathcal{P}_t^{(b)}.$$
An easy computation yields that

$$Q_t^{(b)} = Q_t^{(b)} \otimes \varphi_t^* + P_t^{(b)} \otimes \psi_t^*$$

where $\psi(\sigma) = \varphi(\sigma) + \sigma\varphi'(\sigma)$. Note that, from its very definition, a parabolic operator $Q_t^{(b)}$ belongs at least to $\mathbb{G}\mathbb{C}^2$, for $b \ge 2$. Note also that due to the normalization of $\varphi$, then for every $f \in L^p(\mathbb{R})$ supported on $[0, \infty)$, we have the convergence in $L^p$

$$\varphi_t^*(f) \xrightarrow{t \to 0} f.$$

So, the operators $P_t$ tend weakly as $t$ goes to 0 to the identity on $L_{[0,\infty)}^p(M)$, the set of functions $f \in L^p(M)$ with time support included in $[0, \infty)$, with $p \in [1, \infty)$. The same convergence holds on the set of functions $f \in C^0(M)$ with time support included in $[0, \infty)$. The following **Calderón reproducing formula** follows as a consequence. For every continuous function $f \in L^\infty(M)$ with time support in $[0, \infty)$, then

$$f = \int_0^1 Q_t^{(b)} f \frac{dt}{t} + P_1^{(b)} f. \quad (A.5)$$

Noting that the measure $dt/t$ gives unit mass to intervals of the form $[2^{-i-1}, 2^{-i}]$, and considering the operator $Q_t^{(b)}$ as a kind of multiplier roughly localized at 'frequencies' of size $t^{-1/2}$, Calderón's formula appears as nothing else but a continuous time analogue of the Littlewood-Paley decomposition of $f$, with $dt/t$ in the role of the counting measure.

## A.2. Parabolic Hölder spaces and Schauder estimates.
We recall in this section the definitions and basic properties of the space and space-time-weighted Hölder spaces, with possibly negative regularity index. We also recall the fundamental regularization properties of the heat operator, quantified by Schauder estimates.

Let us start recalling the following well-known facts about Hölder space on $M$, and single out a good class of weights on the parabolic space $M$. Given $0 < \alpha \le 1$, the classical metric Hölder space $H^\alpha$ is defined as the set of real-valued functions $f$ on $M$ with finite $H^\alpha$-norm, defined by the formula

$$\|f\|_{H^\alpha} := \|f\|_{L^\infty(M)} + \sup_{x \neq y \in M} \frac{|f(x) - f(y)|}{d(x, y)^\alpha} < \infty.$$

**DEFINITION.** For $\alpha \in (-3, 3)$, define $C^\alpha := C^\alpha(M)$ as the closure of the set of bounded and smooth functions for the $C^\alpha$-norm, defined by the formula

$$\|f\|_{C^\alpha} := \|e^{-L} f\|_{L^\infty(M)} + \sup_{0 < t \le 1} t^{-\alpha/2} \|Q_t^{(b)} f\|_{L^\infty(M)}.$$
An easy computation yields that

$$Q_t^{(b)} = Q_t^{(b)} \otimes \varphi_t^* + P_t^{(b)} \otimes \psi_t^*$$

where $\psi(\sigma) = \varphi(\sigma) + \sigma\varphi'(\sigma)$. Note that, from its very definition, a parabolic operator $Q_t^{(b)}$ belongs at least to $\mathbb{G}\mathbb{C}^2$, for $b \ge 2$. Note also that due to the normalization of $\varphi$, then for every $f \in L^p(\mathbb{R})$ supported on $[0, \infty)$, we have the convergence in $L^p$

$$\varphi_t^*(f) \xrightarrow{t \to 0} f.$$

So, the operators $P_t$ tend weakly as $t$ goes to 0 to the identity on $L_{[0,\infty)}^p(M)$, the set of functions $f \in L^p(M)$ with time support included in $[0, \infty)$, with $p \in [1, \infty)$. The same convergence holds on the set of functions $f \in C^0(M)$ with time support included in $[0, \infty)$. The following **Calderón reproducing formula** follows as a consequence. For every continuous function $f \in L^\infty(M)$ with time support in $[0, \infty)$, then

$$f = \int_0^1 Q_t^{(b)} f \frac{dt}{t} + P_1^{(b)} f. \quad (A.5)$$

Noting that the measure $dt/t$ gives unit mass to intervals of the form $[2^{-i-1}, 2^{-i}]$, and considering the operator $Q_t^{(b)}$ as a kind of multiplier roughly localized at ‘frequencies’ of size $t^{-1/2}$, Calderón’s formula appears as nothing else but a continuous time analogue of the Littlewood-Paley decomposition of $f$, with $dt/t$ in the role of the counting measure.

## A.2. Parabolic Hölder spaces and Schauder estimates.
We recall in this section the definitions and basic properties of the space and space-time-weighted Hölder spaces, with possibly negative regularity index. We also recall the fundamental regularization properties of the heat operator, quantified by Schauder estimates.

Let us start recalling the following well-known facts about Hölder space on $M$, and single out a good class of weights on the parabolic space $M$. Given $0 < \alpha \le 1$, the classical metric Hölder space $H^\alpha$ is defined as the set of real-valued functions $f$ on $M$ with finite $H^\alpha$-norm, defined by the formula

$$\|f\|_{H^\alpha} := \|f\|_{L^\infty(M)} + \sup_{x \neq y \in M} \frac{|f(x) - f(y)|}{d(x, y)^\alpha} < \infty.$$

**DEFINITION.** For $\alpha \in (-3, 3)$, define $C^\alpha := C^\alpha(M)$ as the closure of the set of bounded and smooth functions for the $C^\alpha$-norm, defined by the formula

$$\|f\|_{C^\alpha} := \|e^{-L} f\|_{L^\infty(M)} + \sup_{0<t \le 1} t^{-\alpha/2} \|Q_t^{(b)} f\|_{L^\infty(M)}.$$
This norm does not depend on the integer $b > |\alpha|/2$, and the two spaces $H^\alpha$ and $C^\alpha$ coincide and have equivalent norms when $0 < \alpha < 1$ – see for instance [1, Proposition 2.5]. These notions have parabolic counterparts which we now introduce. Recall we work with the parabolic space $\mathcal{M} = [0, T] \times M$, for a finite time horizon $T$. The introduction of a time weight in the next definition thus has no effect on the space involved, nor on its topology. Its introduction happens however to be a convenient freedom which allows to simplify a number of arguments. Let then a nonnegative parameter $\kappa$ be given and denote by $w$ the weight

$$w(\tau) := e^{\kappa\tau}. \qquad (A.6)$$

For $0 < \alpha \le 1$, the metric parabolic Hölder space $\mathcal{H}^\alpha = \mathcal{H}^\alpha(\mathcal{M})$ is defined as the set of all functions on $\mathcal{M}$ with finite $\mathcal{H}^\alpha$-norm, defined by the formula

$$\|f\|_{\mathcal{H}^{\alpha}} := \|w^{-1}f\|_{L^{\infty}(\mathcal{M})} + \sup_{0 < \rho((\tau,x), (\sigma, y)) \le 1; \tau \ge \sigma} w^{-1}(\tau) \frac{|f(\tau, x) - f(\sigma, y)|}{\rho((\tau, x), (\sigma, y))^{\alpha}}.$$

As in the above space setting one can recast this definition in a more functional setting, using the parabolic standard operators. A set of distributions was introduced in [2], whose precise definition is irrelevant here.

**DEFINITION.** For $\alpha \in (-3, 3)$, we define the parabolic Hölder space

$$C^{\alpha} = C^{\alpha}(\mathcal{M}) = C_{w}^{\alpha}(\mathcal{M}) = C_{w}^{\alpha}$$

as the closure, in the set of distributions, of the set of bounded and continuous functions on $\mathcal{M}$ for the $C^\alpha - w$-norm, defined by

$$\|f\|_{C_w^\alpha} := \sup_{\substack{Q \in \text{StGC}^k \\ 0 \le k \le 2b}} \|w^{-1} Q_1(f)\|_{L^\infty(\mathcal{M})} + \sup_{\substack{Q \in \text{StGC}^k \\ |\alpha| - k \le 2b}} \sup_{0 < t \le 1} t^{-\alpha/2} \|w^{-1} Q_t(f)\|_{L^\infty(\mathcal{M})}.$$

We write $C_w^\alpha$ if we want to emphasize the dependence of the norm on $w$. The following result was proved in [2], building on Calderón’s formula (A.5).

**PROPOSITION 29.** Choose any finite nonnegative parameter $\kappa$ in the definition (A.6) of the weight $w$. Given $\alpha \in (0, 2)$, set

$$\mathcal{E}^{\alpha} := (C_{\tau}^{\alpha/2} L_{x}^{\infty}) \cap (L_{\tau}^{\infty} C_{x}^{\alpha}),$$

and endow this space with its natural norm. Then $\mathcal{E}^\alpha$ is continuously embedded into $C_w^\alpha$. Furthermore, if $\alpha \in (0, 1)$, the spaces $\mathcal{E}^\alpha, C_w^\alpha$ and $\mathcal{H}^\alpha$ are equal, with equivalent norms.
This norm does not depend on the integer $b > |\alpha|/2$, and the two spaces $H^\alpha$ and $C^\alpha$ coincide and have equivalent norms when $0 < \alpha < 1$ – see for instance [1, Proposition 2.5]. These notions have parabolic counterparts which we now introduce. Recall we work with the parabolic space $\mathcal{M} = [0, T] \times M$, for a finite time horizon $T$. The introduction of a time weight in the next definition thus has no effect on the space involved, nor on its topology. Its introduction happens however to be a convenient freedom which allows to simplify a number of arguments. Let then a nonnegative parameter $\kappa$ be given and denote by $w$ the weight

$$w(\tau) := e^{\kappa\tau}. \qquad (A.6)$$

For $0 < \alpha \le 1$, the metric parabolic Hölder space $\mathcal{H}^\alpha = \mathcal{H}^\alpha(\mathcal{M})$ is defined as the set of all functions on $\mathcal{M}$ with finite $\mathcal{H}^\alpha$-norm, defined by the formula

$$\|f\|_{\mathcal{H}^{\alpha}} := \|w^{-1}f\|_{L^{\infty}(\mathcal{M})} + \sup_{0 < \rho((\tau,x), (\sigma, y)) \le 1; \tau \ge \sigma} w^{-1}(\tau) \frac{|f(\tau, x) - f(\sigma, y)|}{\rho((\tau, x), (\sigma, y))^{\alpha}}.$$

As in the above space setting one can recast this definition in a more functional setting, using the parabolic standard operators. A set of distributions was introduced in [2], whose precise definition is irrelevant here.

**DEFINITION.** For $\alpha \in (-3, 3)$, we define the parabolic Hölder space

$$C^{\alpha} = C^{\alpha}(\mathcal{M}) = C_{w}^{\alpha}(\mathcal{M}) = C_{w}^{\alpha}$$

as the closure, in the set of distributions, of the set of bounded and continuous functions on $\mathcal{M}$ for the $C^\alpha - w$-norm, defined by

$$\|f\|_{C_w^\alpha} := \sup_{\substack{Q \in \text{StGC}^k \\ 0 \le k \le 2b}} \|w^{-1} Q_1(f)\|_{L^\infty(\mathcal{M})} + \sup_{\substack{Q \in \text{StGC}^k \\ |\alpha| - k \le 2b}} \sup_{0 < t \le 1} t^{-\alpha/2} \|w^{-1} Q_t(f)\|_{L^\infty(\mathcal{M})}.$$

We write $C_w^\alpha$ if we want to emphasize the dependence of the norm on $w$. The following result was proved in [2], building on Calderón’s formula (A.5).

**PROPOSITION 29.** Choose any finite nonnegative parameter $\kappa$ in the definition (A.6) of the weight $w$. Given $\alpha \in (0, 2)$, set

$$\mathcal{E}^{\alpha} := (C_{\tau}^{\alpha/2} L_{x}^{\infty}) \cap (L_{\tau}^{\infty} C_{x}^{\alpha}),$$

and endow this space with its natural norm. Then $\mathcal{E}^\alpha$ is continuously embedded into $C_w^\alpha$. Furthermore, if $\alpha \in (0, 1)$, the spaces $\mathcal{E}^\alpha, C_w^\alpha$ and $\mathcal{H}^\alpha$ are equal, with equivalent norms.
The weighted version $(L_\tau^\infty C_x^\alpha)_w$ of $L_\tau^\infty C_x^\alpha$ is the same space, equipped with the norm

$$ \|f\|_{(L_\tau^\infty C_x^\alpha)_w} := \sup_{0 \le \tau \le T} e^{-\kappa \tau} \|f(\tau, \cdot)\|_{C^\alpha}. $$

We use the following regularization properties of the heat operator associated with $L$, in the proof of global in time well-posedness for the (PAM) equation. It is proved under this form in [2, Section 3.4].

**THEOREM 30 (Schauder estimates).** • For any choice of parameters $\beta$ and $\varepsilon > 0$, such that $-2 + 2\varepsilon < \beta < 0$, we have

$$ \|\mathcal{L}^{-1}(v)\|_{C_w^{\beta+2-2\varepsilon}} \lesssim_T \kappa^{-\varepsilon} \|v\|_{(L_T^\infty C_x^\beta)_w}. $$

• Given $\beta \in (0, 2)$ and $\varepsilon \in [0, 1)$, we have the continuity estimate

$$ \|\mathcal{L}^{-1}(v)\|_{C_w^{\beta+2-2\varepsilon}} \lesssim \kappa^{-\varepsilon} \|v\|_{C_w^{\beta}}, $$

for an implicit constant in the inequality independent of $\kappa$.

Before turning to the definition of an intertwined pair of parabolic paraproducts we close this section with another useful continuity property involving the Hölder spaces $C_\omega^\alpha$; it is used in the proof of the continuity properties of the swap an merging operators, Proposition 38 in Appendix C. Recall the manifold $M$ is compact.

**PROPOSITION 31.** Given $\alpha \in (0, 1)$, a space-time weight $\omega$, some integer $a \ge 0$ and a standard family $\mathcal{P} \in \text{StGC}^a$, there exists a constant $c$ depending only on the weight $\omega$, such that

$$ \omega(\tau)^{-1}|(\mathcal{P}_t f)(e) - (\mathcal{P}_s f)(e')| \le c(s + t + \rho(e, e')^2)^{\alpha/2} \|f\|_{C_\omega^\alpha}, $$

uniformly in $s, t \in (0, 1]$ and $e = (\tau, x)$ and $e' = (\sigma, y) \in M$, with $\tau \ge \sigma$.

It is possible, and necessary for our purpose here, to make the link between the notions of regularity defined in terms of the operator $L$ and the usual notion of regularity given by the differentiable structure of the manifold, for regularity indices in the range $(1, 2)$. Since the collection of vector fields $(V_i)_{1 \le i \le \ell_0}$ spans smoothly each tangent space, for every function $f \in C^1(M)$ and every $x \in M$

$$ \nabla f(x) = \sum_{1 \le i \le \ell_0} \gamma_i(x) (V_i f)(x) V_i(x), $$
The weighted version $(L_\tau^\infty C_x^\alpha)_w$ of $L_\tau^\infty C_x^\alpha$ is the same space, equipped with the norm

$$ \|f\|_{(L_\tau^\infty C_x^\alpha)_w} := \sup_{0 \le \tau \le T} e^{-\kappa \tau} \|f(\tau, \cdot)\|_{C^\alpha}. $$

We use the following regularization properties of the heat operator associated with $L$, in the proof of global in time well-posedness for the (PAM) equation. It is proved under this form in [2, Section 3.4].

**THEOREM 30 (Schauder estimates).** • For any choice of parameters $\beta$ and $\varepsilon > 0$, such that $-2 + 2\varepsilon < \beta < 0$, we have

$$ \|\mathcal{L}^{-1}(v)\|_{C_w^{\beta+2-2\varepsilon}} \lesssim_T \kappa^{-\varepsilon} \|v\|_{(L_T^\infty C_x^\beta)_w}. $$

• Given $\beta \in (0, 2)$ and $\varepsilon \in [0, 1)$, we have the continuity estimate

$$ \|\mathcal{L}^{-1}(v)\|_{C_w^{\beta+2-2\varepsilon}} \lesssim \kappa^{-\varepsilon} \|v\|_{C_w^\beta}, $$

for an implicit constant in the inequality independent of $\kappa$.

Before turning to the definition of an intertwined pair of parabolic paraproducts we close this section with another useful continuity property involving the Hölder spaces $C_\omega^\alpha$; it is used in the proof of the continuity properties of the swap an merging operators, Proposition 38 in Appendix C. Recall the manifold $M$ is compact.

**PROPOSITION 31.** Given $\alpha \in (0, 1)$, a space-time weight $\omega$, some integer $a \ge 0$ and a standard family $\mathcal{P} \in \text{StGC}^\alpha$, there exists a constant $c$ depending only on the weight $\omega$, such that

$$ \omega(\tau)^{-1}|(\mathcal{P}_t f)(e) - (\mathcal{P}_s f)(e')| \le c(s+t+\rho(e, e')^2)^{\alpha/2} \|f\|_{C_\omega^\alpha}, $$

uniformly in $s, t \in (0, 1]$ and $e = (\tau, x)$ and $e' = (\sigma, y) \in M$, with $\tau \ge \sigma$.

It is possible, and necessary for our purpose here, to make the link between the notions of regularity defined in terms of the operator $L$ and the usual notion of regularity given by the differentiable structure of the manifold, for regularity indices in the range $(1, 2)$. Since the collection of vector fields $(V_i)_{1 \le i \le \ell_0}$ spans smoothly each tangent space, for every function $f \in C^1(M)$ and every $x \in M$

$$ \nabla f(x) = \sum_{1 \le i \le \ell_0} \gamma_i(x) (V_i f)(x) V_i(x), $$
for a collection $(\gamma_i)_{1 \le i \le \ell_0}$ of smooth coefficients. For two points $x, y \in M$ and $f \in C^2(M)$ we have

$$|f(x) - f(y) - \langle \nabla f(x), \pi_{x,y} \rangle_{T_x M}| \lesssim_f d(x, y)^2,$$

where $T_x M$ is the canonical tangent space of $M$ at the point $x \in M$, and $\pi_{x,y}$ is a tangent vector of $T_x M$ of length $d(x, y)$, whose associated geodesic reaches $y$ at time 1. Such a tangent vector $\pi_{xy}$ is unique if $d(x, y)$ is no greater than $r_M$, the injectivity radius of the compact manifold $M$. Combining these two facts, we have

$$\left| f(x) - f(y) - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) V_{\ell}(f)(x) \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_x M} \right| \lesssim_f d(x, y)^2.$$

We can then define for $\alpha \in (1, 2)$, the space $H^\alpha$ defined by the norm

$$\|f\|_{H^\alpha} := \|f\|_{H^1(M)} \\ + \sup_{\substack{x \neq y \in M \\ d(x,y) < r_M}} \frac{|f(x) - f(y) - \sum_{\ell=1}^{\ell_0} \gamma_\ell(x) V_\ell(f)(x) \langle V_\ell(x), \pi_{x,y} \rangle_{T_x M}|}{d(x,y)^\alpha} \\ < \infty. \qquad (A.7)$$

Following [1, Proposition 2.5], it can be easily proved that for $\alpha \in (1, 2)$ then $C^\alpha$ is continuously embedded into $H^\alpha$: Uniformly in $x, y \in M$ with $d(x, y) \le r_M$, one has

$$\left| f(x) - f(y) - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) V_{\ell}(f)(x) \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_x M} \right| \lesssim_d d(x, y)^{\alpha} \|f\|_{C^{\alpha}}. \quad (A.8)$$

The parabolic counterpart goes as follows, taking into account the fact that because of the parabolic scaling, a regularity of order $\alpha < 2$ can be encoded in finite increments in time, with no need of a higher order expansion, and a first order expansion in space. The precise statement takes the following form.

**PROPOSITION 32.** Given $\alpha \in (1, 2)$ and $f \in C^\alpha$, there exists a positive implicit constant such that for every $e = (\tau, x)$, $e' = (\sigma, y)$ in $M$, with $\rho(e, e') \le r_M$ then

$$\left| f(e) - f(e') - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(e) V_{\ell}(f)(e) \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_x M} \right| \lesssim_f \|f\|_{C^\alpha} \rho(e, e')^{\alpha}.$$

*Proof.* We do not give all the details of the proof since it follows exactly the proof of Proposition 2.5 in [1]. Here is a guideline.
for a collection $(\gamma_i)_{1 \le i \le \ell_0}$ of smooth coefficients. For two points $x, y \in M$ and $f \in C^2(M)$ we have

$$|f(x) - f(y) - \langle \nabla f(x), \pi_{x,y} \rangle_{T_x M}| \lesssim_f d(x, y)^2,$$

where $T_x M$ is the canonical tangent space of $M$ at the point $x \in M$, and $\pi_{x,y}$ is a tangent vector of $T_x M$ of length $d(x, y)$, whose associated geodesic reaches $y$ at time 1. Such a tangent vector $\pi_{xy}$ is unique if $d(x, y)$ is no greater than $r_M$, the injectivity radius of the compact manifold $M$. Combining these two facts, we have

$$\left| f(x) - f(y) - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) V_{\ell}(f)(x) \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_x M} \right| \lesssim_f d(x, y)^2.$$

We can then define for $\alpha \in (1, 2)$, the space $H^\alpha$ defined by the norm

$$\|f\|_{H^\alpha} := \|f\|_{H^1(M)} + \sup_{\substack{x \neq y \in M \\ d(x,y) < r_M}} \frac{|f(x) - f(y) - \sum_{\ell=1}^{\ell_0} \gamma_\ell(x) V_\ell(f)(x) \langle V_\ell(x), \pi_{x,y} \rangle_{T_x M}|}{d(x,y)^\alpha} < \infty. \quad (A.7)$$

Following [1, Proposition 2.5], it can be easily proved that for $\alpha \in (1, 2)$ then $C^\alpha$ is continuously embedded into $H^\alpha$: Uniformly in $x, y \in M$ with $d(x, y) \le r_M$, one has

$$\left| f(x) - f(y) - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) V_{\ell}(f)(x) \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_x M} \right| \lesssim_d d(x, y)^{\alpha} \|f\|_{C^{\alpha}}. \quad (A.8)$$

The parabolic counterpart goes as follows, taking into account the fact that because of the parabolic scaling, a regularity of order $\alpha < 2$ can be encoded in finite increments in time, with no need of a higher order expansion, and a first order expansion in space. The precise statement takes the following form.

**PROPOSITION 32.** Given $\alpha \in (1, 2)$ and $f \in C^\alpha$, there exists a positive implicit constant such that for every $e = (\tau, x)$, $e' = (\sigma, y)$ in $M$, with $\rho(e, e') \le r_M$ then

$$\left| f(e) - f(e') - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(e) V_{\ell}(f)(e) \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_x M} \right| \lesssim_f \|f\|_{C^\alpha} \rho(e, e')^{\alpha}.$$

*Proof.* We do not give all the details of the proof since it follows exactly the proof of Proposition 2.5 in [1]. Here is a guideline.
We decompose $f$ at the scale $r = \rho(e, e')$ with

$$f = (f - \mathcal{P}_{r^2}^2 f) + c \int_{r^2}^{\infty} \mathcal{P}_t Q_t(f) \frac{dt}{t}.$$

We plug this decomposition in the left hand side of the desired inequality. One gets estimate on the contribution of the first part $(f - \mathcal{P}_{r^2}^2 f)$ using that

$$\|f - \mathcal{P}_{r^2}^2 f\|_{\infty} \lesssim r^{\alpha} \|f\|_{C^{\alpha}}, \quad \|V_i(f - \mathcal{P}_{r^2}^2 f)\|_{\infty} \lesssim r^{\alpha-1} \|f\|_{C^{\alpha}}$$

and recalling that $\pi_{x,y}$ has length $d(x, y) \le r$. For the second part, we integrate the contributions along $t > r^2$, applying (A.8) at order 2 to the kernel $K_{\mathcal{P}_t}$ to obtain, for $e = (\tau, x)$ and $e' = (\sigma, y)$, the estimate

$$\left| K_{\mathcal{P}_t}(e, e'') - K_{\mathcal{P}_t}(e', e'') - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) K_{V_i \mathcal{P}_t}(e, e'') \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_t M} \right| \\ \lesssim \left( \frac{r}{\sqrt{t}} \right)^2 \mathcal{G}_t(e, e''),$$

where $\mathcal{G}_t$ is the Gaussian kernel. Integrating this estimate against $Q_t(f)$ gives a factor $(r/\sqrt{t})^{2\alpha/2}$ which can then be integrated along $t \in (r^2, \infty)$ since $\alpha \in (1, 2)$. $\square$

**A.3. Parabolic paraproducts.** We give here a quick presentation of the pair of intertwined paraproducts introduced in [2], following the semigroup approach developed first in [1]. The starting point for the introduction of the operator $\Pi$ is Calderón's reproducing formula (A.5). Using iteratively the Leibniz rule for the differentiation operators $V_i$ or $\partial_\tau$, we have the following decomposition

$$fg = \sum_{\mathcal{I}_b} a_{k, \ell}^{I, J} \int_0^1 (A_{k, \ell}^{I, J}(f, g) + A_{k, \ell}^{I, J}(g, f)) \frac{dt}{t} + \sum_{\mathcal{I}_b} b_{k, \ell}^{I, J} \int_0^1 B_{k, \ell}^{I, J}(f, g) \frac{dt}{t},$$

where

* $\mathcal{I}_b$ is the set of all tuples $(I, J, k, \ell)$ with the tuples $I, J$ and the integers $k, \ell$ satisfying the constraint

$$\frac{|I| + |J|}{2} + k + \ell = \frac{b}{2};$$

* $a_{k, \ell}^{I, J}, b_{k, \ell}^{I, J}$ are bounded sequences of numerical coefficients;
We decompose $f$ at the scale $r = \rho(e, e')$ with

$$f = (f - \mathcal{P}_{r^2}^2 f) + c \int_{r^2}^{\infty} \mathcal{P}_t Q_t(f) \frac{dt}{t}.$$

We plug this decomposition in the left hand side of the desired inequality. One gets estimate on the contribution of the first part $(f - \mathcal{P}_{r^2}^2 f)$ using that

$$\|f - \mathcal{P}_{r^2}^2 f\|_{\infty} \lesssim r^{\alpha} \|f\|_{C^{\alpha}}, \quad \|V_i(f - \mathcal{P}_{r^2}^2 f)\|_{\infty} \lesssim r^{\alpha-1} \|f\|_{C^{\alpha}}$$

and recalling that $\pi_{x,y}$ has length $d(x, y) \le r$. For the second part, we integrate the contributions along $t > r^2$, applying (A.8) at order 2 to the kernel $K_{\mathcal{P}_t}$ to obtain, for $e = (\tau, x)$ and $e' = (\sigma, y)$, the estimate

$$\left| K_{\mathcal{P}_t}(e, e'') - K_{\mathcal{P}_t}(e', e'') - \sum_{\ell=1}^{\ell_0} \gamma_{\ell}(x) K_{V_i \mathcal{P}_t}(e, e'') \langle V_{\ell}(x), \pi_{x,y} \rangle_{T_t M} \right| \lesssim \left( \frac{r}{\sqrt{t}} \right)^2 \mathcal{G}_t(e, e''),$$

where $\mathcal{G}_t$ is the Gaussian kernel. Integrating this estimate against $Q_t(f)$ gives a factor $(r/\sqrt{t})^{2\alpha/2}$ which can then be integrated along $t \in (r^2, \infty)$ since $\alpha \in (1, 2)$. □

**A.3. Parabolic paraproducts.** We give here a quick presentation of the pair of intertwined paraproducts introduced in [2], following the semigroup approach developed first in [1]. The starting point for the introduction of the operator $\Pi$ is Calderón's reproducing formula (A.5). Using iteratively the Leibniz rule for the differentiation operators $V_i$ or $\partial_\tau$, we have the following decomposition

$$fg = \sum_{\mathcal{I}_b} a_{k, \ell}^{I, J} \int_0^1 (A_{k, \ell}^{I, J}(f, g) + A_{k, \ell}^{I, J}(g, f)) \frac{dt}{t} + \sum_{\mathcal{I}_b} b_{k, \ell}^{I, J} \int_0^1 B_{k, \ell}^{I, J}(f, g) \frac{dt}{t},$$

where

* $\mathcal{I}_b$ is the set of all tuples $(I, J, k, \ell)$ with the tuples $I, J$ and the integers $k, \ell$ satisfying the constraint

$$\frac{|I| + |J|}{2} + k + \ell = \frac{b}{2};$$

* $a_{k, \ell}^{I, J}, b_{k, \ell}^{I, J}$ are bounded sequences of numerical coefficients;
* for $(I, J, k, \ell) \in \mathcal{I}_b, \mathcal{A}_{k,\ell}^{I,J}(f,g)$ has the form

$$ \mathcal{A}_{k, \ell}^{I, J}(f, g) := \mathcal{P}_{t}^{(b)}(t^{|I|/2+k} V_{I} \partial_{\tau}^{k}) (\mathcal{S}_{t}^{(b/2)} f \cdot (t^{|J|/2+\ell} V_{J} \partial_{\tau}^{\ell}) \mathcal{P}_{t}^{(b)} g) $$

with $S^{(b/2)} \in \mathrm{GC}^{b/2}$;

* for $(I, J, k, \ell) \in \mathcal{I}_b, \mathcal{B}_{k,\ell}^{I,J}(f,g)$ has the form

$$ \mathcal{B}_{k, \ell}^{I, J}(f, g) := \mathcal{S}_{t}^{(b/2)} (\{(t^{|I|/2+k} V_I \partial_{\tau}^k) \mathcal{P}_t^{(b)} f\} \cdot \{(t^{|J|/2+\ell} V_J \partial_{\tau}^{\ell}) \mathcal{P}_t^{(b)} g\}) $$

with $S^{(b/2)} \in \mathrm{GC}^{b/2}$.

**DEFINITION.** Given $f$ in $\bigcup_{s \in (0,1)} \mathcal{C}^s$ and $g \in L^\infty(\mathcal{M})$, we define the **paraproduct** $\mathbf{P}_g^{(b)} f$ by the formula

$$ \mathbf{P}_g^{(b)} f := \int_0^1 \left\{ \sum_{\mathcal{I}_b; |I|/2+k>b/4} a_{k,\ell}^{I,J} \mathcal{A}_{k,\ell}^{I,J}(f,g) + \sum_{\mathcal{I}_b; |I|/2+k>b/4} b_{k,\ell}^{I,J} \mathcal{B}_{k,\ell}^{I,J}(f,g) \right\} \frac{dt}{t} $$

and the **resonant term** $\Pi^{(b)}(f,g)$ by the formula

$$ \Pi^{(b)}(f,g) := \int_0^1 \left\{ \sum_{\mathcal{I}_b; |I|/2+k \le b/4} a_{k,\ell}^{I,J} (\mathcal{A}_{k,\ell}^{I,J}(f,g) + \mathcal{A}_{k,\ell}^{I,J}(g,f)) \right. \\ \left. + \sum_{\mathcal{I}_b; |I|/2+k=|J|/2+\ell=b/4} b_{k,\ell}^{I,J} \mathcal{B}_{k,\ell}^{I,J}(f,g) \right\} \frac{dt}{t} $$

With these notations, Calderón's formula becomes

$$ fg = \mathbf{P}_g^{(b)} f + \mathbf{P}_f^{(b)} g + \Pi^{(b)}(f,g) + \Delta_{-1}(f,g) $$

with the 'low-frequency part'

$$ \Delta_{-1}(f,g) := \mathcal{P}_1^{(b)}(\mathcal{P}_1^{(b)} f \cdot \mathcal{P}_1^{(b)} g). $$

If $b$ is chosen large enough, then all the operators involved in the paraproduct and resonant terms have a kernel pointwisely bounded by a kernel $\mathcal{G}_t$ at the right scaling. Moreover,

(a) the paraproduct term $\mathbf{P}_g^{(b)} f$ is a finite linear combination of operators of the form

$$ \int_0^1 Q_t^{1, \bullet} (Q_t^2 f \cdot \mathcal{P}_t^1 g) \frac{dt}{t} $$

with $Q^1, Q^2 \in \mathrm{StGC}^{b/4}$, and $\mathcal{P}^1 \in \mathrm{StGC}^{[0,b]}$,
• for $(I, J, k, \ell) \in \mathcal{I}_b, \mathcal{A}_{k,\ell}^{I,J}(f,g)$ has the form

$$ \mathcal{A}_{k, \ell}^{I, J}(f, g) := \mathcal{P}_{t}^{(b)}(t^{|I|/2+k} V_{I} \partial_{\tau}^{k}) (\mathcal{S}_{t}^{(b/2)} f \cdot (t^{|J|/2+\ell} V_{J} \partial_{\tau}^{\ell}) \mathcal{P}_{t}^{(b)} g) $$

with $S^{(b/2)} \in \mathrm{GC}^{b/2}$;

• for $(I, J, k, \ell) \in \mathcal{I}_b, \mathcal{B}_{k,\ell}^{I,J}(f,g)$ has the form

$$ \mathcal{B}_{k, \ell}^{I, J}(f, g) := \mathcal{S}_{t}^{(b/2)} (\{(t^{|I|/2+k} V_I \partial_\tau^k) \mathcal{P}_t^{(b)} f\} \cdot \{(t^{|J|/2+\ell} V_J \partial_\tau^\ell) \mathcal{P}_t^{(b)} g\}) $$

with $S^{(b/2)} \in \mathrm{GC}^{b/2}$.

**DEFINITION.** Given $f$ in $\bigcup_{s \in (0,1)} \mathcal{C}^s$ and $g \in L^\infty(\mathcal{M})$, we define the **paraproduct** $\mathbf{P}_g^{(b)} f$ by the formula

$$ \mathbf{P}_g^{(b)} f := \int_0^1 \left\{ \sum_{\mathcal{I}_b; |I|/2+k>b/4} a_{k,\ell}^{I,J} \mathcal{A}_{k,\ell}^{I,J}(f,g) + \sum_{\mathcal{I}_b; |I|/2+k>b/4} b_{k,\ell}^{I,J} \mathcal{B}_{k,\ell}^{I,J}(f,g) \right\} \frac{dt}{t}, $$

and the **resonant term** $\Pi^{(b)}(f,g)$ by the formula

$$ \Pi^{(b)}(f,g) := \int_0^1 \left\{ \sum_{\mathcal{I}_b; |I|/2+k \le b/4} a_{k,\ell}^{I,J} (\mathcal{A}_{k,\ell}^{I,J}(f,g) + \mathcal{A}_{k,\ell}^{I,J}(g,f)) \right. \\ \left. + \sum_{\mathcal{I}_b; |I|/2+k=|J|/2+\ell=b/4} b_{k,\ell}^{I,J} \mathcal{B}_{k,\ell}^{I,J}(f,g) \right\} \frac{dt}{t}. $$

With these notations, Calderón's formula becomes

$$ fg = \mathbf{P}_g^{(b)} f + \mathbf{P}_f^{(b)} g + \Pi^{(b)}(f,g) + \Delta_{-1}(f,g) $$

with the 'low-frequency part'

$$ \Delta_{-1}(f,g) := \mathcal{P}_1^{(b)}(\mathcal{P}_1^{(b)} f \cdot \mathcal{P}_1^{(b)} g). $$

If $b$ is chosen large enough, then all the operators involved in the paraproduct and resonant terms have a kernel pointwisely bounded by a kernel $G_t$ at the right scaling. Moreover,

(a) the paraproduct term $\mathbf{P}_g^{(b)} f$ is a finite linear combination of operators of the form

$$ \int_0^1 Q_t^{1, \bullet} (Q_t^2 f \cdot \mathcal{P}_t^1 g) \frac{dt}{t} $$

with $Q^1, Q^2 \in \mathrm{StGC}^{b/4}$, and $\mathcal{P}^1 \in \mathrm{StGC}^{[0,b]}$,
(b) the resonant term $\Pi^{(b)}(f, g)$ is a finite linear combination of operators of the form

$$ \int_0^1 \mathcal{P}_t^{1\bullet}(Q_t^1 f \cdot Q_t^2 g) \frac{dt}{t} $$
with $Q^1, Q^2 \in \text{StGC}^{b/4}$ and $\mathcal{P}^1 \in \text{StGC}^{[0,b]}$.

We invite the reader to see what happens of all this when working in the flat torus with its associated Laplacian. Note also that $\mathcal{P}_f^{(b)}(\mathbf{1}) = \Pi^{(b)}(f, \mathbf{1}) = 0$, and that we have the identity

$$ \mathbf{P}_1^{(b)} f = f - \mathcal{P}_1^{(b)}(\mathcal{P}_1^{(b)} f), $$

as a consequence of our choice of renormalizing constant. Therefore, the paraproduct with the constant function $\mathbf{1}$ is equal to the identity operator, up to the strongly regularizing operator $\mathcal{P}_1^{(b)} \circ \mathcal{P}_1^{(b)}$. The regularity properties of the paraproduct and resonant operators can be described as follows; it behaves as its classical, Fourier-based, counterpart (2.1).

**PROPOSITION 33.** (a) For every real-valued regularity exponent $\alpha, \beta$, and every positive regularity exponent $\gamma$, we have

$$ \|\Delta_{-1}(f, g)\|_{C^\gamma} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

(b) For every $\alpha \in (-3, 3)$ and $f \in C^\alpha$, we have

$$ \|\mathbf{P}_g^{(b)} f\|_{C^\alpha} \lesssim \|g\|_{\infty} \|f\|_{C^\alpha} $$
for every $g \in L^\infty$, and

$$ \|\mathbf{P}_g^{(b)} f\|_{C^{\alpha+\beta}} \lesssim \|g\|_{C^\beta} \|f\|_{C^\alpha} $$
for every $g \in C^\beta$ with $\beta < 0$ and $\alpha + \beta \in (-3, 3)$.

(c) For every $\alpha, \beta \in (-3, 3)$ with $\alpha + \beta > 0$, we have the continuity estimate

$$ \|\Pi^{(b)}(f, g)\|_{C^{\alpha+\beta}} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

**DEFINITION.** We define a **modified paraproduct** $\tilde{\mathbf{P}}^{(b)}$ setting

$$ \tilde{\mathbf{P}}_g^{(b)} f := \mathcal{L}^{-1}(\mathbf{P}_g^{(b)}(\mathcal{L} f)). $$

The next proposition shows that if one chooses the parameter $\ell_1$ that appears in the reference kernels $\mathcal{G}_t$, and the exponent $b$ in the definition of the paraproduct, both large enough, then the modified paraproduct $\tilde{\mathbf{P}}^{(b)}$ has the same algebraic/analytic properties as $\mathbf{P}^{(b)}$.
(b) the resonant term $\Pi^{(b)}(f, g)$ is a finite linear combination of operators of the form

$$ \int_0^1 \mathcal{P}_t^{1\bullet}(Q_t^1 f \cdot Q_t^2 g) \frac{dt}{t} $$
with $Q^1, Q^2 \in \text{StGC}^{b/4}$ and $\mathcal{P}^1 \in \text{StGC}^{[0,b]}$.

We invite the reader to see what happens of all this when working in the flat torus with its associated Laplacian. Note also that $\mathbf{P}_f^{(b)}(\mathbf{1}) = \Pi^{(b)}(f, \mathbf{1}) = 0$, and that we have the identity

$$ \mathbf{P}_1^{(b)} f = f - \mathcal{P}_1^{(b)}(\mathcal{P}_1^{(b)} f), $$

as a consequence of our choice of renormalizing constant. Therefore, the paraproduct with the constant function $\mathbf{1}$ is equal to the identity operator, up to the strongly regularizing operator $\mathcal{P}_1^{(b)} \circ \mathcal{P}_1^{(b)}$. The regularity properties of the paraproduct and resonant operators can be described as follows; it behaves as its classical, Fourier-based, counterpart (2.1).

**PROPOSITION 33.** (a) For every real-valued regularity exponent $\alpha, \beta$, and every positive regularity exponent $\gamma$, we have

$$ \|\Delta_{-1}(f, g)\|_{C^\gamma} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

(b) For every $\alpha \in (-3, 3)$ and $f \in C^\alpha$, we have

$$ \begin{gathered} \|\mathbf{P}_g^{(b)} f\|_{C^\alpha} \lesssim \|g\|_{\infty} \|f\|_{C^\alpha} \\ \text{for every } g \in L^\infty, \text{ and} \end{gathered} $$

$$ \begin{gathered} \|\mathbf{P}_g^{(b)} f\|_{C^{\alpha+\beta}} \lesssim \|g\|_{C^\beta} \|f\|_{C^\alpha} \\ \text{for every } g \in C^\beta \text{ with } \beta < 0 \text{ and } \alpha + \beta \in (-3, 3). \end{gathered} $$

(c) For every $\alpha, \beta \in (-3, 3)$ with $\alpha + \beta > 0$, we have the continuity estimate

$$ \|\Pi^{(b)}(f, g)\|_{C^{\alpha+\beta}} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta}. $$

**DEFINITION.** We define a **modified paraproduct** $\tilde{\mathbf{P}}^{(b)}$ setting

$$ \tilde{\mathbf{P}}_g^{(b)} f := \mathcal{L}^{-1}(\mathbf{P}_g^{(b)}(\mathcal{L} f)). $$

The next proposition shows that if one chooses the parameter $\ell_1$ that appears in the reference kernels $\mathcal{G}_t$, and the exponent $b$ in the definition of the paraproduct, both large enough, then the modified paraproduct $\tilde{\mathbf{P}}^{(b)}$ has the same algebraic/analytic properties as $\mathbf{P}^{(b)}$.
**PROPOSITION 34.** • For a choice of large enough constants $\ell_1$ and b, the modified paraproduct $\tilde{P}_g f$ is a finite linear combination of operators of the form

$$ \int_0^1 Q_t^{1\bullet}(Q_t^2 f \cdot P_t^1 g) \frac{dt}{t} $$

with $Q^1 \in \mathrm{GC}^{b/8-2}$, $Q^2 \in \mathrm{StGC}^{b/4}$ and $P^1 \in \mathrm{StGC}^{[0,b]}$.

• For every $\alpha \in (-3, 3)$ and $\varepsilon \in (0, 1)$ with $\alpha - \varepsilon \in (-3, 3)$ and $f \in C^\alpha$, we have

$$ \| \tilde{P}_g^{(b)} f \|_{C_w^{\alpha-\varepsilon}} \lesssim \kappa^{-\varepsilon} \| w^{-1} g \|_{\infty} \| f \|_{C^\alpha}, $$

for every $g \in L^\infty$.

The notation $Q^{1\bullet}$ does not make sense for a generic element $Q^1$ of $\mathrm{GC}$. The operator that appears in this formula is actually of the form $Q_t^1 = Q_t t^{-1} \mathcal{L}^{-1}$, with $Q \in \mathrm{StGC}$, for which the notation $Q^{1\bullet}$ makes sense. Note that the norm $\|f\|_{C^\alpha}$ above has no weight. Note here the normalization identity

$$ \tilde{P}_1^{(b)} f = f - \mathcal{L}^{-1} \circ P_1^{(b)} \circ P_1^{(b)} (\mathcal{L} f) $$

for every distribution in $f \in S'_o$; it reduces to

$$ \tilde{P}_1^{(b)} f = f - P_1^{(b)} P_1^{(b)}(f) $$

if $f_{\tau=0} = 0$.

Following the definition of the **inner difference operator** $\mathcal{D}$ given in Section 3.2, we extend it to the parabolic setting defining $\mathcal{D}(=\mathcal{D}_e)$ by the formula

$$ \iint_{\mathbb{M}^2} (\mathcal{D} f)(e')g(e) \nu(de)\nu(de') := \iint_{\mathbb{M}^2} (f(e') - f(e))g(e) \nu(de)\nu(de'); $$

with this notation, the crucial motivating relation

$$ P_f(\tilde{P}_a g) - P_{fa} g = P_f(\tilde{P}_{\mathcal{D}a} g) $$

holds true.

Last, we prove an elementary property of the modified paraproduct that provides some pointwise information on the solutions to singular partial differential equations.

**PROPOSITION 35.** Let $\alpha, \beta$ be positive regularity exponents, and let $u, Z \in C^\alpha$ with $v \in C^\beta$ be given, with $Z(0, \cdot) = 0$. Assume that

$$ u - \tilde{P}_v Z \in C^{\alpha+\beta}, $$
**PROPOSITION 34.** • For a choice of large enough constants $\ell_1$ and b, the modified paraproduct $\tilde{P}_g f$ is a finite linear combination of operators of the form

$$ \int_0^1 Q_t^{1\bullet}(Q_t^2 f \cdot P_t^1 g) \frac{dt}{t} $$

with $Q^1 \in \mathrm{GC}^{b/8-2}$, $Q^2 \in \mathrm{StGC}^{b/4}$ and $P^1 \in \mathrm{StGC}^{[0,b]}$.

• For every $\alpha \in (-3, 3)$ and $\varepsilon \in (0, 1)$ with $\alpha - \varepsilon \in (-3, 3)$ and $f \in C^\alpha$, we have

$$ \|\tilde{P}_g^{(b)} f\|_{C_w^{\alpha-\varepsilon}} \lesssim \kappa^{-\varepsilon} \|w^{-1}g\|_{\infty} \|f\|_{C^\alpha}, $$

for every $g \in L^\infty$.

The notation $Q^{1\bullet}$ does not make sense for a generic element $Q^1$ of GC. The operator that appears in this formula is actually of the form $Q_t^1 = Q_t t^{-1} \mathcal{L}^{-1}$, with $Q \in \mathrm{StGC}$, for which the notation $Q^{1\bullet}$ makes sense. Note that the norm $\|f\|_{C^\alpha}$ above has no weight. Note here the normalization identity

$$ \tilde{P}_1^{(b)} f = f - \mathcal{L}^{-1} \circ P_1^{(b)} \circ P_1^{(b)} (\mathcal{L} f) $$

for every distribution in $f \in S'_o$; it reduces to

$$ \tilde{P}_1^{(b)} f = f - P_1^{(b)} P_1^{(b)}(f) $$

if $f_{\tau=0} = 0$.

Following the definition of the **inner difference operator** $\mathcal{D}$ given in Section 3.2, we extend it to the parabolic setting defining $\mathcal{D}(=\mathcal{D}_e)$ by the formula

$$ \iint_{\mathbb{M}^2} (\mathcal{D} f)(e')g(e) \nu(de)\nu(de') := \iint_{\mathbb{M}^2} (f(e') - f(e))g(e) \nu(de)\nu(de'); $$

with this notation, the crucial motivating relation

$$ P_f(\tilde{P}_a g) - P_{fa} g = P_f(\tilde{P}_{\mathcal{D}a} g) $$

holds true.

Last, we prove an elementary property of the modified paraproduct that provides some pointwise information on the solutions to singular partial differential equations.

**PROPOSITION 35.** Let $\alpha, \beta$ be positive regularity exponents, and let $u, Z \in C^\alpha$ with $v \in C^\beta$ be given, with $Z(0, \cdot) = 0$. Assume that

$$ u - \tilde{P}_v Z \in C^{\alpha+\beta}, $$
and define $\gamma := \min(\alpha + \beta, 1)$. If $\alpha + \beta \neq 1$, we have

$$|u(e) - u(e') - v(e)(Z(e) - Z(e'))| \lesssim \rho(e, e')^{\gamma},$$

uniformly in $e, e' \in \mathcal{M}$ with $\rho(e, e') \le 1$. If $\alpha + \beta = 1$, we have an additional logarithmic loss

$$|u(e) - u(e') - v(e)(Z(e) - Z(e'))| \lesssim \rho(e, e') \log(1 + \rho(e, e')^{-1}).$$

*Proof.* Due to the assumption, one has

$$|u(e) - u(e') - v(e)(Z(e) - Z(e'))| \lesssim \rho(e, e')^{\beta} + (\star)$$

with

$$(\star) := |(\tilde{P}_v Z)(e) - (\tilde{P}_v Z)(e') - v(e)(Z(e) - Z(e'))|.$$

Using Calderón reproducing formula, or the normalization, yields

$$\tilde{P}_1 Z = Z$$

since $Z(0, \cdot) = 0$, and we see that $(\star)$ is equal to

$$\left| \int_0^1 Q_t^*(Q_t Z P_t v)(e) - Q_t^*(Q_t Z P_t v)(e') - v(e)Q_t^*(Q_t Z)(e) + v(e)Q_t^*(Q_t Z)(e') \frac{dt}{t} \right|,$$

so

$$(\star) \lesssim \int_0^1 \left| \int (K_{Q_t^*}(e, a) - K_{Q_t^*}(e', a)) Q_t Z(a) (P_t v(a) - v(e)) v da \right| \frac{dt}{t}.$$

Using the regularity estimates on $v$ and on the kernel of the approximation operators, one sees that

$$\begin{aligned} (\star) & \lesssim \|v\|_{C^\beta} \int_0^1 \int \min\left\{1, \frac{\rho(e, e')}{\sqrt{t}}\right\} \mathcal{G}_t(e, a) |Q_t Z(a)| (t + \rho(a, e))^2^{\gamma/4} v da \frac{dt}{t} \\ & \lesssim \|v\|_{C^\beta} \|Z\|_{C^\alpha} \int_0^1 \int t^{\frac{(2\alpha+\gamma)}{4}} \frac{dt}{t} + \|v\|_{C^\beta} \|Z\|_{C^\alpha} \int_{\rho^2}^1 \int \frac{\rho(e, e')}{\sqrt{t}} t^{\frac{(2\alpha+\gamma)}{4}} \frac{dt}{t} \\ & \lesssim \|v\|_{C^\beta} \|Z\|_{C^\alpha} \rho(e, e')^{\gamma}, \end{aligned}$$

which concludes the proof. □
and define $\gamma := \min(\alpha + \beta, 1)$. If $\alpha + \beta \neq 1$, we have

$$|u(e) - u(e') - v(e)(Z(e) - Z(e'))| \lesssim \rho(e, e')^{\gamma},$$

uniformly in $e, e' \in \mathcal{M}$ with $\rho(e, e') \le 1$. If $\alpha + \beta = 1$, we have an additional logarithmic loss

$$|u(e) - u(e') - v(e)(Z(e) - Z(e'))| \lesssim \rho(e, e') \log(1 + \rho(e, e')^{-1}).$$

*Proof.* Due to the assumption, one has

$$|u(e) - u(e') - v(e)(Z(e) - Z(e'))| \lesssim \rho(e, e')^{\beta} + (\star)$$

with

$$(\star) := |(\tilde{P}_v Z)(e) - (\tilde{P}_v Z)(e') - v(e)(Z(e) - Z(e'))|.$$

Using Calderón reproducing formula, or the normalization, yields

$$\tilde{P}_1 Z = Z$$

since $Z(0, \cdot) = 0$, and we see that $(\star)$ is equal to

$$\left| \int_0^1 Q_t^*(Q_t Z P_t v)(e) - Q_t^*(Q_t Z P_t v)(e') - v(e)Q_t^*(Q_t Z)(e) + v(e)Q_t^*(Q_t Z)(e') \frac{dt}{t} \right|,$$

so

$$(\star) \lesssim \int_0^1 \left| \int (K_{Q_t^*}(e, a) - K_{Q_t^*}(e', a)) Q_t Z(a) (P_t v(a) - v(e)) v da \right| \frac{dt}{t}.$$

Using the regularity estimates on $v$ and on the kernel of the approximation operators, one sees that

$$\begin{align*}
(\star) & \lesssim \|v\|_{C^\beta} \int_0^1 \int \min\left\{1, \frac{\rho(e, e')}{\sqrt{t}}\right\} \mathcal{G}_t(e, a) |Q_t Z(a)| (t + \rho(a, e))^2^{\gamma/4} v da \frac{dt}{t} \\
& \lesssim \|v\|_{C^\beta} \|Z\|_{C^\alpha} \int_0^1 \int t^{\frac{(2\alpha+\gamma)}{4}} \frac{dt}{t} + \|v\|_{C^\beta} \|Z\|_{C^\alpha} \int_{\rho^2}^1 \int \frac{\rho(e, e')}{\sqrt{t}} t^{\frac{(2\alpha+\gamma)}{4}} \frac{dt}{t} \\
& \lesssim \|v\|_{C^\beta} \|Z\|_{C^\alpha} \rho(e, e')^{\gamma},
\end{align*}$$

which concludes the proof. □
## Appendix B. Paracontrolled expansion formula

We give in this section a detailed and rigorous proof of Theorem 4. The parameter b is fixed, and we note $Π$ for $Π^{(b)}$.

**THEOREM 36 (Higher order Taylor expansion).** Let $f: \mathbf{R} \mapsto \mathbf{R}$ be a $C^4$ function, and let $u$ be a real-valued and $C^\alpha$ function on $M$, with $\alpha \in (0, 1)$. Then

$$f(u) = P_{f'(u)}(u) + \frac{1}{2}\{P_{f^{(2)}(u)}(u^2) - 2P_{f^{(2)}(u)}(u)\} \\ + \frac{1}{3!}\{P_{f^{(3)}(u)}(u^3) - 3P_{f^{(3)}(u)}(u^2) + 3P_{f^{(3)}(u)}(u^2)\} + f(u)^2 \quad (B.1)$$

for some remainder $f(u)^2 \in C^{4\alpha}$. If moreover $f$ is of class $C^5$, then the remainder term $f(u)^2$ is locally Lipschitz with respect to $u$, in the sense that

$$\|f(u)^2 - f(v)^2\|_{C^{4\alpha}} \lesssim (1 + \|u\|_{C^\alpha} + \|v\|_{C^\alpha})^4 \|u - v\|_{C^\alpha}. \quad (B.2)$$

*Proof.* Let us give a detailed proof of the third order expansion, that claims that

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

is a $3\alpha$-Hölder function. We invite the reader to follow what comes next in the light of the proof given in Section 2 in the time-independent, flat, model setting of the torus.

As, by definition, the paraproduct operator $P_g(\cdot)$ is a finite sum of different terms, each of them of the form

$$A_g^1(\cdot) := \int_0^1 Q_t^{1\bullet}(Q_t^2(\cdot)P_t^1(g)) \frac{dt}{t},$$

with $Q^1, Q^2$ at least to $\text{StGC}^3$, it is sufficient to prove that the following function

$$(\star) := f(u) - \int_0^1 \left[ Q_t^{1\bullet}(Q_t^2(u)P_t^1(f'(u))) + \frac{1}{2}Q_t^{1\bullet}(Q_t^2(u^2)P_t^1(f^{(2)}(u))) - Q_t^{1\bullet}(Q_t^2(u)P_t^1(f^{(2)}(u))\right] \frac{dt}{t}$$

is an element of $C^{3\alpha}$. Using Calderón's reproducing formula together with the normalization of the paraproduct, we have

$$f(u) \approx \int_0^1 Q_t^{1\bullet} Q_t^2(f(u)P_t^1(1)) \frac{dt}{t}$$
## Appendix B. Paracontrolled expansion formula

We give in this section a detailed and rigorous proof of Theorem 4. The parameter b is fixed, and we note $Π$ for $Π^{(b)}$.

**THEOREM 36 (Higher order Taylor expansion).** Let $f: \mathbf{R} \mapsto \mathbf{R}$ be a $C^4$ function, and let $u$ be a real-valued and $C^\alpha$ function on $M$, with $\alpha \in (0, 1)$. Then

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

for some remainder $f(u)^2 \in C^{4\alpha}$. If moreover $f$ is of class $C^5$, then the remainder term $f(u)^2$ is locally Lipschitz with respect to $u$, in the sense that

$$\|f(u)^2 - f(v)^2\|_{C^{4\alpha}} \lesssim (1 + \|u\|_{C^\alpha} + \|v\|_{C^\alpha})^4 \|u - v\|_{C^\alpha}. \quad (B.2)$$

*Proof.* Let us give a detailed proof of the third order expansion, that claims that

$$(\star) := f(u) - \mathbf{P}_{f'(u)}(u) - \frac{1}{2}\{\mathbf{P}_{f^{(2)}(u)}(u^2) - 2\mathbf{P}_{f^{(2)}(u)}(u)\}$$

is a $3\alpha$-Hölder function. We invite the reader to follow what comes next in the light of the proof given in Section 2 in the time-independent, flat, model setting of the torus.

As, by definition, the paraproduct operator $\mathbf{P}_g(\cdot)$ is a finite sum of different terms, each of them of the form

$$A_g^1(\cdot) := \int_0^1 Q_t^{1\bullet}(Q_t^2(\cdot)P_t^1(g)) \frac{dt}{t},$$

with $Q^1, Q^2$ at least to $\text{StGC}^3$, it is sufficient to prove that the following function

$$(\star) := f(u) - \int_0^1 \left[ Q_t^{1\bullet}(Q_t^2(u)P_t^1(f'(u))) + \frac{1}{2}Q_t^{1\bullet}(Q_t^2(u^2)P_t^1(f^{(2)}(u))) - Q_t^{1\bullet}(Q_t^2(u)P_t^1(f^{(2)}(u))\right] \frac{dt}{t}$$

is an element of $C^{3\alpha}$. Using Calderón's reproducing formula together with the normalization of the paraproduct, we have

$$f(u) \approx \int_0^1 Q_t^{1\bullet} Q_t^2(f(u)P_t^1(1)) \frac{dt}{t}$$
up to a remainder quantity corresponding to the low-frequency part that is as smooth as we want. So one can write (★) under the form

$$ (★) = \int_{0}^{1} Q_{t}^{1★}(\varepsilon_{t}) \frac{dt}{t}, \qquad (B.3) $$

with

$$ \varepsilon_t := Q_t^2(f(u))P_t^1(1) - Q_t^2(u)P_t^1(f'(u)) \\ - \frac{1}{2}Q_t^2(u^2)P_t^1(f^{(2)}(u)) + Q_t^2(u)P_t^1(f^{(2)}(u))u. $$

Due to the orthogonality/cancellation property of the operators $Q_t^{1★}$, it suffices for us to get an $L^{\infty}$ control of $\varepsilon_t$. Using the kernel representation of the different operators, we have for every $e \in \mathcal{M}$

$$ \varepsilon_t(e) = \iint_{M^2} K_{Q_t^2}(e, e') K_{P_t^1}(e, e'') \left\{ f(u(e')) - u(e')f'(u(e'')) \right. \\ \left. - \frac{1}{2}u^2(e')f^{(2)}(u(e'')) + u(e')f^{(2)}(u(e''))u(e'') \right\} \nu(de')\nu(de''). $$

Note also that we have from the usual Taylor formula for $f$

$$ \begin{aligned} & f(u(e')) - u(e')f'(u(e'')) - \frac{1}{2}u^2(e')f^{(2)}(u(e'')) + u(e')f^{(2)}(u(e''))u(e'') \\ & = \iiint_{[0,1]^3} f^{(3)}(u(e'')) + s_3s_2s_1(u(e') - u(e''))s_2s_1(u(e') - u(e''))^3 ds_3 ds_2 ds_1 \\ & \quad + f(u(e'')) + u(e'')f'(u(e'')) + \frac{1}{2}u^2(e'')f^{(2)}(u(e'')). \end{aligned} $$

When we integrate against $K_{Q_t^2}(e, e')K_{P_t^1}(e, e'')$ a quantity depending only in $e''$ has no contribution, since the latter kernel satisfies a cancellation property along the $e'$-variable. So we have exactly

$$ \begin{aligned} \varepsilon_t(e) = \iint_{M^2} K_{Q_t^2}(e, e') K_{P_t^1}(e, e'') & \\ \left( \iiint_{[0,1]^3} f^{(3)}(u(e'')) + s_3s_2s_1(u(e') - u(e''))s_2s_1(u(e') - u(e''))^3 ds_3 ds_2 ds_1 \right) & \\ \times \nu(de')\nu(de''). & \end{aligned} $$

Since $K_{Q_t^2}$ and $K_{P_t^1}$ are both pointwisely dominated by the Gaussian kernel $G_t$, and using the fact that $f^{(3)}$ is bounded on the range of $u$, we obtain the uniform control

$$ \begin{aligned} |\varepsilon_t(e)| & \lesssim \iint_{M^2} \mathcal{G}_t(e, e')\mathcal{G}_t(e, e'')(u(e') - u(e''))^3 \nu(de')\nu(de'') \\ & \lesssim \|u\|_{C^a}^3 t^{3\alpha/2}, \end{aligned} $$
up to a remainder quantity corresponding to the low-frequency part that is as smooth as we want. So one can write (★) under the form

$$ (★) = \int_{0}^{1} Q_{t}^{1★}(\varepsilon_{t}) \frac{dt}{t}, \qquad (B.3) $$

with

$$ \varepsilon_t := Q_t^2(f(u))P_t^1(1) - Q_t^2(u)P_t^1(f'(u)) \\ - \frac{1}{2}Q_t^2(u^2)P_t^1(f^{(2)}(u)) + Q_t^2(u)P_t^1(f^{(2)}(u))u. $$

Due to the orthogonality/cancellation property of the operators $Q_t^{1★}$, it suffices for us to get an $L^{\infty}$ control of $\varepsilon_t$. Using the kernel representation of the different operators, we have for every $e \in \mathcal{M}$

$$ \varepsilon_t(e) = \iint_{M^2} K_{Q_t^2}(e, e') K_{P_t^1}(e, e'') \left\{ f(u(e')) - u(e')f'(u(e'')) \right. \\ \left. - \frac{1}{2}u^2(e')f^{(2)}(u(e'')) + u(e')f^{(2)}(u(e''))u(e'') \right\} \nu(de')\nu(de''). $$

Note also that we have from the usual Taylor formula for $f$

$$ \begin{aligned} & f(u(e')) - u(e')f'(u(e'')) - \frac{1}{2}u^2(e')f^{(2)}(u(e'')) + u(e')f^{(2)}(u(e''))u(e'') \\ &= \iiint_{[0,1]^3} f^{(3)}(u(e'')) + s_3s_2s_1(u(e') - u(e''))s_2s_1(u(e') - u(e''))^3 ds_3 ds_2 ds_1 \\ & \quad + f(u(e'')) + u(e'')f'(u(e'')) + \frac{1}{2}u^2(e'')f^{(2)}(u(e'')). \end{aligned} $$

When we integrate against $K_{Q_t^2}(e, e')K_{P_t^1}(e, e'')$ a quantity depending only in $e''$ has no contribution, since the latter kernel satisfies a cancellation property along the $e'$-variable. So we have exactly

$$ \begin{aligned} \varepsilon_t(e) &= \iint_{M^2} K_{Q_t^2}(e, e') K_{P_t^1}(e, e'') \\ & \quad \left( \iiint_{[0,1]^3} f^{(3)}(u(e'')) + s_3 s_2 s_1 (u(e') - u(e'')) s_2 s_1 (u(e') - u(e''))^3 ds_3 ds_2 ds_1 \right) \\ & \quad \times \nu(de') \nu(de''). \end{aligned} $$

Since $K_{Q_t^2}$ and $K_{P_t^1}$ are both pointwisely dominated by the Gaussian kernel $G_t$, and using the fact that $f^{(3)}$ is bounded on the range of $u$, we obtain the uniform control

$$ \begin{aligned} |\varepsilon_t(e)| & \lesssim \iint_{M^2} \mathcal{G}_t(e, e') \mathcal{G}_t(e, e'') (u(e') - u(e''))^3 \nu(de') \nu(de'') \\ & \lesssim \|u\|_{C^a}^3 t^{3\alpha/2}, \end{aligned} $$
from which the fact that (⋆) belongs to $C^{3\alpha}$ follows from (B.3). We used for that purpose the identity

$$u(e') - u(e'') = (u(e') - u(e)) + (u(e) - u(e'')),$$

together with Proposition 29 on the characterization of parabolic regularity in terms of increments, to see that

$$|u(e') - u(e'')| \lesssim (d(e', e) + d(e'', e))^{\alpha} \|f\|_{C^{\alpha}}.$$

The fourth order expansion of the statement is proved by a very similar reasoning left to the reader. One proceeds exactly as in the proof of Theorem 4 to prove the stability estimate (B.2). □

Observe the fact that one can give a paracontrolled expansion formula with the $\tilde{P}$ operator in place of the $P$ operator, as a consequence of Proposition 34 on the structure of the modified paraproduct $\tilde{P}$, and the proof of Theorem 36.

## Appendix C. Continuity results

Recall the definitions of the corrector

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

the (modified) commutators

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

$$R(f, g, h) = P_f(\tilde{P}_g h) - P_{fg}h,$$

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

and their iterates, introduced in Section 3. We use also their $\circ$ variants, built with $P$ in the role of $\tilde{P}$; we have for instance

$$R^{\circ}(f, g, h) = P_f(P_g h) - P_{fg}h.$$

All these operators are initially defined on the space of smooth functions. We prove in this section the continuity results on these operators stated in Section 3.

Let $\chi$ stand for a smooth nonnegative function on $[0, \infty)$, equal to 1 in a neighbourhood of 0, with $\chi(r) = 0$ for $r \ge r_M$. Given $e = (t, x) \in \mathcal{M}$, set

$$\delta_{\ell}(e, e') := \chi(d(x', x)) \langle V_{\ell}(x), \pi_{xx'} \rangle_{T_{x}M},$$

for $e' = (t', x') \in \mathcal{M}$ and $\exp_x(\pi_{xx'}) = x'$, and $1 \le \ell \le \ell_0$, and

$$\Pi_{(1)}^{\ell}(g, h) := \Pi(P_{\delta_{\ell}(e, \cdot)} g, h)(e).$$
from which the fact that (⋆) belongs to $C^{3\alpha}$ follows from (B.3). We used for that purpose the identity

$$u(e') - u(e'') = (u(e') - u(e)) + (u(e) - u(e'')), $$

together with Proposition 29 on the characterization of parabolic regularity in terms of increments, to see that

$$|u(e') - u(e'')| \lesssim (d(e', e) + d(e'', e))^{\alpha} \|f\|_{C^{\alpha}}.$$

The fourth order expansion of the statement is proved by a very similar reasoning left to the reader. One proceeds exactly as in the proof of Theorem 4 to prove the stability estimate (B.2). □

Observe the fact that one can give a paracontrolled expansion formula with the $\tilde{P}$ operator in place of the $P$ operator, as a consequence of Proposition 34 on the structure of the modified paraproduct $\tilde{P}$, and the proof of Theorem 36.

## Appendix C. Continuity results

Recall the definitions of the corrector

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

the (modified) commutators

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

$$R(f, g, h) = P_f(\tilde{P}_g h) - P_{fg}h,$$

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

and their iterates, introduced in Section 3. We use also their $\circ$ variants, built with $P$ in the role of $\tilde{P}$; we have for instance

$$R^{\circ}(f, g, h) = P_f(P_g h) - P_{fg}h.$$

All these operators are initially defined on the space of smooth functions. We prove in this section the continuity results on these operators stated in Section 3.

Let $\chi$ stand for a smooth nonnegative function on $[0, \infty)$, equal to 1 in a neighbourhood of 0, with $\chi(r) = 0$ for $r \ge r_M$. Given $e = (t, x) \in \mathcal{M}$, set

$$\delta_{\ell}(e, e') := \chi(d(x', x)) \langle V_{\ell}(x), \pi_{xx'} \rangle_{T_{x}M},$$

for $e' = (t', x') \in \mathcal{M}$ and $\exp_x(\pi_{xx'}) = x'$, and $1 \le \ell \le \ell_0$, and

$$\Pi_{(1)}^{\ell}(g, h) := \Pi(P_{\delta_{\ell}(e, \cdot)} g, h)(e).$$
DEFINITION 37. The **refined paraproduct** is defined by the formula

$$C_{(1)}(f, g, h)(e) := C(f, g, h)(e) - \sum_{\ell=1}^{\ell_0} \gamma_{\ell} (V_{\ell} f) \Pi_{(1)}^{\ell}(g, h)(e).$$

Recall from Section 3.1.1 the simple definition of the refined corrector in the model setting of the flat torus.

**C.1. Boundedness of commutators/correctors.** We start by looking at the case of the swap and merging operators $S$ and $R$. We do not emphasize in the next statement the choice of parameter $\kappa$ in the time weight. This has no consequence on the use of these continuity results for the study of singular PDEs, as we only use Schauder estimates in weighted spaces to deal with the terms from the enhancement of the noise in the study of the linear (PAM) equation, not for all the well-defined terms built from the corrector, commutator and their iterates.

PROPOSITION 38. • Let $\alpha, \beta, \gamma$ be Hölder regularity exponents with

$$\gamma \in (-3, 3), \quad \beta \in (0, 1) \quad \text{and} \quad \alpha \in (-\infty, 0).$$

Then if

$$\beta + \gamma < 3, \quad \text{and} \quad \delta := \alpha + \beta + \gamma \in (-3, 3),$$

we have

$$\|S(f, g, h)\|_{C^\delta} + \|S^\circ(f, g, h)\|_{C^\delta} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}, \quad (C.1)$$

so the modified commutator on paraproducts extends naturally into a trilinear continuous map from $C^\alpha \times C^\beta \times C^\gamma$ to $C^\delta$.

• If $\alpha = 0$ then the product $fg$ has a sense for $f \in L^\infty(M)$ and $g \in C^\beta$, with $0 < \beta < 1$, and we have

$$\|\mathbf{R}^\circ(f, g, h)\|_{C^{\beta+\gamma}} + \|\mathbf{R}(f, g, h)\|_{C^{\beta+\gamma}} \lesssim |f|_{L^\infty} |g|_{C^\beta} |h|_{C^\gamma}. \quad (C.2)$$

• If $\alpha, \beta \in (0, 1)$ and $\gamma \in (-3, 3)$, then we have

$$\|\mathbf{R}^\circ(f, g, h)\|_{C^{\alpha+\beta+\gamma}} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}. \quad (C.3)$$

*Proof.* Recall the operators $P_g^{(b)}(\cdot)$, respectively $\tilde{P}_g^{(b)}(\cdot)$, are given by a finite sum of operators of the form

$$A_f^1(\cdot) := \int_0^1 Q_t^{1\bullet}(Q_t^2(\cdot)P_t^1(f)) \frac{dt}{t},$$
DEFINITION 37. The **refined paraproduct** is defined by the formula

$$C_{(1)}(f, g, h)(e) := C(f, g, h)(e) - \sum_{\ell=1}^{\ell_0} \gamma_{\ell} (V_{\ell} f) \Pi_{(1)}^{\ell}(g, h)(e).$$

Recall from Section 3.1.1 the simple definition of the refined corrector in the model setting of the flat torus.

**C.1. Boundedness of commutators/correctors.** We start by looking at the case of the swap and merging operators $S$ and $R$. We do not emphasize in the next statement the choice of parameter $\kappa$ in the time weight. This has no consequence on the use of these continuity results for the study of singular PDEs, as we only use Schauder estimates in weighted spaces to deal with the terms from the enhancement of the noise in the study of the linear (PAM) equation, not for all the well-defined terms built from the corrector, commutator and their iterates.

PROPOSITION 38. • Let $\alpha, \beta, \gamma$ be Hölder regularity exponents with

$$\gamma \in (-3, 3), \quad \beta \in (0, 1) \quad \text{and} \quad \alpha \in (-\infty, 0).$$

Then if

$$\beta + \gamma < 3, \quad \text{and} \quad \delta := \alpha + \beta + \gamma \in (-3, 3),$$

we have

$$\|S(f, g, h)\|_{C^{\delta}} + \|S^{\circ}(f, g, h)\|_{C^{\delta}} \lesssim \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \|h\|_{C^{\gamma}}, \quad (C.1)$$

so the modified commutator on paraproducts extends naturally into a trilinear continuous map from $C^{\alpha} \times C^{\beta} \times C^{\gamma}$ to $C^{\delta}$.

• If $\alpha = 0$ then the product $fg$ has a sense for $f \in L^{\infty}(M)$ and $g \in C^{\beta}$, with $0 < \beta < 1$, and we have

$$\|\mathbf{R}^{\circ}(f, g, h)\|_{C^{\beta+\gamma}} + \|\mathbf{R}(f, g, h)\|_{C^{\beta+\gamma}} \lesssim |f|_{L^{\infty}} |g|_{C^{\beta}} |h|_{C^{\gamma}}. \quad (C.2)$$

• If $\alpha, \beta \in (0, 1)$ and $\gamma \in (-3, 3)$, then we have

$$\|\mathbf{R}^{\circ}(f, g, h)\|_{C^{\alpha+\beta+\gamma}} \lesssim \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \|h\|_{C^{\gamma}}. \quad (C.3)$$

*Proof.* Recall the operators $P_g^{(b)}(\cdot)$, respectively $\tilde{P}_g^{(b)}(\cdot)$, are given by a finite sum of operators of the form

$$A_f^1(\cdot) := \int_0^1 Q_t^{1\bullet}(Q_t^2(\cdot)P_t^1(f)) \frac{dt}{t},$$
respectively

$$ \tilde{\mathcal{A}}_f^1(\cdot) := \int_0^1 \tilde{\mathcal{Q}}_t^{1\bullet}(\tilde{\mathcal{Q}}_t^2(\cdot)\mathcal{P}_t^1(f)) \frac{dt}{t}, $$

where $\mathcal{Q}^1, \mathcal{Q}^2, \tilde{\mathcal{Q}}^2$ belong at least to $\text{StGC}^3$ and $\tilde{\mathcal{Q}}^1$ is an element of $\text{GC}^3$. We describe similarly the operator $\mathcal{P}_f^{(b)}(\cdot)$ as a finite sum of operators of the form

$$ \mathcal{A}_f^2(\cdot) := \int_0^1 \mathcal{Q}_t^{3\bullet}(\mathcal{Q}_t^4(\cdot)\mathcal{P}_t^2(f)) \frac{dt}{t}. $$

Thus, we need to study a generic modified commutator

$$ \mathcal{A}_f^2(\tilde{\mathcal{A}}_g^1(h)) - \mathcal{A}_g^1(\mathcal{A}_f^2(h)), $$

and introduce for that purpose the intermediate quantity

$$ \mathcal{E}(f, g, h) := \int_0^1 \mathcal{Q}_s^{3\bullet}(\mathcal{Q}_s^4(h) \cdot \mathcal{P}_s^1(g) \cdot \mathcal{P}_s^2(f)) \frac{ds}{s}. $$

(We proceed similarly for the study of $\mathcal{S}^\circ$.) Note here that due to the normalization $\mathcal{P}_1 \simeq \text{Id}$, up to some strongly regularizing operator, there is no loss of generality in assuming that

$$ \int_0^1 \tilde{\mathcal{Q}}_t^{1\bullet} \tilde{\mathcal{Q}}_t^2 \frac{dt}{t} = \int_0^1 \mathcal{Q}_t^{1\bullet} \mathcal{Q}_t^2 \frac{dt}{t} = \int_0^1 \mathcal{Q}_t^{3\bullet} \mathcal{Q}_t^4 \frac{dt}{t} = \text{Id}. \quad (C.4) $$

**Step 1. Study of $\mathcal{A}_f^2(\tilde{\mathcal{A}}_g^1(h)) - \mathcal{E}(f, g, h)$**. We shall use a family $\mathcal{Q}$ in $\text{StGC}^a$, for some $a > |\delta|$, to control the Hölder norm of that quantity. By definition, and using the normalization (C.4), for every $r \in (0, 1)$, the quantity $\mathcal{Q}_r(\mathcal{A}_f^2(\tilde{\mathcal{A}}_g^1(h)) - \mathcal{E}(f, g, h))$ is equal to

$$ \int_0^1 \int_0^1 \mathcal{Q}_r \mathcal{Q}_s^{3\bullet} \{ \mathcal{Q}_s^4 \tilde{\mathcal{Q}}_t^{1\bullet} (\tilde{\mathcal{Q}}_t^2(h) \mathcal{P}_t^1(g)) \cdot \mathcal{P}_s^2(f) \} \frac{ds \, dt}{st} \\ - \int_0^1 \mathcal{Q}_r \mathcal{Q}_s^{3\bullet} (\mathcal{Q}_s^4(h) \cdot \mathcal{P}_s^1(g) \cdot \mathcal{P}_s^2(f)) \frac{ds}{s} \\ = \int_0^1 \int_0^1 \mathcal{Q}_r \mathcal{Q}_s^{3\bullet} \{ \mathcal{Q}_s^4 \tilde{\mathcal{Q}}_t^{1\bullet} (\tilde{\mathcal{Q}}_t^2(h) (\mathcal{P}_t^1(g) - \mathcal{P}_s^1(g))) \cdot \mathcal{P}_s^2(f) \} \frac{ds \, dt}{st}, $$

where in the last line the variable of $\mathcal{P}_s^1(g)$ is that of $\mathcal{Q}_s^{3\bullet}$, and so it is frozen through the action of $\tilde{\mathcal{Q}}_s^4 \tilde{\mathcal{Q}}_t^{1\bullet}$. Then using that $g \in C^\beta$ with $\beta \in (0, 1)$, we know by Proposition 31 that we have, for $\tau \ge \sigma$,

$$ |(\mathcal{P}_s^1 g)(x, \tau) - (\mathcal{P}_t^1 g)(y, \sigma)| \lesssim (s+t+\rho((x, \tau), (y, \sigma))^2)^{\alpha/2} \|g\|_{C^\beta}. $$
respectively

$$ \tilde{\mathcal{A}}_f^1(\cdot) := \int_0^1 \tilde{\mathcal{Q}}_t^1(\cdot) (\tilde{\mathcal{Q}}_t^2(\cdot) \mathcal{P}_t^1(f)) \frac{dt}{t}, $$

where $\mathcal{Q}^1, \mathcal{Q}^2, \tilde{\mathcal{Q}}^2$ belong at least to $\text{StGC}^3$ and $\tilde{\mathcal{Q}}^1$ is an element of $\text{GC}^3$. We describe similarly the operator $\mathcal{P}_f^{(b)}(\cdot)$ as a finite sum of operators of the form

$$ \mathcal{A}_f^2(\cdot) := \int_0^1 \mathcal{Q}_t^{3\bullet}(\mathcal{Q}_t^4(\cdot) \mathcal{P}_t^2(f)) \frac{dt}{t}. $$

Thus, we need to study a generic modified commutator

$$ \mathcal{A}_f^2(\tilde{\mathcal{A}}_g^1(h)) - \mathcal{A}_g^1(\mathcal{A}_f^2(h)), $$

and introduce for that purpose the intermediate quantity

$$ \mathcal{E}(f, g, h) := \int_0^1 \mathcal{Q}_s^{3\bullet}(\mathcal{Q}_s^4(h) \cdot \mathcal{P}_s^1(g) \cdot \mathcal{P}_s^2(f)) \frac{ds}{s}. $$

(We proceed similarly for the study of $\mathcal{S}^\circ$.) Note here that due to the normalization $\mathcal{P}_1 \simeq \text{Id}$, up to some strongly regularizing operator, there is no loss of generality in assuming that

$$ \int_0^1 \tilde{\mathcal{Q}}_t^{1\bullet} \tilde{\mathcal{Q}}_t^2 \frac{dt}{t} = \int_0^1 \mathcal{Q}_t^{1\bullet} \mathcal{Q}_t^2 \frac{dt}{t} = \int_0^1 \mathcal{Q}_t^{3\bullet} \mathcal{Q}_t^4 \frac{dt}{t} = \text{Id}. \quad (C.4) $$

**Step 1. Study of $\mathcal{A}_f^2(\tilde{\mathcal{A}}_g^1(h)) - \mathcal{E}(f, g, h)$**. We shall use a family $\mathcal{Q}$ in $\text{StGC}^a$, for some $a > |\delta|$, to control the Hölder norm of that quantity. By definition, and using the normalization (C.4), for every $r \in (0, 1)$, the quantity $\mathcal{Q}_r(\mathcal{A}_f^2(\tilde{\mathcal{A}}_g^1(h)) - \mathcal{E}(f, g, h))$ is equal to

$$ \begin{aligned} & \int_0^1 \int_0^1 \mathcal{Q}_r \mathcal{Q}_s^{3\bullet} \{ \mathcal{Q}_s^4 \tilde{\mathcal{Q}}_t^{1\bullet} (\tilde{\mathcal{Q}}_t^2(h) \mathcal{P}_t^1(g)) \cdot \mathcal{P}_s^2(f) \} \frac{ds \, dt}{st} \\ & \quad - \int_0^1 \mathcal{Q}_r \mathcal{Q}_s^{3\bullet} (\mathcal{Q}_s^4(h) \cdot \mathcal{P}_s^1(g) \cdot \mathcal{P}_s^2(f)) \frac{ds}{s} \\ & = \int_0^1 \int_0^1 \mathcal{Q}_r \mathcal{Q}_s^{3\bullet} \{ \mathcal{Q}_s^4 \tilde{\mathcal{Q}}_t^{1\bullet} (\tilde{\mathcal{Q}}_t^2(h) (\mathcal{P}_t^1(g) - \mathcal{P}_s^1(g))) \cdot \mathcal{P}_s^2(f) \} \frac{ds \, dt}{st}, \end{aligned} $$

where in the last line the variable of $\mathcal{P}_s^1(g)$ is that of $\mathcal{Q}_s^{3\bullet}$, and so it is frozen through the action of $\tilde{\mathcal{Q}}_s^4 \tilde{\mathcal{Q}}_t^1$. Then using that $g \in C^\beta$ with $\beta \in (0, 1)$, we know by Proposition 31 that we have, for $\tau \ge \sigma$,

$$ |(\mathcal{P}_s^1 g)(x, \tau) - (\mathcal{P}_t^1 g)(y, \sigma)| \lesssim (s+t+\rho((x, \tau), (y, \sigma))^2)^{\alpha/2} \|g\|_{C^\beta}. $$
Note that it follows from Equation (A.1) that the kernel of $Q_s^4 \tilde{Q}_t^{*1}$ is pointwisely bounded by $G_{t+s}$, and allowing different constants in the definition of $G$, we have

$$G_{t+s}((x, \tau), (y, \sigma))(s+t+d(x, y)^2)^{\beta/2} \lesssim (s+t)^{\beta/2} G_{t+s}((x, \tau), (y, \sigma)). \quad (C.5)$$

So using the cancellation property of the operators $Q$, respectively $Q^i$ and $\tilde{Q}^i$, at an order no less than $a$, respectively 3, we deduce that

$$\begin{aligned} \| Q_r (A_f^2 (\tilde{A}_g^1(h)) - \mathcal{E}(f, g, h)) \|_{\infty} & \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \\ & \qquad \int_0^1 \int_0^1 \left( \frac{sr}{(s+r)^2} \right)^{a/2} \left( \frac{st}{(s+t)^2} \right)^{3/2} \\ & \qquad \times t^{\gamma/2} (s+t)^{\beta/2} s^{\alpha/2} \frac{ds dt}{st}, \end{aligned}$$

where we used that $\alpha$ is negative to control $P_s^2(f)$. The integral over $t \in (0, 1)$ can be computed since $\gamma > -3$ and $\beta + \gamma < 3$, and we have

$$\begin{aligned} \| Q_r (A_f^2 (\tilde{A}_g^1(h)) - \mathcal{E}(f, g, h)) \|_{\infty} & \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \\ & \qquad \int_0^1 \int_0^1 \left( \frac{sr}{(s+r)^2} \right)^{a/2} s^{\delta/2} \frac{ds}{s} \\ & \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} r^{\delta/2}, \end{aligned}$$

uniformly in $r \in (0, 1)$ because $|a| > \delta$. That concludes the estimate for the high-frequency part. We repeat the same reasoning for the low-frequency part by replacing $Q_r$ with $Q_1$ and conclude that

$$\| A_f^2 (\tilde{A}_g^1(h)) - \mathcal{E}(f, g, h) \|_{C^\delta} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}.$$

**Step 2.** Study of $A_g^1(A_f^2(h)) - \mathcal{E}(f, g, h)$. This term is almost the same as that of Step 1 and can be treated in exactly the same way. Note that $Q_r(A_g^1(A_f^2(h)) - \mathcal{E}(f, g, h))$ is equal, for every $r \in (0, 1)$, to

$$\begin{aligned} & \int_0^1 \int_0^1 Q_r Q_t^{1\bullet} (Q_2^2 Q_s^{3\bullet} (Q_s^4(h) P_s^2(f)) \cdot P_t^1(g)) \frac{ds dt}{st} \\ & \quad - \int_0^1 Q_r Q_s^{3\bullet} (Q_s^4(h) \cdot P_s^1(g) \cdot P_s^2(f)) \frac{ds}{s} \\ & = \int_0^1 \int_0^1 Q_r Q_t^{1\bullet} \{ Q_2^2 Q_s^{3\bullet} (Q_s^4(h) (P_t^1(g) - P_s^1(g)) \cdot P_s^2(f)) \} \frac{ds dt}{st}, \end{aligned}$$

where in the last line the variable of $P_t^1(g)$ is that of $Q_t^{1\bullet}$, so it is frozen through the action of $Q_s^{3\bullet}$. The same proof as in Step 1 can be repeated, which gives the first statement of the theorem.
Note that it follows from Equation (A.1) that the kernel of $Q_s^4 \tilde{Q}_t^{*1}$ is pointwisely bounded by $G_{t+s}$, and allowing different constants in the definition of $G$, we have

$$G_{t+s}((x, \tau), (y, \sigma))(s+t+d(x, y)^2)^{\beta/2} \lesssim (s+t)^{\beta/2} G_{t+s}((x, \tau), (y, \sigma)). \quad (C.5)$$

So using the cancellation property of the operators $Q$, respectively $Q^i$ and $\tilde{Q}^i$, at an order no less than $a$, respectively 3, we deduce that

$$\begin{aligned} & \|Q_r(A_f^2(\tilde{A}_g^1(h)) - \mathcal{E}(f, g, h))\|_{\infty} \\ & \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \int_0^1 \int_0^1 \left(\frac{sr}{(s+r)^2}\right)^{a/2} \left(\frac{st}{(s+t)^2}\right)^{3/2} \\ & \qquad \times t^{\gamma/2} (s+t)^{\beta/2} s^{\alpha/2} \frac{ds dt}{st}, \end{aligned}$$

where we used that $\alpha$ is negative to control $P_s^2(f)$. The integral over $t \in (0, 1)$ can be computed since $\gamma > -3$ and $\beta + \gamma < 3$, and we have

$$\begin{aligned} & \|Q_r(A_f^2(\tilde{A}_g^1(h)) - \mathcal{E}(f, g, h))\|_{\infty} \\ & \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} \int_0^1 \int_0^1 \left(\frac{sr}{(s+r)^2}\right)^{a/2} s^{\delta/2} \frac{ds}{s} \\ & \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma} r^{\delta/2}, \end{aligned}$$

uniformly in $r \in (0, 1)$ because $|a| > \delta$. That concludes the estimate for the high-frequency part. We repeat the same reasoning for the low-frequency part by replacing $Q_r$ with $Q_1$ and conclude that

$$\|A_f^2(\tilde{A}_g^1(h)) - \mathcal{E}(f, g, h)\|_{C^\delta} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}.$$

**Step 2.** Study of $A_g^1(A_f^2(h)) - \mathcal{E}(f, g, h)$. This term is almost the same as that of Step 1 and can be treated in exactly the same way. Note that $Q_r(A_g^1(A_f^2(h)) - \mathcal{E}(f, g, h))$ is equal, for every $r \in (0, 1)$, to

$$\begin{aligned} & \int_0^1 \int_0^1 Q_r Q_t^{1\bullet}(Q_t^2 Q_s^{3\bullet}(Q_s^4(h) P_s^2(f)) \cdot P_t^1(g)) \frac{ds dt}{st} \\ & - \int_0^1 Q_r Q_s^{3\bullet}(Q_s^4(h) \cdot P_s^1(g) \cdot P_s^2(f)) \frac{ds}{s} \\ & = \int_0^1 \int_0^1 Q_r Q_t^{1\bullet}\{Q_t^2 Q_s^{3\bullet}(Q_s^4(h)(P_t^1(g) - P_s^1(g)) \cdot P_s^2(f))\} \frac{ds dt}{st}, \end{aligned}$$

where in the last line the variable of $P_t^1(g)$ is that of $Q_t^{1\bullet}$, so it is frozen through the action of $Q_s^{3\bullet}$. The same proof as in Step 1 can be repeated, which gives the first statement of the theorem.
**Step 3.** *Proof of the second statement.* For the second statement, Step 1 still holds. So it only remains to compare $\mathcal{E}(f, g, h)$ with $\mathcal{A}_{fg}^2(h)$. This amounts to compare $\mathcal{P}_t^2(fg)$ with $\mathcal{P}_t^1(fg)\mathcal{P}_t^2(f)$. Using the regularity of $g \in C^\beta$ and the uniform boundedness of $f \in L^\infty$, we get

$$ \|\mathcal{P}_t^2(fg) - \mathcal{P}_t^1(g)\mathcal{P}_t^2(uf)\|_{L^\infty} \lesssim t^{\beta/2} $$

which allows us to conclude.

**Step 4.** *Proof of the third statement.* The key observation is that $\mathbb{R}^\circ$ satisfies more cancellation that $\mathbb{R}$, namely

$$ \mathbb{R}^{\circ}(\mathbf{1}, g, h) = \mathbb{R}^{\circ}(f, \mathbf{1}, h) = 0. $$

Taking advantage of that fact, for any $Q \in \text{StGC}^3$, we have for $s \in (0, 1)$ and $e \in \mathcal{M}$, the identity

$$ Q_s(\mathbb{R}^{\circ}(f, g, h))(e) = Q_s(\mathbb{R}^{\circ}(f - f(e), g - g(e), h))(e). $$

The difference structure has been taken into account in this cancellation/identity and we can now estimate each piece in the definition of $\mathbb{R}^\circ$ separately. Using previous arguments, one has

$$
\begin{aligned}
& |Q_s(\mathbf{P}_{f-f(e)}\mathbf{P}_{g-g(e)}(h))(e)| \\
& \lesssim \int_{[0,1]^3} \iint_{\mathcal{M}^2} \left(\frac{st_1}{(s+t_1)^2}\right)^3 \left(\frac{t_1t_2}{(t_1+t_2)^2}\right)^3 \left(\frac{t_2t_3}{(t_2+t_3)^2}\right)^3 \\
& \qquad \times \mathcal{G}_{s+t_1}(e, e')\mathcal{G}_{t_1+t_2}(e, e'') \\
& \qquad |f(e') - f(e)| |g(e) - g(e'')| \nu(de')\nu(de'') t_3^{\gamma/2} \|h\|_{C^\gamma} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \\
& \lesssim_{f,g,h} \int_{[0,1]^3} \iint_{\mathcal{M}^2} \left(\frac{st_1}{(s+t_1)^2}\right)^3 \left(\frac{t_1t_2}{(t_1+t_2)^2}\right)^3 \left(\frac{t_2t_3}{(t_2+t_3)^2}\right)^3 \\
& \qquad \times \mathcal{G}_{s+t_1}(e, e')\mathcal{G}_{t_1+t_2}(e, e'') \\
& \qquad \rho(e, e')^\alpha \rho(e, e'')^\beta \nu(de')\nu(de'') t_3^{\gamma/2} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \\
& \lesssim_{f,g,h} \int_{[0,1]^3} \left(\frac{st_1}{(s+t_1)^2}\right)^3 \left(\frac{t_1t_2}{(t_1+t_2)^2}\right)^3 \left(\frac{t_2t_3}{(t_2+t_3)^2}\right)^3 \\
& \qquad \times (s+t_1+t_2)^{(\alpha+\beta)/2} t_3^{\gamma/2} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \\
& \lesssim s^{(\alpha+\beta+\gamma)/2} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma},
\end{aligned}
$$
**Step 3.** *Proof of the second statement.* For the second statement, Step 1 still holds. So it only remains to compare $\mathcal{E}(f, g, h)$ with $\mathcal{A}_{fg}^2(h)$. This amounts to compare $\mathcal{P}_t^2(fg)$ with $\mathcal{P}_t^1(fg\mathcal{P}_t^2(f))$. Using the regularity of $g \in C^\beta$ and the uniform boundedness of $f \in L^\infty$, we get

$$ \|\mathcal{P}_t^2(fg) - \mathcal{P}_t^1(g)\mathcal{P}_t^2(uf)\|_{L^\infty} \lesssim t^{\beta/2} $$

which allows us to conclude.

**Step 4.** *Proof of the third statement.* The key observation is that $\mathbb{R}^\circ$ satisfies more cancellation that $\mathbb{R}$, namely

$$ \mathbb{R}^{\circ}(\mathbf{1}, g, h) = \mathbb{R}^{\circ}(f, \mathbf{1}, h) = 0. $$

Taking advantage of that fact, for any $Q \in \text{StGC}^3$, we have for $s \in (0, 1)$ and $e \in \mathcal{M}$, the identity

$$ Q_s(\mathbb{R}^{\circ}(f, g, h))(e) = Q_s(\mathbb{R}^{\circ}(f - f(e), g - g(e), h))(e). $$

The difference structure has been taken into account in this cancellation/identity and we can now estimate each piece in the definition of $\mathbb{R}^\circ$ separately. Using previous arguments, one has

$$
\begin{aligned}
& |Q_s(\mathbf{P}_{f-f(e)}\mathbf{P}_{g-g(e)}(h))(e)| \\
& \lesssim \int_{[0,1]^3} \iint_{\mathcal{M}^2} \left(\frac{st_1}{(s+t_1)^2}\right)^3 \left(\frac{t_1t_2}{(t_1+t_2)^2}\right)^3 \left(\frac{t_2t_3}{(t_2+t_3)^2}\right)^3 \\
& \qquad \times \mathcal{G}_{s+t_1}(e, e')\mathcal{G}_{t_1+t_2}(e, e'') \\
& \qquad |f(e') - f(e)| |g(e) - g(e'')| \nu(de')\nu(de'') t_3^{\gamma/2} \|h\|_{C^\gamma} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \\
& \lesssim_{f,g,h} \int_{[0,1]^3} \iint_{\mathcal{M}^2} \left(\frac{st_1}{(s+t_1)^2}\right)^3 \left(\frac{t_1t_2}{(t_1+t_2)^2}\right)^3 \left(\frac{t_2t_3}{(t_2+t_3)^2}\right)^3 \\
& \qquad \times \mathcal{G}_{s+t_1}(e, e')\mathcal{G}_{t_1+t_2}(e, e'') \\
& \qquad \rho(e, e')^\alpha \rho(e, e'')^\beta \nu(de')\nu(de'') t_3^{\gamma/2} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \\
& \lesssim_{f,g,h} \int_{[0,1]^3} \left(\frac{st_1}{(s+t_1)^2}\right)^3 \left(\frac{t_1t_2}{(t_1+t_2)^2}\right)^3 \left(\frac{t_2t_3}{(t_2+t_3)^2}\right)^3 \\
& \qquad \times (s+t_1+t_2)^{(\alpha+\beta)/2} t_3^{\gamma/2} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \\
& \lesssim s^{(\alpha+\beta+\gamma)/2} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma},
\end{aligned}
$$
uniformly in $e \in \mathcal{M}$ and $s \in (0, 1)$, which concludes the proof of the desired estimate for this first term. (The intermediate implicit constants in the upper bounds are constant multiples of $\|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}$.) The second estimate is obtained similarly, observing that

$$|(f(e') - f(e))(g(e') - g(e))| \lesssim \rho(e', e)^{\alpha+\beta} \|f\|_{C^\alpha} \|g\|_{C^\beta},$$

and gives in the end

$$\|Q_s(\mathbf{R}^\circ(f, g, h))\|_{L^\infty} \lesssim s^{(\alpha+\beta+\gamma)/2} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}. \quad \square$$

**REMARK 39.** The above proof actually shows the following property of the operator

$$S_{f,h} := g \mapsto S(f, g, h) \quad (C.6)$$

where $f \in C^\alpha$ and $h \in C^\gamma$ are fixed. For all families $Q^1, Q^2 \in \mathcal{G}C^\alpha$ for some $a > 0$, the linear operator $Q_t^1 S_{f,h} Q_s^{2\bullet}$ has a kernel pointwisely bounded by

$$(t+s)^{(\beta+\gamma)/2} \left( \frac{st}{(s+t)^2} \right)^{a/2} G_{t+s}(e, e') \|f\|_{C^\alpha} \|h\|_{C^\gamma}.$$

**PROPOSITION 40.** Let $\alpha, \beta, \gamma$ be Hölder regularity exponents with $\alpha \in (0, 1)$, $\beta \in (-3, 3)$ and $\gamma \in (-\infty, 3]$. Set

$$\delta := (\alpha + \beta) \wedge 3 + \gamma.$$

If

$$0 < \alpha + \beta + \gamma < 1 \quad \text{and} \quad \beta + \gamma < 0$$

then the corrector $\mathcal{C}$ extends continuously into a trilinear map from $C^\alpha \times C^\beta \times C^\gamma$ to $C^\delta$.

* If $\alpha, \beta, \gamma$ are positive then the commutator $\mathcal{D}$ is a continuous trilinear map from $C^\alpha \times C^\beta \times C^\gamma$ to $C^\delta$.

*Proof.* The result on $\mathcal{C}$ was already proved in [1, Proposition 3.6] in a more general setting. We only focus here on proving the boundedness of $\mathcal{D}$. As already done above, we represent the operator $\mathcal{P}_f^{(b)}(\cdot)$ under the form

$$\mathcal{A}_f(\cdot) := \int_0^1 Q_t^{1\bullet}(Q_t^2(\cdot)\mathcal{P}_t^1(f)) \frac{dt}{t},$$

and the resonant term $\Pi^{(b)}(g, h)$ as

$$\mathcal{B}(g, h) := \int_0^1 \mathcal{P}_t^{2\bullet}(Q_t^3(g) Q_t^4(h)) \frac{dt}{t}.$$
uniformly in $e \in \mathcal{M}$ and $s \in (0, 1)$, which concludes the proof of the desired estimate for this first term. (The intermediate implicit constants in the upper bounds are constant multiples of $\|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}$.) The second estimate is obtained similarly, observing that

$$|(f(e') - f(e))(g(e') - g(e))| \lesssim \rho(e', e)^{\alpha+\beta} \|f\|_{C^\alpha} \|g\|_{C^\beta},$$

and gives in the end

$$\|Q_s(\mathbf{R}^\circ(f, g, h))\|_{L^\infty} \lesssim s^{(\alpha+\beta+\gamma)/2} \|f\|_{C^\alpha} \|g\|_{C^\beta} \|h\|_{C^\gamma}. \quad \square$$

**REMARK 39.** The above proof actually shows the following property of the operator

$$S_{f,h} := g \mapsto S(f, g, h) \quad (C.6)$$

where $f \in C^\alpha$ and $h \in C^\gamma$ are fixed. For all families $Q^1, Q^2 \in \mathcal{G}C^\alpha$ for some $a > 0$, the linear operator $Q_t^1 S_{f,h} Q_s^{2\bullet}$ has a kernel pointwisely bounded by

$$(t+s)^{(\beta+\gamma)/2} \left( \frac{st}{(s+t)^2} \right)^{a/2} G_{t+s}(e, e') \|f\|_{C^\alpha} \|h\|_{C^\gamma}.$$

**PROPOSITION 40.** Let $\alpha, \beta, \gamma$ be Hölder regularity exponents with $\alpha \in (0, 1)$, $\beta \in (-3, 3)$ and $\gamma \in (-\infty, 3]$. Set

$$\delta := (\alpha + \beta) \wedge 3 + \gamma.$$

If

$$0 < \alpha + \beta + \gamma < 1 \quad \text{and} \quad \beta + \gamma < 0$$

then the corrector $\mathcal{C}$ extends continuously into a trilinear map from $C^\alpha \times C^\beta \times C^\gamma$ to $C^\delta$.

* If $\alpha, \beta, \gamma$ are positive then the commutator $\mathcal{D}$ is a continuous trilinear map from $C^\alpha \times C^\beta \times C^\gamma$ to $C^\delta$.

*Proof.* The result on $\mathcal{C}$ was already proved in [1, Proposition 3.6] in a more general setting. We only focus here on proving the boundedness of $\mathcal{D}$. As already done above, we represent the operator $\mathcal{P}_f^{(b)}(\cdot)$ under the form

$$\mathcal{A}_f(\cdot) := \int_0^1 Q_t^{1\bullet}(Q_t^2(\cdot)P_t^1(f)) \frac{dt}{t},$$

and the resonant term $\Pi^{(b)}(g, h)$ as

$$\mathcal{B}(g, h) := \int_0^1 \mathcal{P}_t^{2\bullet}(Q_t^3(g)Q_t^4(h)) \frac{dt}{t}.$$
Thus, we need to study a generic modified commutator

$$ (\star) := \mathcal{B}(A_f(g), h) - A_f(\mathcal{B}(g, h)) \\ = \int_0^1 \int_0^1 \mathcal{P}_t^{2\bullet}(Q_t^3 Q_s^{1\bullet}(Q_s^2(g) \mathcal{P}_t^1(f)) Q_t^4(h)) \frac{ds}{s} \frac{dt}{t} \\ - \int_0^1 \int_0^1 Q_s^{1\bullet}(Q_s^2 \mathcal{P}_t^{2\bullet}(Q_t^3(g) Q_t^4(h)) \mathcal{P}_s^1(f)) \frac{ds}{s} \frac{dt}{t}, $$

and introduce for that purpose the intermediate quantity

$$ \mathcal{E}(f, g, h) := \int_0^1 \mathcal{P}_t^{2\bullet}(\mathcal{P}_t^1(f) Q_t^3(g) Q_t^4(h)) \frac{dt}{t}. $$

Then we compare the two quantities with $\mathcal{E}(f, g, h)$, such as done previously. Each of these two comparisons makes appear an exact commutation on the function $f$, due to our choice of normalization for our paraproducts. Using the $C^\alpha$ regularity on $f$ together with the cancellation property of the $Q$ operators, we get

$$ \begin{aligned} \|Q_r(\star)\|_{L^\infty} & \lesssim \int_0^1 \int_0^1 \left(\frac{r}{r+t}\right)^3 \left(\frac{st}{(s+t)^2}\right)^3 s^{\beta/2} t^{\gamma/2} (s+t)^{\alpha/2} \frac{dt}{t} \frac{ds}{s} \\ & \quad + \int_0^1 \int_0^1 \left(\frac{rs}{(r+s)^2}\right)^3 \left(\frac{s}{s+t}\right)^3 t^{\beta/2} t^{\gamma/2} (s+t)^{\alpha/2} \frac{dt}{t} \frac{ds}{s} \\ & \lesssim \int_0^1 \left(\frac{r}{r+t}\right)^3 t^{(\alpha+\beta+\gamma)/2} \frac{dt}{t} + \int_0^1 \left(\frac{r}{r+t}\right)^3 t^{\beta/2} t^{\gamma/2} (r+t)^{\alpha/2} \frac{dt}{t} \\ & \lesssim r^{\delta/2}, \end{aligned} $$

which shows that $(\star)$ belongs to $C^\delta$.

<figure><img src="image_4.png" /></figure>

**C.2. Boundedness of iterated commutators/correctors.** We now turn to the study of the continuity properties of the iterated versions of commutators/correctors, and start with the (modified) iterated commutator on paraproducts.

**PROPOSITION 41.** If $\gamma \in (0, 1)$, $\alpha, \nu \in (0, 1/2)$ and $\beta \in (-3, 3)$ then we have

$$ \|\mathbb{R}^{\circ}(u, f, P_a g) - P_a \mathbb{R}^{\circ}(u, f, g)\|_{C^{\alpha+\beta+\gamma+\nu}} \lesssim \|f\|_{C^{\alpha}} \|g\|_{C^{\beta}} \|u\|_{C^{\nu}} \|a\|_{C^{\gamma}}. \quad (C.7) $$

This statement is a combination of both (C.1) and (C.3) of Proposition 38; we let the reader write the proof. We now state the continuity result for the 4-linear
Thus, we need to study a generic modified commutator

$$ (\star) := \mathcal{B}(A_f(g), h) - A_f(\mathcal{B}(g, h)) \\ = \int_0^1 \int_0^1 \mathcal{P}_t^{2\bullet}(Q_t^3 Q_s^{1\bullet}(Q_s^2(g) \mathcal{P}_t^1(f)) Q_t^4(h)) \frac{ds}{s} \frac{dt}{t} \\ - \int_0^1 \int_0^1 Q_s^{1\bullet}(Q_s^2 \mathcal{P}_t^{2\bullet}(Q_t^3(g) Q_t^4(h)) \mathcal{P}_s^1(f)) \frac{ds}{s} \frac{dt}{t}, $$

and introduce for that purpose the intermediate quantity

$$ \mathcal{E}(f, g, h) := \int_0^1 \mathcal{P}_t^{2\bullet}(\mathcal{P}_t^1(f) Q_t^3(g) Q_t^4(h)) \frac{dt}{t}. $$

Then we compare the two quantities with $\mathcal{E}(f, g, h)$, such as done previously. Each of these two comparisons makes appear an exact commutation on the function $f$, due to our choice of normalization for our paraproducts. Using the $C^\alpha$ regularity on $f$ together with the cancellation property of the $Q$ operators, we get

$$ \begin{aligned} \|Q_r(\star)\|_{L^\infty} & \lesssim \int_0^1 \int_0^1 \left(\frac{r}{r+t}\right)^3 \left(\frac{st}{(s+t)^2}\right)^3 s^{\beta/2} t^{\gamma/2} (s+t)^{\alpha/2} \frac{dt}{t} \frac{ds}{s} \\ & \quad + \int_0^1 \int_0^1 \left(\frac{rs}{(r+s)^2}\right)^3 \left(\frac{s}{s+t}\right)^3 t^{\beta/2} t^{\gamma/2} (s+t)^{\alpha/2} \frac{dt}{t} \frac{ds}{s} \\ & \lesssim \int_0^1 \left(\frac{r}{r+t}\right)^3 t^{(\alpha+\beta+\gamma)/2} \frac{dt}{t} + \int_0^1 \left(\frac{r}{r+t}\right)^3 t^{\beta/2} t^{\gamma/2} (r+t)^{\alpha/2} \frac{dt}{t} \\ & \lesssim r^{\delta/2}, \end{aligned} $$

which shows that $(\star)$ belongs to $C^\delta$.

<figure><img src="image_4.png" /></figure>

**C.2. Boundedness of iterated commutators/correctors.** We now turn to the study of the continuity properties of the iterated versions of commutators/correctors, and start with the (modified) iterated commutator on paraproducts.

**PROPOSITION 41.** If $\gamma \in (0, 1)$, $\alpha, \nu \in (0, 1/2)$ and $\beta \in (-3, 3)$ then we have

$$ \|\mathbb{R}^\circ(u, f, P_a g) - P_a \mathbb{R}^\circ(u, f, g)\|_{C^{\alpha+\beta+\gamma+\nu}} \lesssim \|f\|_{C^\alpha} \|g\|_{C^\beta} \|u\|_{C^\nu} \|a\|_{C^\gamma}. \quad (C.7) $$

This statement is a combination of both (C.1) and (C.3) of Proposition 38; we let the reader write the proof. We now state the continuity result for the 4-linear
iterated swap operator defined in (3.15); a similar continuity result holds for the 5-linear iterated operator defined in (3.16); its proof is left to the reader.

PROPOSITION 42. Let $\alpha, \beta_1, \beta_2, \gamma$ be Hölder regularity exponents with $\gamma \in (-3, 3)$, $\beta_1, \beta_2 \in (0, 1)$ and $\alpha \in (-\infty, 0)$. Then if

$$ \alpha + \beta_2 + \gamma < 3, \quad \text{and} \quad \delta := \alpha + \beta + \gamma + \nu \in (-3, 3), $$

we have

$$ \|S(f, (g_1, g_2), h)\|_{C^{\delta}} \lesssim \|f\|_{C^{\alpha}} \|g_1\|_{C_1^{\beta}} \|g_2\|_{C_2^{\beta}} \|h\|_{C^{\gamma}}, \quad (C.8) $$

so the commutator defines a quadrilinear continuous map from $C^\alpha \times C^{\beta_1} \times C^{\beta_2} \times C^\gamma$ to $C^\delta$.

*Proof.* Fix some functions $f \in C^\alpha$ and $g_2 \in C^{\beta_2}$; we have

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

With the same notations as in the proof of Proposition 38, for which we have relations (C.4), we write

$$ \begin{aligned} P_{g_1}[S(f, g_2, h)] &= \int_0^1 Q_t^{1\bullet}(Q_t^2[S(f, g_2, h)] \cdot P_t^1 g_1) \frac{dt}{t} \\ &= \int_0^1 \int_0^1 Q_t^{1\bullet}(Q_t^2[S(f, \tilde{Q}_s^{1\bullet}\tilde{Q}_s^2 g_2, h)] \cdot P_t^1 g_1) \frac{ds}{s} \frac{dt}{t}. \end{aligned} $$

Expanding $S(f, \tilde{P}_{g_1} g_2, h)$ correspondingly, we get with $S_{f,h}$ defined in (C.6),

$$ S(f, (g_1, g_2), h) = \int_0^1 \int_0^1 Q_t^{1\bullet} \{ Q_t^2 S_{f,h}(\tilde{Q}_s^{1\bullet}(\tilde{Q}_s^2 g_2 \cdot (P_t^1 g_1 - P_s^1 g_1))) \} \frac{ds}{s} \frac{dt}{t}, \quad (C.9) $$

where the variable of $P_t^1 g_1 h$ is that of $Q_t^{1\bullet}$. Since $g_1$ belongs to $C^{\beta_1}$, with $\beta_1 \in (0, 1)$, we know from Proposition 31 that

$$ |(P_t^1 g_1)(e) - (P_s^1 g_1)(e')| \lesssim (t + s + \rho(e, e')^2)^{\beta_1/2} \|g_1\|_{C^{\beta_1}}, $$

for all $e, e' \in \mathcal{M}$. As above, fix a collection $\mathcal{Q}$ of $\text{StGC}^a$, for some $a > 3$, to control Hölder norms. We need to estimate

$$ \|Q_r S(f, (g_1, g_2), h)\|_{L^\infty(\mathcal{M})}. $$

Using decomposition (C.9), we have

$$ \|Q_r S(f, (g_1, g_2), h)\|_{L^\infty(\mathcal{M})} \lesssim \int_0^1 \int_0^1 \left( \frac{rt}{(r+t)^2} \right)^{a/2} I_{s,t} \frac{ds}{s} \frac{dt}{t}, \quad (C.10) $$
iterated swap operator defined in (3.15); a similar continuity result holds for the 5-linear iterated operator defined in (3.16); its proof is left to the reader.

**PROPOSITION 42.** Let $\alpha, \beta_1, \beta_2, \gamma$ be Hölder regularity exponents with $\gamma \in (-3, 3)$, $\beta_1, \beta_2 \in (0, 1)$ and $\alpha \in (-\infty, 0)$. Then if

$$ \alpha + \beta_2 + \gamma < 3, \quad \text{and} \quad \delta := \alpha + \beta + \gamma + \nu \in (-3, 3), $$

we have

$$ \|S(f, (g_1, g_2), h)\|_{C^{\delta}} \lesssim \|f\|_{C^{\alpha}} \|g_1\|_{C_1^{\beta}} \|g_2\|_{C_2^{\beta}} \|h\|_{C^{\gamma}}, \quad (C.8) $$

so the commutator defines a quadrilinear continuous map from $C^\alpha \times C^{\beta_1} \times C^{\beta_2} \times C^\gamma$ to $C^\delta$.

*Proof.* Fix some functions $f \in C^\alpha$ and $g_2 \in C^{\beta_2}$; we have

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

With the same notations as in the proof of Proposition 38, for which we have relations (C.4), we write

$$ \begin{aligned} P_{g_1}[S(f, g_2, h)] &= \int_0^1 Q_t^{1\bullet}(Q_t^2[S(f, g_2, h)] \cdot P_t^1 g_1) \frac{dt}{t} \\ &= \int_0^1 \int_0^1 Q_t^{1\bullet}(Q_t^2[S(f, \tilde{Q}_s^{1\bullet}\tilde{Q}_s^2 g_2, h)] \cdot P_t^1 g_1) \frac{ds}{s} \frac{dt}{t}. \end{aligned} $$

Expanding $S(f, \tilde{P}_{g_1} g_2, h)$ correspondingly, we get with $S_{f,h}$ defined in (C.6),

$$ S(f, (g_1, g_2), h) = \int_0^1 \int_0^1 Q_t^{1\bullet} \{ Q_t^2 S_{f,h}(\tilde{Q}_s^{1\bullet}(\tilde{Q}_s^2 g_2 \cdot (P_t^1 g_1 - P_s^1 g_1))) \} \frac{ds}{s} \frac{dt}{t}, \quad (C.9) $$

where the variable of $P_t^1 g_1 h$ is that of $Q_t^{1\bullet}$. Since $g_1$ belongs to $C^{\beta_1}$, with $\beta_1 \in (0, 1)$, we know from Proposition 31 that

$$ |(P_t^1 g_1)(e) - (P_s^1 g_1)(e')| \lesssim (t + s + \rho(e, e')^2)^{\beta_1/2} \|g_1\|_{C^{\beta_1}}, $$

for all $e, e' \in \mathcal{M}$. As above, fix a collection $\mathcal{Q}$ of StGC$^a$, for some $a > 3$, to control Hölder norms. We need to estimate

$$ \|Q_r S(f, (g_1, g_2), h)\|_{L^\infty(\mathcal{M})}. $$

Using decomposition (C.9), we have

$$ \|Q_r S(f, (g_1, g_2), h)\|_{L^\infty(\mathcal{M})} \lesssim \int_0^1 \int_0^1 \left( \frac{rt}{(r+t)^2} \right)^{a/2} I_{s,t} \frac{ds}{s} \frac{dt}{t}, \quad (C.10) $$