where

$$I_{s,t} := \sup_{e \in \mathcal{M}} Q_t^2 S_{f,h} (\tilde{Q}_s^1 (\tilde{Q}_s^2 g_2 \cdot (P_t^1 g_1(e) - P_s^1 g_1)))(e).$$

Due to Remark 39, we have a pointwise estimate of the kernel of the operator $Q_t^2 S_{f,h}(Q_s^1(\cdot))$, so with the pointwise regularity estimate on $h$ and (C.5), we deduce that

$$ \begin{aligned} I_{s,t} &\lesssim (s+t)^{(\alpha+\beta_1+\gamma)/2} \|\tilde{Q}_s^2 g_2\|_{L^\infty} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|h\|_{C^\gamma} \\ &\lesssim (s+t)^{\delta/2} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}. \end{aligned} $$

It follows from that estimate and the fact that $|\sigma| < a$, that

$$\|Q_r S(f, (g_1, g_2), h)\|_{L^\infty(\mathcal{M})} \lesssim r^{\delta/2} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma},$$

uniformly in $r \in (0, 1)$. A similar analysis of the low frequency of $S(f, (g_1, g_2), h)$ can be done, which completes the proof of the Hölder estimate. $\square$

**PROPOSITION 43.** Pick $\beta_1, \beta_2 \in (0, 1)$ with $\beta_1 + \beta_2 < 1$, and $\gamma \in (-3, 3)$. We have

$$ \begin{aligned} & \|\mathbf{R}(f, \tilde{\mathbf{P}}_{g_1} g_2, h) - \mathbf{P}_{g_1} \mathbf{R}(f, g_2, h)\|_{C^{\beta_1+\beta_2+\gamma}} \\ & \quad + \|\mathbf{R}^\circ(f, \mathbf{P}_{g_1} g_2, h) - \mathbf{P}_{g_1} \mathbf{R}^\circ(f, g_2, h)\|_{C^{\beta_1+\beta_2+\gamma}} \\ & \lesssim \|f\|_{L^\infty} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}. \end{aligned} \quad (C.11) $$

*Proof.* Fix a parameter $r \in (0, 1)$ and look for a control (uniformly in $e \in \mathcal{M}$) of
$$ (\star) := Q_r[\mathbf{R}(f, \tilde{\mathbf{P}}_{g_1} g_2, h) - \mathbf{P}_{g_1} \mathbf{R}(f, g_2, h)](e). $$

We follow the arguments of Proposition 38 – more precisely of (C.2) and (C.3), since

$$ (\star) = Q_r[\mathbf{R}(f, \tilde{\mathbf{P}}_{g_1} g_2, h)](e) - \int_0^1 Q_r Q_s^\bullet [\mathcal{P}_s g_1 \cdot \mathcal{Q}_s[\mathbf{R}(f, g_2, h)]](e), $$

and we have seen there how to control the composition of a $Q$ operator with $\mathbf{R}$. So we repeat the exact same reasoning with replacing the function $f$ by $\tilde{\mathbf{P}}_{g_1} f$ and now the commutator (with the extra paraproduct $\mathbf{P}_{g_1}$) brings terms of the form

$$ [\mathcal{P}_{g_1} g_2(e') - \mathbf{P}_{g_1} g_2(e'')] - \mathcal{P}_r(g_1)(e''')(g_2(e') - g_2(e'')) $$

for points $e', e'', e'''$ of the parabolic space $\mathcal{M}$, multiplied by Gaussian kernels localizing points at the scale $\rho(e, e') + \rho(e, e'') + \rho(e, e''') \lesssim r^{1/2}$. Using Proposition 35, we deduce that since $g_1 \in C^{\beta_1}$ and $g_2 \in C^{\beta_2}$ then

$$ |[\mathcal{P}_{g_1} g_2(e') - \mathbf{P}_{g_1} g_2(e'')] - \mathcal{P}_r(g_1)(e''')(g_2(e') - g_2(e''))| \lesssim r^{(\beta_1+\beta_2)/2}. $$

The result follows by repeating the computations done in proving (C.3). $\square$
where

$$I_{s,t} := \sup_{e \in \mathcal{M}} Q_t^2 S_{f,h} (\tilde{Q}_s^1 (\tilde{Q}_s^2 g_2 \cdot (P_t^1 g_1(e) - P_s^1 g_1)))(e).$$

Due to Remark 39, we have a pointwise estimate of the kernel of the operator $Q_t^2 S_{f,h}(Q_s^1(\cdot))$, so with the pointwise regularity estimate on $h$ and (C.5), we deduce that

$$ \begin{aligned} I_{s,t} &\lesssim (s+t)^{(\alpha+\beta_1+\gamma)/2} \|\tilde{Q}_s^2 g_2\|_{L^\infty} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|h\|_{C^\gamma} \\ &\lesssim (s+t)^{\delta/2} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}. \end{aligned} $$

It follows from that estimate and the fact that $|\sigma| < a$, that

$$\|Q_r S(f, (g_1, g_2), h)\|_{L^\infty(\mathcal{M})} \lesssim r^{\delta/2} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma},$$

uniformly in $r \in (0, 1)$. A similar analysis of the low frequency of $S(f, (g_1, g_2), h)$ can be done, which completes the proof of the Hölder estimate. $\square$

**PROPOSITION 43.** Pick $\beta_1, \beta_2 \in (0, 1)$ with $\beta_1 + \beta_2 < 1$, and $\gamma \in (-3, 3)$. We have

$$ \begin{aligned} & \|\mathbf{R}(f, \tilde{\mathbf{P}}_{g_1} g_2, h) - \mathbf{P}_{g_1} \mathbf{R}(f, g_2, h)\|_{C^{\beta_1+\beta_2+\gamma}} \\ & \quad + \|\mathbf{R}^\circ(f, \mathbf{P}_{g_1} g_2, h) - \mathbf{P}_{g_1} \mathbf{R}^\circ(f, g_2, h)\|_{C^{\beta_1+\beta_2+\gamma}} \\ & \lesssim \|f\|_{L^\infty} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}. \end{aligned} \quad (C.11) $$

*Proof.* Fix a parameter $r \in (0, 1)$ and look for a control (uniformly in $e \in \mathcal{M}$) of

$$ (\star) := Q_r[\mathbf{R}(f, \tilde{\mathbf{P}}_{g_1} g_2, h) - \mathbf{P}_{g_1} \mathbf{R}(f, g_2, h)](e). $$

We follow the arguments of Proposition 38 – more precisely of (C.2) and (C.3), since

$$ (\star) = Q_r[\mathbf{R}(f, \tilde{\mathbf{P}}_{g_1} g_2, h)](e) - \int_0^1 Q_r Q_s^\bullet [\mathcal{P}_s g_1 \cdot \mathcal{Q}_s[\mathbf{R}(f, g_2, h)]](e), $$

and we have seen there how to control the composition of a $Q$ operator with $\mathbf{R}$. So we repeat the exact same reasoning with replacing the function $f$ by $\tilde{\mathbf{P}}_{g_1} f$ and now the commutator (with the extra paraproduct $\mathbf{P}_{g_1}$) brings terms of the form

$$ [\mathcal{P}_{g_1} g_2(e') - \mathbf{P}_{g_1} g_2(e'')] - \mathcal{P}_r(g_1)(e''')(g_2(e') - g_2(e'')) $$

for points $e', e'', e'''$ of the parabolic space $\mathcal{M}$, multiplied by Gaussian kernels localizing points at the scale $\rho(e, e') + \rho(e, e'') + \rho(e, e''') \lesssim r^{1/2}$. Using Proposition 35, we deduce that since $g_1 \in C^{\beta_1}$ and $g_2 \in C^{\beta_2}$ then

$$ \|[\mathcal{P}_{g_1} g_2(e') - \mathbf{P}_{g_1} g_2(e'')] - \mathcal{P}_r(g_1)(e''')(g_2(e') - g_2(e''))\| \lesssim r^{(\beta_1+\beta_2)/2}. $$

The result follows by repeating the computations done in proving (C.3). $\square$
We now look at the iterated corrector. The proof of continuity for the lower and upper iterates are almost the same and the reader can see clearly on the model case of iterated integrals what the difference is.

**PROPOSITION 44.** Let $\alpha, \beta_1 \in (0, 1)$, $\beta_2 \in (-3, 3)$ and $\gamma \in (-\infty, 3]$. Assume that $\alpha + \beta_1 + \beta_2 < 3$ with

$$ \delta := \alpha + \beta_1 + \beta_2 + \gamma \in (0, 1), \quad \alpha + \beta_2 + \gamma < 0 \quad \text{and} \quad \beta_1 + \beta_2 + \gamma < 0. $$

Then the 4-linear upper iterated corrector $\mathcal{C}$ is a continuous map from $\mathcal{C}^\alpha \times \mathcal{C}^{\beta_1} \times \mathcal{C}^{\beta_2} \times \mathcal{C}^\gamma$ to $\mathcal{C}^\delta$.

*Proof.* Fix $f \in \mathcal{C}^\alpha$ and $h \in \mathcal{C}^\gamma$ and set

$$ \bar{\mathcal{C}}(\cdot) := \mathcal{C}(f, \cdot, h), $$

so

$$ \mathcal{C}(f, (g_1, g_2), h) = \bar{\mathcal{C}}(\tilde{\mathcal{Q}}_{g_1} g_2) - g_1 \bar{\mathcal{C}}(g_2). $$

Using the same notation as previously, and omitting for convenience the indices on the different collections $\mathcal{Q}$ and $\mathcal{P}$, we write

$$ \begin{aligned} \bar{\mathcal{C}}(\tilde{\mathcal{P}}_{g_1} g_2) &= \int_0^1 \bar{\mathcal{C}} \tilde{\mathcal{Q}}_s^\bullet (\tilde{\mathcal{Q}}_s g_2 \cdot \mathcal{P}_s g_1) \frac{ds}{s}, \\ g_1 \bar{\mathcal{C}}(g_2) &= g_1 \bar{\mathcal{C}}(\tilde{\mathcal{P}}_1(g_2)) = g_1 \int_0^1 \bar{\mathcal{C}} \tilde{\mathcal{Q}}_s^\bullet (\tilde{\mathcal{Q}}_s g_2 \cdot \mathcal{P}_s \mathbf{1}) \frac{ds}{s}. \end{aligned} $$

Note that due to the conservation property of the heat semigroup associated with $L$, the quantity $\mathcal{P}_s \mathbf{1}$ is either constant equal to 1 or to 0, depending on whether $\mathcal{P}_s$ encodes some cancellation or not. Thus, given $e = (x, \tau) \in \mathcal{M}$, and setting

$$ F_{s,e} := \tilde{\mathcal{Q}}_s g_2 \cdot (\mathcal{P}_s g_1 - \mathcal{P}_s(\mathbf{1}) \cdot g_2(e)), $$

we have

$$ \mathcal{C}(f, (g_1, g_2), h)(e) = \bar{\mathcal{C}}(\tilde{\mathcal{P}}_{g_1} g_2)(e) - \mathcal{P}_s(1)(e) \bar{\mathcal{C}}(g_2)(e) = \int_0^1 \bar{\mathcal{C}}(\tilde{\mathcal{Q}}_s^\bullet F_{s,e})(e) \frac{ds}{s}. $$

As before, we can use that $g_1 \in \mathcal{C}^{\beta_1}$, with $\beta_1 \in (0, 1)$. We have for $e, e' \in \mathcal{M}$

$$ |g_1(e) - g_1(e')| \lesssim \rho(e, e')^\beta \|g_1\|_{\mathcal{C}^{\beta_1}}, $$

and therefore, using the ‘Gaussian bounds’ for $\mathcal{P}_s$,

$$ |(\mathcal{P}_s g_1)(e') - (\mathcal{P}_s \mathbf{1})(e') g_1(e)| \lesssim (s + \rho(e, e')^2)^{\beta/2} \|g_1\|_{\mathcal{C}^{\beta_1}}. $$
We now look at the iterated corrector. The proof of continuity for the lower and upper iterates are almost the same and the reader can see clearly on the model case of iterated integrals what the difference is.

**PROPOSITION 44.** Let $\alpha, \beta_1 \in (0, 1)$, $\beta_2 \in (-3, 3)$ and $\gamma \in (-\infty, 3]$. Assume that $\alpha + \beta_1 + \beta_2 < 3$ with

$$ \delta := \alpha + \beta_1 + \beta_2 + \gamma \in (0, 1), \quad \alpha + \beta_2 + \gamma < 0 \quad \text{and} \quad \beta_1 + \beta_2 + \gamma < 0. $$

Then the 4-linear upper iterated corrector $\mathcal{C}$ is a continuous map from $\mathcal{C}^\alpha \times \mathcal{C}^{\beta_1} \times \mathcal{C}^{\beta_2} \times \mathcal{C}^\gamma$ to $\mathcal{C}^\delta$.

*Proof.* Fix $f \in \mathcal{C}^\alpha$ and $h \in \mathcal{C}^\gamma$ and set

$$ \bar{\mathcal{C}}(\cdot) := \mathcal{C}(f, \cdot, h), $$

so

$$ \mathcal{C}(f, (g_1, g_2), h) = \bar{\mathcal{C}}(\tilde{\mathcal{Q}}_{g_1} g_2) - g_1 \bar{\mathcal{C}}(g_2). $$

Using the same notation as previously, and omitting for convenience the indices on the different collections $\mathcal{Q}$ and $\mathcal{P}$, we write

$$ \begin{aligned} \bar{\mathcal{C}}(\tilde{\mathcal{P}}_{g_1} g_2) &= \int_0^1 \bar{\mathcal{C}} \tilde{\mathcal{Q}}_s \cdot (\tilde{\mathcal{Q}}_s g_2 \cdot \mathcal{P}_s g_1) \frac{ds}{s}, \\ g_1 \bar{\mathcal{C}}(g_2) &= g_1 \bar{\mathcal{C}}(\tilde{\mathcal{P}}_1(g_2)) = g_1 \int_0^1 \bar{\mathcal{C}} \tilde{\mathcal{Q}}_s \cdot (\tilde{\mathcal{Q}}_s g_2 \cdot \mathcal{P}_s \mathbf{1}) \frac{ds}{s}. \end{aligned} $$

Note that due to the conservation property of the heat semigroup associated with $L$, the quantity $\mathcal{P}_s \mathbf{1}$ is either constant equal to 1 or to 0, depending on whether $\mathcal{P}_s$ encodes some cancellation or not. Thus, given $e = (x, \tau) \in \mathcal{M}$, and setting

$$ F_{s,e} := \tilde{\mathcal{Q}}_s g_2 \cdot (\mathcal{P}_s g_1 - \mathcal{P}_s(\mathbf{1}) \cdot g_2(e)), $$

we have

$$ \mathcal{C}(f, (g_1, g_2), h)(e) = \bar{\mathcal{C}}(\tilde{\mathcal{P}}_{g_1} g_2)(e) - \mathcal{P}_s(1)(e) \bar{\mathcal{C}}(g_2)(e) = \int_0^1 \bar{\mathcal{C}}(\tilde{\mathcal{Q}}_s \cdot \mathcal{F}_{s,e})(e) \frac{ds}{s}. $$

As before, we can use that $g_1 \in \mathcal{C}^{\beta_1}$, with $\beta_1 \in (0, 1)$. We have for $e, e' \in \mathcal{M}$

$$ |g_1(e) - g_1(e')| \lesssim \rho(e, e')^{\beta} \|g_1\|_{\mathcal{C}^{\beta_1}}, $$

and therefore, using the ‘Gaussian bounds’ for $\mathcal{P}_s$,

$$ |(\mathcal{P}_s g_1)(e') - (\mathcal{P}_s \mathbf{1})(e') g_1(e)| \lesssim (s + \rho(e, e')^2)^{\beta/2} \|g_1\|_{\mathcal{C}^{\beta_1}}. $$
As done in the proof of Proposition 40, we introduce an intermediate quantity of the form

$$S(a, b, c) := \int_{0}^{1} \mathcal{P}_{t}(Q_{t}b \cdot Q_{t}c \cdot \mathcal{P}_{t}a) \frac{dt}{t},$$

and write

$$ \begin{aligned} \bar{\mathcal{C}}(\tilde{Q}_{s}^{\bullet} F_{s,e})(e) &= \Pi(\tilde{\mathbb{P}}_{f}(\tilde{Q}_{s}^{\bullet} F_{s,e}), h)(e) - S(f, \tilde{Q}_{s}^{\bullet} F_{s,e}, h)(e) \\ &\quad + S(f, \tilde{Q}_{s}^{\bullet} F_{s,e}, h)(e) - f(e) \cdot \Pi(\tilde{Q}_{s}^{\bullet} F_{s,e}, h)(e) \\ &=: I_{1}(s) + I_{2}(s). \end{aligned} \quad (C.12) $$

• We start with the estimate for $I_2$. One can then write with generic notations for the resonant term $\Pi$

$$(S(f, g_2, h) - f \cdot \Pi(g_2, h))(e) = \int_0^1 \mathcal{P}_t(Q_t g_2 \cdot Q_t h \cdot (\mathcal{P}_t f - f(e)))(e) \frac{dt}{t},$$

and it is known that the integrand is pointwisely bounded by $t^{(\alpha+\nu_1+\nu_2)/2}$. Since this argument only uses pointwise estimates, we can replace $b$ by $Q_s^\bullet F_{s,e}$. Therefore, by writing

$$\int_0^1 I_2(s) \frac{ds}{s} = \int_0^1 \int_0^1 \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (\mathcal{P}_t f - f(e)))(e) \frac{dt}{t} \frac{ds}{s}$$

and using

$$\| Q_t \tilde{Q}_s^\bullet \phi \|_{L^\infty(\mathcal{M})} \lesssim \left( \frac{st}{(s+t)^2} \right)^{3/2} \| \phi \|_{L^\infty(\mathcal{M})}, \quad (C.13)$$

with $\phi = F_{s,e}$, we obtain

$$ \begin{aligned} & \left\| \int_0^1 I_2(s) \frac{ds}{s} \right\|_{L^\infty(\mathcal{M})} \\ & \le \int_0^1 \int_0^1 \|e \mapsto \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e)\|_{L^\infty} \frac{dt}{t} \frac{ds}{s} \\ & \lesssim \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|f\|_{C^\alpha} \|h\|_{C^\gamma} \\ & \quad \times \int_0^1 \int_0^1 \left( \frac{st}{(s+t)^2} \right)^{3/2} G_{t+s}(e, e')(s+\rho(e, e')^2)^{\beta_1/2} s^{\beta_2/2} t^{(\alpha+\gamma)/2} \frac{ds}{s} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \\ & \quad \times s^{\beta_2/2} (s+t)^{\beta_1/2} t^{(\alpha+\gamma)/2} \frac{ds}{s} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}, \end{aligned} $$

since $\alpha + \beta_1 + \beta_2 + \gamma > 0$.
As done in the proof of Proposition 40, we introduce an intermediate quantity of the form

$$S(a, b, c) := \int_{0}^{1} \mathcal{P}_{t}(Q_{t}b \cdot Q_{t}c \cdot \mathcal{P}_{t}a) \frac{dt}{t},$$

and write

$$ \begin{aligned} \bar{\mathcal{C}}(\tilde{Q}_{s}^{\bullet} F_{s,e})(e) &= \Pi(\tilde{\mathbb{P}}_{f}(\tilde{Q}_{s}^{\bullet} F_{s,e}), h)(e) - S(f, \tilde{Q}_{s}^{\bullet} F_{s,e}, h)(e) \\ &\quad + S(f, \tilde{Q}_{s}^{\bullet} F_{s,e}, h)(e) - f(e) \cdot \Pi(\tilde{Q}_{s}^{\bullet} F_{s,e}, h)(e) \\ &=: I_{1}(s) + I_{2}(s). \end{aligned} \quad (C.12) $$

• We start with the estimate for $I_2$. One can then write with generic notations for the resonant term $\Pi$

$$(S(f, g_2, h) - f \cdot \Pi(g_2, h))(e) = \int_0^1 \mathcal{P}_t(Q_t g_2 \cdot Q_t h \cdot (\mathcal{P}_t f - f(e)))(e) \frac{dt}{t},$$

and it is known that the integrand is pointwisely bounded by $t^{(\alpha+\nu_1+\nu_2)/2}$. Since this argument only uses pointwise estimates, we can replace $b$ by $Q_s^\bullet F_{s,e}$. Therefore, by writing

$$\int_0^1 I_2(s) \frac{ds}{s} = \int_0^1 \int_0^1 \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (\mathcal{P}_t f - f(e))(e) \frac{dt}{t} \frac{ds}{s}$$

and using

$$\| Q_t \tilde{Q}_s^\bullet \phi \|_{L^\infty(\mathcal{M})} \lesssim \left( \frac{st}{(s+t)^2} \right)^{3/2} \| \phi \|_{L^\infty(\mathcal{M})}, \quad (C.13)$$

with $\phi = F_{s,e}$, we obtain

$$ \begin{aligned} & \left\| \int_0^1 I_2(s) \frac{ds}{s} \right\|_{L^\infty(\mathcal{M})} \\ & \le \int_0^1 \int_0^1 \|e \mapsto \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e) \|_{L^\infty} \frac{dt}{t} \frac{ds}{s} \\ & \lesssim \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|f\|_{C^\alpha} \|h\|_{C^\gamma} \\ & \quad \times \int_0^1 \int_0^1 \left( \frac{st}{(s+t)^2} \right)^{3/2} G_{t+s}(e, e')(s+\rho(e, e')^2)^{\beta_1/2} s^{\beta_2/2} t^{(\alpha+\gamma)/2} \frac{ds}{s} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \\ & \quad \times s^{\beta_2/2} (s+t)^{\beta_1/2} t^{(\alpha+\gamma)/2} \frac{ds}{s} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}, \end{aligned} $$

since $\alpha + \beta_1 + \beta_2 + \gamma > 0$.
• Let us now estimate the regularity of $I_2(s)$. Let $e, e' \in \mathcal{M}$ with $\rho(e, e') \le 1$. We split the integral in $t$ into two parts, corresponding to $t < \rho(e, e')^2$ or $t > \rho(e, e')^2$. In the first case, note that

$$ \int_0^{\rho(e,e')^2} t^{(\alpha+\beta_1+\beta_2+\gamma)/2} \frac{dt}{t} \lesssim \rho(e, e')^{\alpha+\beta_1+\beta_2+\gamma}, $$

so that by repeating the arguments above, we get the desired estimate. In the case $t > \rho^2$ with $\rho := \rho(e, e')$, write for $s \in (0, 1)$

$$ \begin{aligned} & \int_{\rho^2}^{1} \{ \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e) \\ & \qquad - \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e'} \cdot Q_t h \cdot (f(e') - \mathcal{P}_t f))(e') \} \frac{dt}{t} \\ &= \int_{\rho^2}^{1} \{ \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e) \\ & \qquad - \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e') \} \frac{dt}{t} \\ & \qquad + (g_1(e) - g_1(e')) \int_{\rho^2}^{1} \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet \tilde{Q}_s b \cdot Q_t h \cdot (f(e') - \mathcal{P}_t f))(e') \frac{dt}{t} \\ & \qquad - (f(e) - f(e')) \int_{\rho^2}^{1} \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h)(e') \frac{dt}{t}. \end{aligned} \quad (C.14) $$

For the second and third term, we can assume $s \simeq t$ by (C.13). One obtains

$$ \begin{aligned} & |g_1(e) - g_1(e')| \int_{\rho^2}^{1} |\mathcal{P}_t(Q_t \tilde{Q}_s^\bullet \tilde{Q}_s g_2 \cdot Q_t h \cdot (f(e') - \mathcal{P}_t f))(e')| \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^{\beta_1} \int_{\rho^2}^{1} t^{(\alpha+\beta_1+\gamma)/2} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^{\alpha+\beta_1+\beta_2+\gamma}, \end{aligned} $$

since $\alpha + \beta_2 + \gamma$ is negative, and

$$ \begin{aligned} & |f(e) - f(e')| \int_{\rho^2}^{1} |\mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h)(e')| \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^\alpha \int_{\rho^2}^{1} t^{(\beta_1+\beta_2+\gamma)/2} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^{\alpha+\beta_1+\beta_2+\gamma}, \end{aligned} $$
• Let us now estimate the regularity of $I_2(s)$. Let $e, e' \in \mathcal{M}$ with $\rho(e, e') \le 1$. We split the integral in $t$ into two parts, corresponding to $t < \rho(e, e')^2$ or $t > \rho(e, e')^2$. In the first case, note that

$$ \int_0^{\rho(e,e')} t^{(\alpha+\beta_1+\beta_2+\gamma)/2} \frac{dt}{t} \lesssim \rho(e, e')^{\alpha+\beta_1+\beta_2+\gamma}, $$

so that by repeating the arguments above, we get the desired estimate. In the case $t > \rho^2$ with $\rho := \rho(e, e')$, write for $s \in (0, 1)$

$$ \begin{aligned} & \int_{\rho^2}^{1} \{ \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e) \\ & \qquad - \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e'} \cdot Q_t h \cdot (f(e') - \mathcal{P}_t f))(e') \} \frac{dt}{t} \\ &= \int_{\rho^2}^{1} \{ \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e) \\ & \qquad - \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h \cdot (f(e) - \mathcal{P}_t f))(e') \} \frac{dt}{t} \\ & \qquad + (g_1(e) - g_1(e')) \int_{\rho^2}^{1} \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet \tilde{Q}_s b \cdot Q_t h \cdot (f(e') - \mathcal{P}_t f))(e') \frac{dt}{t} \\ & \qquad - (f(e) - f(e')) \int_{\rho^2}^{1} \mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h)(e') \frac{dt}{t}. \end{aligned} \quad (C.14) $$

For the second and third term, we can assume $s \simeq t$ by (C.13). One obtains

$$ \begin{aligned} & |g_1(e) - g_1(e')| \int_{\rho^2}^{1} |\mathcal{P}_t(Q_t \tilde{Q}_s^\bullet \tilde{Q}_s g_2 \cdot Q_t h \cdot (f(e') - \mathcal{P}_t f))(e')| \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^{\beta_1} \int_{\rho^2}^{1} t^{(\alpha+\beta_1+\gamma)/2} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^{\alpha+\beta_1+\beta_2+\gamma}, \end{aligned} $$

since $\alpha + \beta_2 + \gamma$ is negative, and

$$ \begin{aligned} & |f(e) - f(e')| \int_{\rho^2}^{1} |\mathcal{P}_t(Q_t \tilde{Q}_s^\bullet F_{s,e} \cdot Q_t h)(e')| \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^\alpha \int_{\rho^2}^{1} t^{(\beta_1+\beta_2+\gamma)/2} \frac{dt}{t} \\ & \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \rho^{\alpha+\beta_1+\beta_2+\gamma}, \end{aligned} $$
since $\beta_1 + \beta_2 + \gamma$ is also negative. For the first term in (C.14), we now repeat the arguments of the proof of Proposition 40, which rely on the Lipschitz regularity of the heat kernel as well as the fact that $(\alpha + \beta_1 + \beta_2 + \gamma) \in (0, 1)$. Summarizing, we have shown that for $e, e' \in \mathcal{M}$ with $\rho(e, e') \le 1$

$$ \left| \int_0^1 (I_2(s)(e) - I_2(s)(e')) \frac{ds}{s} \right| \lesssim \rho(e, e')^{\alpha+\beta_1+\beta_2+\gamma} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}. $$

Let us now come to $I_1(s)$ as defined in (C.12). Set $\phi := \tilde{Q}_s^\bullet F_{s,e}$, and write

$$ |\Pi(\tilde{P}_f(\phi), h) - S(f, g_2, h)| \le \int_0^1 |\mathcal{P}_t(A_t(\phi, f) \cdot Q_t h)| \frac{dt}{t} $$

with

$$ A_t(\phi, f) := Q_t \left( \int_0^1 \mathcal{P}_t \tilde{Q}_r^\bullet (\tilde{Q}_r \phi \cdot \mathcal{P}_r f) \frac{dr}{r} - \mathcal{P}_t f \mathcal{P}_t \phi \right). $$

Following the proof of Proposition 40, and using (C.13), one obtains

$$ \begin{aligned} & \|A_t(\tilde{Q}_s^\bullet F_{s,e}, u)\|_{L^\infty(\mathcal{M})} \\ & \lesssim \int_0^1 \left( \frac{rt}{(r+t)^2} \right)^{3/2} \left( \frac{sr}{(s+r)^2} \right)^{3/2} \\ & \qquad \times s^{\beta_2/2} (r+t)^{(\alpha+\beta_1)/2} \frac{dr}{r} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}}, \end{aligned} $$

hence

$$ \left\| \int_0^1 I_1(s) \frac{ds}{s} \right\|_{L^\infty(\mathcal{M})} \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \\ \times \int_0^1 \int_0^1 \int_0^1 \left( \frac{rt}{(r+t)^2} \right)^{3/2} \left( \frac{sr}{(s+r)^2} \right)^{3/2} s^{\beta_2/2} (r+t)^{(\alpha+\beta_1)/2} t^{\gamma/2} \frac{dr}{r} \frac{ds}{s} \frac{dt}{t}, $$

and the triple integral is finite since $(\alpha + \beta_1 + \beta_2 + \gamma)$ is positive.

• For the regularity estimate of $I_1(s)$, consider

$$ \int_0^1 \{ \mathcal{P}_t(A_t(\tilde{Q}_s^\bullet F_{s,e}, f) \cdot Q_t h)(e) - \mathcal{P}_t(A_t(\tilde{Q}_s^\bullet F_{s,e'}, f) \cdot Q_t h)(e') \} \frac{dt}{t}. $$

The estimate of this expression is similar, though simpler, compared to the one of $I_2(s)$, as here $e$ is frozen only in one spot. As before, one deals with this terms using the heat kernel regularity of $\mathcal{P}_t$ and the regularity estimate for $a$. $\square$
since $\beta_1 + \beta_2 + \gamma$ is also negative. For the first term in (C.14), we now repeat the arguments of the proof of Proposition 40, which rely on the Lipschitz regularity of the heat kernel as well as the fact that $(\alpha + \beta_1 + \beta_2 + \gamma) \in (0, 1)$. Summarizing, we have shown that for $e, e' \in \mathcal{M}$ with $\rho(e, e') \le 1$

$$ \left| \int_0^1 (I_2(s)(e) - I_2(s)(e')) \frac{ds}{s} \right| \\ \lesssim \rho(e, e')^{\alpha+\beta_1+\beta_2+\gamma} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma}. $$

Let us now come to $I_1(s)$ as defined in (C.12). Set $\phi := \tilde{Q}_s^\bullet F_{s,e}$, and write

$$ |\Pi(\tilde{P}_f(\phi), h) - S(f, g_2, h)| \le \int_0^1 |\mathcal{P}_t(A_t(\phi, f) \cdot Q_t h)| \frac{dt}{t} $$

with

$$ A_t(\phi, f) := Q_t \left( \int_0^1 \mathcal{P}_t \tilde{Q}_r^\bullet (\tilde{Q}_r \phi \cdot \mathcal{P}_r f) \frac{dr}{r} - \mathcal{P}_t f \mathcal{P}_t \phi \right). $$

Following the proof of Proposition 40, and using (C.13), one obtains

$$ \begin{aligned} \|A_t(\tilde{Q}_s^\bullet F_{s,e}, u)\|_{L^\infty(\mathcal{M})} & \lesssim \int_0^1 \left(\frac{rt}{(r+t)^2}\right)^{3/2} \left(\frac{sr}{(s+r)^2}\right)^{3/2} \\ & \quad \times s^{\beta_2/2} (r+t)^{(\alpha+\beta_1)/2} \frac{dr}{r} \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}}, \end{aligned} $$

hence

$$ \left\| \int_0^1 I_1(s) \frac{ds}{s} \right\|_{L^\infty(\mathcal{M})} \lesssim \|f\|_{C^\alpha} \|g_1\|_{C^{\beta_1}} \|g_2\|_{C^{\beta_2}} \|h\|_{C^\gamma} \\ \times \int_0^1 \int_0^1 \int_0^1 \left(\frac{rt}{(r+t)^2}\right)^{3/2} \left(\frac{sr}{(s+r)^2}\right)^{3/2} s^{\beta_2/2} (r+t)^{(\alpha+\beta_1)/2} t^{\gamma/2} \frac{dr}{r} \frac{ds}{s} \frac{dt}{t}, $$

and the triple integral is finite since $(\alpha + \beta_1 + \beta_2 + \gamma)$ is positive.

• For the regularity estimate of $I_1(s)$, consider

$$ \int_0^1 \{ \mathcal{P}_t(A_t(\tilde{Q}_s^\bullet F_{s,e}, f) \cdot Q_t h)(e) - \mathcal{P}_t(A_t(\tilde{Q}_s^\bullet F_{s,e'}, f) \cdot Q_t h)(e') \} \frac{dt}{t}. $$

The estimate of this expression is similar, though simpler, compared to the one of $I_2(s)$, as here $e$ is frozen only in one spot. As before, one deals with this terms using the heat kernel regularity of $\mathcal{P}_t$ and the regularity estimate for $a$. $\square$
Acknowledgements

The authors warmly thank the reviewers for their thorough readings of previous versions of the present work.

**Conflict of Interest:** None

References

[1] I. Bailleul and F. Bernicot, ‘Heat semigroup and singular PDEs’, *J. Funct. Anal.* **270** (2016), 3344–3452.

[2] I. Bailleul, F. Bernicot and D. Frey, ‘Space-time paraproducts for paracontrolled calculus, 3D-PAM and multiplicative Burgers equations’, *Ann. Sci. Éc. Norm. Supér.* **(4)** **51**(6) (2018), 1399–1457.

[3] I. Bailleul, A. Debussche and M. Hofmanovà, ‘Quasilinear generalised parabolic Anderson model equation’, *Stoch. Part. Diff. Eq.: Anal. Comput.* **7**(1) (2018), 40–63.

[4] I. Bailleul and M. Hoshino, ‘Paracontrolled calculus and regularity structures (I)’, *J. Math. Soc. Japan*, to appear. Preprint, 2020, arXiv:1812.07919.

[5] Y. Bruned, A. Chandra, I. Chevyrev and M. Hairer, ‘Renormalising SPDEs in regularity structures’, *J. Eur. Math. Soc.* (2020), (to appear).

[6] Y. Bruned, F. Gabriel, M. Hairer and L. Zambotti, ‘Geometric stochastic heat equations’, Preprint, 2019, arXiv:1902.02884.

[7] Y. Bruned, M. Hairer and L. Zambotti, ‘Algebraic renormalisation of regularity structures’, *Invent. Math.* **215**(3) (2019), 1039–1156.

[8] M. Caruana, T. Lévy and T. Lyons, *Differential Equations Driven by Rough Paths*, Lecture Notes in Mathematics 1908 (Springer-Verlag, Berlin Heidelberg, 2006).

[9] R. Catellier and K. Chouk, ‘Paracontrolled distributions and the 3-dimensional stochastic quantization equation’, *Ann. Probab.* **46**(5) (2018), 2621–2679.

[10] A. Chandra and M. Hairer, ‘An analytic BPHZ theorem for regularity structures’, Preprint, 2016, arXiv:1612.08138.

[11] J.-Y. Chemin, ‘Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires’, *Duke Math. J.* **56**(3) (1988), 431–469.

[12] J.-Y. Chemin, ‘Interaction contrôlée dans les équations aux dérivées partielles non linéaires’, *Bull. Soc. Math. France* **116**(3) (1988), 341–383.

[13] K. Chouk and R. Allez, ‘The continuous Anderson hamiltonian in dimension two’, Preprint, 2015, arXiv:1511.02718.

[14] K. Chouk and G. Cannizzaro, ‘Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential’, *Ann. Probab.* **46**(3) (2018), 1710–1763.

[15] A. Dahlqvist, J. Diehl and B. Driver, ‘The parabolic Anderson model on Riemann surfaces’, *Probab. Theory Related Fields* **174**(1-2) (2019), 369–444.

[16] X. T. Duong and A. McIntosh, ‘Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients’, *J. Geom. Anal.* **6**(2) (1996), 181–205.

[17] M. Gubinelli, P. Imkeller and N. Perkowski, ‘Paracontrolled distributions and singular PDEs’, *Forum Math. Pi* **3**(e6) (2015), 75 pages.

[18] M. Gubinelli and N. Perkowski, ‘KPZ reloaded’, *Comm. Math. Phys.* **349**(1) (2017), 165–269.
Acknowledgements

The authors warmly thank the reviewers for their thorough readings of previous versions of the present work.

**Conflict of Interest:** None

References

[1] I. Bailleul and F. Bernicot, ‘Heat semigroup and singular PDEs’, *J. Funct. Anal.* **270** (2016), 3344–3452.

[2] I. Bailleul, F. Bernicot and D. Frey, ‘Space-time paraproducts for paracontrolled calculus, 3D-PAM and multiplicative Burgers equations’, *Ann. Sci. Éc. Norm. Supér.* **(4)** **51**(6) (2018), 1399–1457.

[3] I. Bailleul, A. Debussche and M. Hofmanovà, ‘Quasilinear generalised parabolic Anderson model equation’, *Stoch. Part. Diff. Eq.: Anal. Comput.* **7**(1) (2018), 40–63.

[4] I. Bailleul and M. Hoshino, ‘Paracontrolled calculus and regularity structures (I)’, *J. Math. Soc. Japan*, to appear. Preprint, 2020, arXiv:1812.07919.

[5] Y. Bruned, A. Chandra, I. Chevyrev and M. Hairer, ‘Renormalising SPDEs in regularity structures’, *J. Eur. Math. Soc.* (2020), (to appear).

[6] Y. Bruned, F. Gabriel, M. Hairer and L. Zambotti, ‘Geometric stochastic heat equations’, Preprint, 2019, arXiv:1902.02884.

[7] Y. Bruned, M. Hairer and L. Zambotti, ‘Algebraic renormalisation of regularity structures’, *Invent. Math.* **215**(3) (2019), 1039–1156.

[8] M. Caruana, T. Lévy and T. Lyons, *Differential Equations Driven by Rough Paths*, Lecture Notes in Mathematics 1908 (Springer-Verlag, Berlin Heidelberg, 2006).

[9] R. Catellier and K. Chouk, ‘Paracontrolled distributions and the 3-dimensional stochastic quantization equation’, *Ann. Probab.* **46**(5) (2018), 2621–2679.

[10] A. Chandra and M. Hairer, ‘An analytic BPHZ theorem for regularity structures’, Preprint, 2016, arXiv:1612.08138.

[11] J.-Y. Chemin, ‘Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires’, *Duke Math. J.* **56**(3) (1988), 431–469.

[12] J.-Y. Chemin, ‘Interaction contrôlée dans les équations aux dérivées partielles non linéaires’, *Bull. Soc. Math. France* **116**(3) (1988), 341–383.

[13] K. Chouk and R. Allez, ‘The continuous Anderson hamiltonian in dimension two’, Preprint, 2015, arXiv:1511.02718.

[14] K. Chouk and G. Cannizzaro, ‘Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential’, *Ann. Probab.* **46**(3) (2018), 1710–1763.

[15] A. Dahlqvist, J. Diehl and B. Driver, ‘The parabolic Anderson model on Riemann surfaces’, *Probab. Theory Related Fields* **174**(1-2) (2019), 369–444.

[16] X. T. Duong and A. McIntosh, ‘Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients’, *J. Geom. Anal.* **6**(2) (1996), 181–205.

[17] M. Gubinelli, P. Imkeller and N. Perkowski, ‘Paracontrolled distributions and singular PDEs’, *Forum Math. Pi* **3**(e6) (2015), 75 pages.

[18] M. Gubinelli and N. Perkowski, ‘KPZ reloaded’, *Comm. Math. Phys.* **349**(1) (2017), 165–269.
[19] M. Hairer, 'A theory of regularity structures', Invent. Math. 198(2) (2014), 269-504.

[20] M. Hairer, 'The motion of a random string', Preprint, 2016, arXiv:1605.02192.

[21] M. Hairer and E. Pardoux, 'A Wong-Zakai theorem for stochastic PDEs', J. Math. Soc. Japan 67(4) (2015), 1551-1604.

[22] T. Lyons, 'Differential equations driven by rough signals', Rev. Mat. Iberoam. 14(2) (1998), 215-310.

[23] N. Perkowski, 'Paracontrolled distributions and singular SPDEs', https://personal-homepages.mis.mpg.de/perkow/paracontrolled-leipzig.pdf, 2019.

[24] R. Zhu and X. Zhu, 'Three-dimensional Navier-Stokes equations driven by space-time white noise', J. Differential Equations 259(9) (2015), 4443-4506.

[25] R. Zhu and X. Zhu, 'Approximating three-dimensional Navier-Stokes equations driven by space-time white noise', Inf. Dim. Anal., Quantum Probab. Rel. Topics 20(4)1750020 (2017).

[26] R. Zhu and X. Zhu, 'Lattice approximation to the dynamical $\Phi_3^4$ model', Ann. Probab. 46(1) (2018), 397-455.
[19] M. Hairer, 'A theory of regularity structures', Invent. Math. 198(2) (2014), 269-504.

[20] M. Hairer, 'The motion of a random string', Preprint, 2016, arXiv:1605.02192.

[21] M. Hairer and E. Pardoux, 'A Wong-Zakai theorem for stochastic PDEs', J. Math. Soc. Japan 67(4) (2015), 1551-1604.

[22] T. Lyons, 'Differential equations driven by rough signals', Rev. Mat. Iberoam. 14(2) (1998), 215-310.

[23] N. Perkowski, 'Paracontrolled distributions and singular SPDEs', https://personal-homepages.mis.mpg.de/perkow/paracontrolled-leipzig.pdf, 2019.

[24] R. Zhu and X. Zhu, 'Three-dimensional Navier-Stokes equations driven by space-time white noise', J. Differential Equations 259(9) (2015), 4443-4506.

[25] R. Zhu and X. Zhu, 'Approximating three-dimensional Navier-Stokes equations driven by space-time white noise', Inf. Dim. Anal., Quantum Probab. Rel. Topics 20(4)1750020 (2017).

[26] R. Zhu and X. Zhu, 'Lattice approximation to the dynamical $\Phi_3^4$ model', Ann. Probab. 46(1) (2018), 397-455.