\tau&\in\mathsf{T}_{|\beta|}\end{array}\right\\}\stackrel{{\scriptstyle\eqref{jc28}}}{{\Rightarrow}}(Q_{1}^{\perp}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}})\mathsf{z}_{k}\tau\in\mathsf{T}_{\geq
1+(k-1)\alpha+|\beta|}.$

Finally, the contribution to the sum for $j=0$ vanishes, if
$\delta\leq\alpha$, since

$\displaystyle\left.\begin{array}[]{rl}\mathsf{z}_{k}&\in\mathsf{T}_{(k+1)\alpha}\\\
\tau&\in\mathsf{T}_{|\beta|}\end{array}\right\\}\stackrel{{\scriptstyle\eqref{jc28}}}{{\Rightarrow}}\mathsf{z}_{k}\tau\in\mathsf{T}_{\geq
k\alpha+|\beta|}.$

We now show that (for $\delta>\alpha$)

(116)
$\displaystyle\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha}|F_{\eta+\alpha,Q}-F_{Q}^{k}F_{k}F|\lesssim\tilde{M}\sum_{|\gamma|\leq|\beta|}d^{\eta+\alpha-|\gamma|}\mu^{|\gamma|-2}$

and (for $\delta>1$)

(117)
$\displaystyle\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha}|F_{\eta+\alpha,\perp}-F_{\delta^{\prime}}^{k-1}F_{\perp}F_{k}F|\lesssim\tilde{M}\sum_{|\gamma|\leq|\beta|}d^{\eta+\alpha-|\gamma|}\mu^{|\gamma|-2},$

where we recall the definition of $\delta$ and $\delta^{\prime}$ and set
$F_{k}=f(\mathbf{z}).\mathsf{z}_{k}$ as well as

$\displaystyle\begin{array}[]{rlrl}F&=f(\mathbf{z}).\tau,&\qquad
F_{Q}&=f_{\delta}(\mathbf{z}).Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}},\\\
F_{\delta^{\prime}}&=f_{\delta^{\prime}}(\mathbf{z}).Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}},&\qquad
F_{\perp}&=f_{\delta}(\mathbf{z}).Q_{1}^{\perp}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}.\end{array}$

For (116) and $|\beta|<\eta-(k-1)\alpha$, we apply the approximate morphism
property (46) with $\eta+\alpha$ playing the role of $\eta$, $J:=k+2$ and

$\displaystyle\begin{array}[]{l||l|l||l}j&\tau^{j}&\eta_{j}&\hat{\eta}_{j}\\\
\hline\cr
1,\ldots,k&Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}&\alpha&\delta=\eta-|\beta|-(k-2)\alpha\\\
k+1&\tau&|\beta|&\eta-(k-1)\alpha\\\
J&\mathsf{z}_{k}&(k+1)\alpha&\eta-|\beta|+2\alpha\end{array}$

Recall that $\mathsf{z}_{k}\in\mathsf{T}_{(k+1)\alpha}$ and
$\tau\in\mathsf{T}_{|\beta|}$, so that $|\beta|<\eta-(k-1)\alpha$ implies that
$f_{\hat{\eta}_{k+1}}.\tau^{k+1}=F$ and $f_{\hat{\eta}_{J}}.\tau^{J}=F_{k}$.
Since
$(Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}})_{\beta_{j}}=(\mathsf{z}_{k})_{\beta_{j}}=0$
unless $\beta_{x,j}=0$, (46) yields in view of (69) and $N_{0}\leq 1$

$\displaystyle|F_{\eta+\alpha,Q}-F_{Q}^{k}F_{k}F|\lesssim\sum_{\mathsf{A}_{(\eta_{j})}}d^{|\beta_{1}|+\ldots+|\beta_{k}|}\|\tau_{\beta_{k+1}}\|\|(\mathsf{z}_{k})_{\beta_{K}}\|.$

In all non-zero contributions to the right-hand side of (46) we have
$|\beta_{k+1}|=|\beta|$ and $|\beta_{J}|=(k+1)\alpha$. Hence, by definition of
$\mathsf{A}_{(\eta_{j})}$ and (33), it holds

$\displaystyle|\beta_{1}|+\ldots+|\beta_{J}|$
$\displaystyle\geq\eta+\alpha+(k+1)\alpha$
$\displaystyle\Rightarrow\quad|\beta_{1}|+\ldots+|\beta_{k}|$
$\displaystyle\geq\eta+\alpha-|\beta|,$

so that it follows by $d\leq 1$ and $\|(\mathsf{z}_{k})_{\beta_{J}}\|\leq 1$

(118)
$\displaystyle|F_{\eta+\alpha,Q}-F_{Q}^{k}F_{k}F|\lesssim(1+|\nu(\mathbf{z})|)d^{\eta+\alpha-|\beta|}\|\tau_{\beta}\|.$

Since $\tau\in\mathsf{T}_{-}$, we have $\tau_{\beta}=0$ unless
$\langle\beta\rangle\geq 1$, cf. (30), so that by $N_{0}\leq 1$, (8), (9) and
(10) it follows

(119)
$\displaystyle\begin{split}\|\tau_{\beta}\|&=N_{0}^{\langle\beta\rangle}\|\Delta\Pi_{\mathbf{z}\lambda}({\mathbf{z}^{\prime}})\|_{\mathsf{T}_{|\beta|}}=N_{0}^{\langle\beta\rangle}\|\Gamma_{{\mathbf{z}^{\prime}}\mathbf{z}}\Delta\Pi_{{\mathbf{z}^{\prime}}\lambda}({\mathbf{z}^{\prime}})\|_{\mathsf{T}_{|\beta|}}\\\
&\lesssim
N_{0}\sum_{|\gamma|\leq|\beta|}d^{|\beta|-|\gamma|}\|\Delta\Pi_{{\mathbf{z}^{\prime}}\lambda}({\mathbf{z}^{\prime}})\|_{\mathsf{T}_{|\gamma|}}\\\
&\lesssim
N_{0}\sum_{|\gamma|\leq|\beta|}d^{|\beta|-|\gamma|}\mu^{|\gamma|-2}.\end{split}$

Combining this with (118) and
$\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha}N_{0}(1+|\nu(\mathbf{z})|)\lesssim\tilde{M}$,
which is a variant of (112), we arrive at (116).

For (117) and $|\beta|<\eta-(k-2)\alpha-1$, we apply the approximate morphism
property (46) with $\eta+\alpha$ playing the role of $\eta$, $J:=k+2$ and

$\displaystyle\begin{array}[]{l||l|l||l}j&\tau^{j}&\eta_{j}&\hat{\eta}_{j}\\\
\hline\cr
1,\ldots,k-1&Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}&\alpha&\delta^{\prime}=\eta-|\beta|-(k-3)\alpha-1\\\
k&Q_{1}^{\perp}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}&1&\delta=\eta-|\beta|-(k-2)\alpha\\\
k+1&\tau&|\beta|&\eta-(k-2)\alpha-1\\\
J&\mathsf{z}_{k}&(k+1)\alpha&\eta-|\beta|+3\alpha-1\end{array}$

Here, the condition $|\beta|<\eta-(k-2)\alpha-1$ guarantees that
$f_{\hat{\eta}_{k+1}}.\tau^{k+1}=F$ and $f_{\hat{\eta}_{J}}.\tau^{J}=F_{k}$.
Again since
$(Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}})_{\beta_{j}}=(\mathsf{z}_{k})_{\beta_{j}}=0$
unless $\beta_{x,j}=0$, (46) yields in view of (69) and $N_{0}\leq 1$

$\displaystyle|F_{\eta+\alpha,\perp}-F_{\delta^{\prime}}^{k-1}F_{\perp}F_{k}F|\lesssim\sum_{\mathsf{A}_{(\eta_{j})}}|\nu(\mathbf{z})|^{\beta_{k,x}}d^{|\beta_{1}|+\ldots+|\beta_{k}|}\|\tau_{\beta_{k+1}}\|\|(\mathsf{z}_{k})_{\beta_{J}}\|,$

which by the same reasons as above leads to (117). Together, we have shown

$\displaystyle\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha}$
$\displaystyle|f_{\eta+\alpha}.(\tau^{(0)})^{k}\mathsf{z}_{k}\tau-(F_{Q}^{k}F_{k}F+kF_{\delta^{\prime}}^{k-1}F_{\perp}F_{k}F)|$
$\displaystyle\lesssim\tilde{M}\sum_{|\beta|<\eta+\alpha}d^{\eta+\alpha-|\beta|}\mu^{|\beta|-2}.$

Since $N_{0}\leq\|u\|_{0}\leq 1$ and $\tau\in\mathsf{T}_{-}$, so that
$\tau_{\beta}=0$ unless $\langle\beta\rangle\geq 1$, we have with (119)

$\displaystyle|F|$ $\displaystyle=|f(\mathbf{z}).\tau|\lesssim
N_{0}(1+1_{|\beta|>1}|\nu(\mathbf{z})|)\sum_{|\gamma|\leq|\beta|}d^{|\beta|-|\gamma|}\mu^{|\gamma|-2}$
$\displaystyle\leq\|u\|_{0}(1+1_{|\beta|>1}|\nu(\mathbf{z})|)\sum_{|\gamma|\leq|\beta|}d^{|\beta|-|\gamma|}\mu^{|\gamma|-2}.$

By $F_{k}=f.\mathsf{z}_{k}=\frac{1}{k!}a^{(k)}(u(\mathbf{z}))$ and
$\operatorname{dist}_{\mathbf{z}}\leq 1$ we learn

$\displaystyle 1+1_{|\beta|>1}|\nu(\mathbf{z})|$ $\displaystyle\leq
1+1_{|\beta|>1}\operatorname{dist}_{\mathbf{z}}^{-|\beta|}\|\nu\|_{w,1}^{\prime}$
$\displaystyle\lesssim\operatorname{dist}_{\mathbf{z}}^{-|\beta|}(1+1_{|\beta|>1}\|u\|_{w,\eta}^{\frac{1}{\eta}})\lesssim\operatorname{dist}_{\mathbf{z}}^{-|\beta|}(1+\|u\|_{w,\eta})^{\frac{\beta}{\eta}},$

so that estimating (115) is hence reduced to estimate

$\displaystyle\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha-|\beta|}|[a(u)]$
$\displaystyle-\sum_{k\alpha<\eta+\alpha-|\beta|}\frac{1}{k!}a^{(k)}(u(\mathbf{z}))\big{(}F_{Q}^{k}+kF_{\delta^{\prime}}^{k-1}F_{\perp}\big{)}|$
$\displaystyle\lesssim(1+\|u\|_{w,\eta})^{\frac{\eta+\alpha-\beta}{\eta}}d^{\eta+\alpha-|\beta|}.$

By Taylor’s formula, $\frac{1}{k!}a^{(k)}(u(\mathbf{z}))\lesssim 1$,
$\|u\|_{0}\leq 1$ and Lemma 2 we know

$\displaystyle\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha-|\beta|}$
$\displaystyle\bigg{|}[a(u)]-\sum_{k\alpha<\eta+\alpha-|\beta|}\frac{1}{k!}a^{(k)}(u(\mathbf{z}))[u]^{k}\bigg{|}$
$\displaystyle\lesssim(\operatorname{dist}_{\mathbf{z}}^{\alpha}[u]_{\alpha})^{\frac{\eta+\alpha-|\beta|}{\alpha}}d^{\eta+\alpha-|\beta|}\lesssim\|u\|_{w,\eta}^{\frac{\eta+\alpha-|\beta|}{\eta}}d^{\eta+\alpha-|\beta|}.$

Hence, it remains to estimate

$\displaystyle\operatorname{dist}_{\mathbf{z}}^{\eta+\alpha-|\beta|}$
$\displaystyle\big{|}[u]^{k}-(F_{Q}^{k}+kF_{\delta^{\prime}}^{k-1}F_{\perp})\big{|}$
$\displaystyle\lesssim(1+\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{\frac{\eta+\alpha-|\beta|}{\eta}}d^{\eta+\alpha-|\beta|}.$

For this we write
$F_{\delta}:=F_{Q}+F_{\perp}=f_{\delta}(\mathbf{z}).\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}$,
so that (51) shows $|[u]-F_{\delta}|\leq[u]_{\delta}d^{\delta}$. We also
observe that $F_{\delta^{\prime}}$ is active only for $k\geq 2$, in which case
$\eta\leq 1+\mathsf{n}^{\prime}\alpha$, $|\beta|\geq\alpha$ and (32) yield

$\displaystyle\delta^{\prime}=\eta-|\beta|-(k-3)\alpha-1\leq\mathsf{n}^{\prime}\alpha<1,$

so that
$F_{\delta^{\prime}}=f_{\delta^{\prime}}(\mathbf{z}).Q_{1}\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}=f_{\delta^{\prime}}(\mathbf{z}).\tau^{(0)}_{{\mathbf{z}^{\prime}}\mathbf{z}}$
and hence
$|[u]-F_{\delta^{\prime}}|\leq[u]_{\delta^{\prime}}d^{\delta^{\prime}}$. Thus,
we may employ (102) from Step 5.2 in the proof of Lemma 3 with

$\displaystyle\begin{array}[]{l|l|l|l|l|l|l|l|l}p&q&r&s&t&\kappa_{1}&\kappa_{2}&M_{1}&M_{2}\\\
\hline\cr
F_{\delta}&[u]&F_{Q}&F_{\delta^{\prime}}&F_{\perp}&\delta&\delta^{\prime}&[u]_{\delta}&1+[u]_{\delta^{\prime}}\end{array},$

and

$\displaystyle\begin{array}[]{l|l|l|l}q_{\alpha}&r_{\alpha}&s_{\alpha}&t_{1}\\\
\hline\cr[u]_{\alpha}&1&1&1_{\delta>1}(1+|\nu(\mathbf{z})|)\end{array},$

so that the constant

$\displaystyle[u]_{\delta}(1+[u]_{\alpha}^{k-1})+1_{\delta>1}([u]_{\delta^{\prime}}(1+[u]_{\alpha}^{k-2})+(1+[u]_{\alpha}^{k-2}))(1+|\nu|)$

is estimated by
$\operatorname{dist}_{\mathbf{z}}^{-(\eta+\alpha-|\beta|)}(1+\|u\|_{\eta}+\|\nu\|_{\eta}^{\prime})^{\frac{\eta+\alpha-|\beta|}{\eta}}$
in virtue of Lemma 2, $\operatorname{dist}_{\mathbf{z}}\leq 1$, and

$\displaystyle\delta+(k-1)\alpha=\delta^{\prime}+(k-2)\alpha+1=\eta+\alpha-|\beta|.$

This completes the proof of (106).

Step 2\. In this step, we prove for fixed $\mathbf{x}\in B_{1}(0)$ a local
splitting estimate of the form

(120)
$\displaystyle\begin{split}\operatorname{dist}_{\mathbf{x}}^{\eta}&\inf_{a_{0}\in
I,\ell_{0}}\|(\partial_{t}-a_{0}\Delta)U_{\lambda}(\mathbf{x},\cdot)-\ell_{0}\|_{B_{R}(\mathbf{x})}\\\
&\lesssim
M\sum_{|\beta|=\alpha}^{\eta}R^{\eta-|\beta|}\lambda^{|\beta|-2},\end{split}$

where $\lambda\in(0,\frac{1}{10}\operatorname{dist}_{\mathbf{x}})$,
$R\in(0,\frac{1}{2}\operatorname{dist}_{\mathbf{x}})$ and

$\displaystyle M$
$\displaystyle:=\|u\|_{0}+(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})\|u\|_{0}^{1-\frac{\alpha}{\eta}}.$

We will prove this estimate by distinguishing between small and large scales
in $\lambda$. We start by assuming
$\lambda<\operatorname{dist}_{\mathbf{x}}(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{-\frac{1}{\eta}}$.
It will be convenient to introduce one more jet $\mathbf{y}\mapsto
F^{\prime}(\mathbf{x},\mathbf{y})$ defined similarly to $\mathbf{y}\mapsto
F(\mathbf{x},\mathbf{y})$, but with $\eta$ in place of $\eta+\alpha$.
Specifically, with $\ell_{0}:=f_{\eta}(\mathbf{x}).P_{\mathbf{x}}$, where
$P_{\mathbf{x}}$ is defined in Assumption 1,

$\displaystyle F^{\prime}(\mathbf{x},\cdot)$
$\displaystyle:=a(u(\mathbf{x}))\Delta
u+f_{\eta}(\mathbf{x}).\Pi^{-}_{\mathbf{x}}-\xi$
$\displaystyle=a(u(\mathbf{x}))\Delta
U(\mathbf{x},\cdot)+\ell_{0}+f_{\eta}(\mathbf{x}).\partial_{s}\Pi_{\mathbf{x}}-\xi,$

where we have used the definition of $\Pi^{-}_{\mathbf{x}}$, cf. (5).
Therefore, we learn from (105)

$\displaystyle\partial_{s}U_{\lambda}(\mathbf{x},\cdot)$
$\displaystyle=\partial_{s}u_{\lambda}-\partial_{s}f_{\eta}(\mathbf{x}).\Pi_{\mathbf{x}\lambda}$
$\displaystyle=(EF)_{\lambda}+\xi_{\lambda}-f_{\eta}(\mathbf{x}).\partial_{s}\Pi_{\mathbf{x}\lambda}$
$\displaystyle=(EF)_{\lambda}-F_{\lambda}^{\prime}(\mathbf{x},\cdot)+a(u(\mathbf{x}))\Delta
U_{\lambda}(\mathbf{x},\cdot)+\ell_{0}.$

Hence, in order to show (120) for
$\lambda<\operatorname{dist}_{\mathbf{x}}(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{-\frac{1}{\eta}}$,
it suffices to show that
$\operatorname{dist}_{\mathbf{x}}^{\eta}|(EF)_{\lambda}(\mathbf{y})-F_{\lambda}^{\prime}(\mathbf{x},\mathbf{y})\big{|}$
is estimated by the right-hand side of (120) for every $\mathbf{y}\in
B_{R}(\mathbf{x})$. We write

$\displaystyle(EF)_{\lambda}(\mathbf{y})-F_{\lambda}^{\prime}(\mathbf{x},\mathbf{y})$
$\displaystyle=(EF)_{\lambda}(\mathbf{y})-F_{\lambda}(\mathbf{y},\mathbf{y})$
$\displaystyle+F_{\lambda}(\mathbf{y},\mathbf{y})-F_{\lambda}^{\prime}(\mathbf{y},\mathbf{y})$
$\displaystyle+F_{\lambda}^{\prime}(\mathbf{y},\mathbf{y})-F_{\lambda}^{\prime}(\mathbf{x},\mathbf{y}),$

and estimate each of the contributions. For the third contribution, arguing
identically as in Step 5.3, but with $\mathbf{x}$ playing the role of
$\mathbf{z}$, $\mathbf{y}$ playing the role of $\mathbf{z}^{\prime}$,
$\lambda$ playing the role of $\mu$, $F^{\prime}$ playing the role of $F$ and
$\eta$ playing the role of $\eta+\alpha$, we obtain

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}$
$\displaystyle\big{|}F_{\lambda}^{\prime}(\mathbf{x},\mathbf{y})-F_{\lambda}^{\prime}(\mathbf{y},\mathbf{y})\big{|}\lesssim
M\sum_{|\beta|=\alpha}^{\eta-\alpha}R^{\eta-|\beta|}\lambda^{|\beta|-2},$

which is contained in the right-hand side of (120).

Next we observe that by Step 5.3, the hypothesis of Proposition 2 is
satisfied: Take $\mu\in(0,\lambda)$, $\mathbf{z},\mathbf{z}^{\prime}\in
B_{\lambda-\mu}(\mathbf{y})$. Then it holds $\mathbf{z}^{\prime}\in
B_{\frac{1}{2}\operatorname{dist}\mathbf{z}}(\mathbf{z})$, since
$d(\mathbf{z}^{\prime},\mathbf{z})\leq
2(\lambda-\mu)\leq\frac{1}{5}\operatorname{dist}_{\mathbf{x}}$ and

$\displaystyle\operatorname{dist}_{\mathbf{z}}$
$\displaystyle=1-d(\mathbf{z},0)\geq
1-(d(\mathbf{x},0)+d(\mathbf{y},\mathbf{x})+d(\mathbf{z},\mathbf{y}))$
$\displaystyle\geq\operatorname{dist}_{\mathbf{x}}-\frac{1}{2}\operatorname{dist}_{\mathbf{x}}-(\lambda-\mu)\geq\frac{2}{5}\operatorname{dist}_{\mathbf{x}},$

where we have used $\mathbf{y}\in
B_{\frac{1}{2}\operatorname{dist}_{\mathbf{x}}}(\mathbf{x})$ and
$\lambda\in(0,\frac{1}{10}\operatorname{dist}_{\mathbf{x}})$. Moreover,
$\mu<\lambda<\frac{1}{10}\operatorname{dist}_{\mathbf{x}}\leq\frac{1}{4}\operatorname{dist}_{\mathbf{z}}$.
Thus, (27) is fulfilled with
$C:=\operatorname{dist}_{\mathbf{z}}^{-(\eta+\alpha)}\tilde{M}\lesssim\operatorname{dist}_{\mathbf{x}}^{-(\eta+\alpha)}\tilde{M}$,
where $\eta+\alpha-2>0$ plays the role of $\kappa$. Hence the output (28)
implies

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}\big{|}(EF)_{\lambda}(\mathbf{y})-F_{\lambda}(\mathbf{y},\mathbf{y})\big{|}$
$\displaystyle\lesssim\operatorname{dist}_{\mathbf{x}}^{-\alpha}\tilde{M}\lambda^{\eta+\alpha-2},$

which is contained in the right-hand side of (120), since
$\lambda\leq\operatorname{dist}_{\mathbf{x}}\leq 1$ and
$\lambda<\operatorname{dist}_{\mathbf{x}}(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{-\frac{1}{\eta}}$
imply

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{-\alpha}\tilde{M}\lambda^{\alpha}\leq\|u\|_{0}+(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{\frac{\eta+\alpha}{\eta}}\|u\|_{0}^{1-\frac{\alpha}{\eta}}\left(\frac{\lambda}{\operatorname{dist}_{\mathbf{x}}}\right)^{\alpha}\leq
M.$

Furthermore, using (112), we have

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}\big{|}F_{\lambda}(\mathbf{y},\mathbf{y})-F_{\lambda}^{\prime}(\mathbf{y},\mathbf{y})\big{|}$
$\displaystyle=\operatorname{dist}_{\mathbf{x}}^{\eta}\big{|}(f_{\eta+\alpha}-f_{\eta})(\mathbf{y}).\Pi^{-}_{\mathbf{y}}(\mathbf{y})|$
$\displaystyle\lesssim\operatorname{dist}_{\mathbf{x}}^{\eta}N_{0}(1+|\nu(\mathbf{y})|)\lambda^{\eta-2}\lesssim
M\lambda^{\eta-2}.$

Together, this shows (120) for
$\lambda<\operatorname{dist}_{\mathbf{x}}(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{-\frac{1}{\eta}}$.

If
$\lambda\geq\operatorname{dist}_{\mathbf{x}}(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{-\frac{1}{\eta}}$,
we first remark that $|{\beta_{x}}|=1$ and $|\beta|\notin\mathbb{N}_{0}$ imply
$|\beta|\geq 1+\alpha$, so that
$R,\lambda\leq\operatorname{dist}_{\mathbf{x}}\leq 1$ shows for such $\beta$
in virtue of Young’s inequality

$\displaystyle\sum_{|\gamma|\leq|\beta|}R^{|\beta|-|\gamma|}\lambda^{|\gamma|-2}\leq
R^{|\beta|-\alpha}\lambda^{\alpha-2}+\lambda^{|\beta|-2}\leq
R\lambda^{\alpha-2}+\lambda^{\alpha-1}.$

Hence, using also Lemma 2, we obtain

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}$
$\displaystyle\sum_{\mathbb{N}_{0}\not\ni|\beta|<\eta}|\nu(\mathbf{x})|^{|{\beta_{x}}|}\sum_{|\gamma|\leq|\beta|}R^{|\beta|-|\gamma|}\lambda^{|\gamma|-2}=\left(\frac{\operatorname{dist}_{\mathbf{x}}}{\lambda}\right)^{\eta-\alpha}\lambda^{\eta-2}$
$\displaystyle\quad+1_{\eta>1+\alpha}\left(\frac{\operatorname{dist}_{\mathbf{x}}}{\lambda}\right)^{\eta-\alpha-1}(\operatorname{dist}_{\mathbf{x}}|\nu(\mathbf{x})|)(R\lambda^{\eta-3}+\lambda^{\eta-2})$
$\displaystyle\lesssim(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{1-\frac{\alpha}{\eta}}\sum_{|\beta|\leq\eta}R^{\eta-|\beta|}\lambda^{|\beta|-2},$

where we have used that $\eta-\alpha>0$ and the second contribution is only
present if $\eta-\alpha-1>0$. Moreover, Lemma 2 and
$\lambda\geq\operatorname{dist}_{\mathbf{x}}(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})^{-\frac{1}{\eta}}$
yield

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}[u]_{\alpha}\lambda^{\alpha-2}$
$\displaystyle=(\operatorname{dist}_{\mathbf{x}}^{\alpha}[u]_{\alpha})\left(\frac{\lambda}{\operatorname{dist}_{\mathbf{x}}}\right)^{\alpha-\eta}\lambda^{\eta-2}$
$\displaystyle\lesssim(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})\|u\|_{0}^{1-\frac{\alpha}{\eta}}\lambda^{\eta-2}.$

Hence, if we observe

$\displaystyle(\partial_{s}-\Delta)U_{\lambda}(\mathbf{x},\cdot)$
$\displaystyle=(\partial_{s}-\Delta)u_{\lambda}+f_{\eta}(\mathbf{x}).(\partial_{s}-\Delta)\Pi_{\mathbf{x}\lambda}$
$\displaystyle=(\partial_{s}-\Delta)u_{\lambda}+f_{\eta}(\mathbf{x}).\Gamma_{\mathbf{y}\mathbf{x}}(\partial_{s}-\Delta)\Pi_{\mathbf{y}\lambda},$

estimate (82), (8) and (10) imply together with the trivial estimate
$N_{0}\leq
N_{0}^{1-\frac{\alpha}{\eta}}\leq\|u\|_{0}^{1-\frac{\alpha}{\eta}}$, cf. (52),

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}$
$\displaystyle\inf_{a_{0}\in
I,\ell_{0}}\|(\partial_{t}-a_{0}\Delta)U_{\lambda}(\mathbf{x},\cdot)-\ell_{0}\|_{B_{R}(\mathbf{x})}$
$\displaystyle\lesssim\operatorname{dist}_{\mathbf{x}}^{\eta}[u]_{\alpha}\lambda^{\alpha-2}+\operatorname{dist}_{\mathbf{x}}^{\eta}N_{0}\sum_{|\beta|<\eta}|\nu(\mathbf{x})|^{{\beta_{x}}}\sum_{|\gamma|\leq|\beta|}R^{|\beta|-|\gamma|}\lambda^{|\gamma|-2}$
$\displaystyle\lesssim(\|u\|_{\eta}+\|\nu\|_{\eta}^{\prime})\|u\|_{0}^{1-\frac{\alpha}{\eta}}\lambda^{\eta-2},$

which is contained in the right-hand side of (120).

Combining the estimates for small and large scales $\lambda$, we are led to
(120).

Step 3\. In this step, we establish the three-point continuity condition: for
all $\mathbf{x}\in B_{1}(0)$, $\mathbf{y}\in
B_{\frac{1}{2}\operatorname{dist}_{\mathbf{x}}}(\mathbf{x})$ and
$\mathbf{z}\in B_{\frac{1}{2}\operatorname{dist}_{\mathbf{x}}}(\mathbf{y})$ it
holds

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}$
$\displaystyle\big{|}U(\mathbf{x},\mathbf{z})-U(\mathbf{x},\mathbf{y})-U(\mathbf{y},\mathbf{z})-\gamma(\mathbf{x},\mathbf{y})\cdot(z-y)\big{|}$
(121) $\displaystyle\lesssim
M\sum_{|\beta|=\alpha}^{\eta-\alpha}d^{|\beta|}(\mathbf{y},\mathbf{x})d^{\eta-|\beta|}(\mathbf{z},\mathbf{y}),$

where $M>0$ is as in Step 5.3 and $\gamma(\mathbf{x},\mathbf{y})$ is defined
by

(122)
$\gamma(\mathbf{x},\mathbf{y}):=(f_{\eta}(\mathbf{x}).\Gamma_{\mathbf{y}\mathbf{x}}-f_{\eta}(\mathbf{y}).{\rm
id})\mathsf{z}_{x}.$

To establish (121), use (9) to write
$\Pi_{\mathbf{x}}(\mathbf{z})-\Pi_{\mathbf{x}}(\mathbf{y})=\Gamma_{\mathbf{y}\mathbf{x}}\Pi_{\mathbf{y}}(\mathbf{z})$,
so that

$\displaystyle
U(\mathbf{x},\mathbf{z})-U(\mathbf{x},\mathbf{y})-U(\mathbf{y},\mathbf{z})-\gamma(\mathbf{x},\mathbf{y})\cdot(z-y)$
$\displaystyle=f_{\eta}(\mathbf{x}).(\Pi_{\mathbf{x}}(\mathbf{z})-\Pi_{\mathbf{x}}(\mathbf{y}))-f_{\eta}(\mathbf{y}).\Pi_{\mathbf{y}}(\mathbf{z})-\gamma(\mathbf{x},\mathbf{y})\cdot(z-y)$
$\displaystyle=f_{\eta}(\mathbf{x}).\Gamma_{\mathbf{y}\mathbf{x}}\Pi_{\mathbf{y}}(\mathbf{z})-f_{\eta}(\mathbf{y}).\Pi_{\mathbf{y}}(\mathbf{z})-\gamma(\mathbf{x},\mathbf{y})\cdot(z-y)$
$\displaystyle=(f_{\eta}(\mathbf{x}).\Gamma_{\mathbf{y}\mathbf{x}}-f_{\eta}(\mathbf{y}).{\rm
id})({\rm id}-\mathsf{P})\Pi_{\mathbf{y}}(\mathbf{z}),$

where we have used
$\mathsf{P}\Pi_{\mathbf{y}}(\mathbf{z}):=\mathsf{z}_{x}\cdot(z-y)$, cf. (6).
Hence, by the definition of
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}f_{\eta}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}$,
we find

$\displaystyle\operatorname{dist}_{\mathbf{x}}^{\eta}$
$\displaystyle\big{|}U(\mathbf{x},\mathbf{z})-U(\mathbf{x},\mathbf{y})-U(\mathbf{y},\mathbf{z})-\gamma(\mathbf{x},\mathbf{y})\cdot(z-y)\big{|}$
$\displaystyle\lesssim\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}f_{\eta}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}\sum_{|\beta|=\alpha}^{\eta-\alpha}d^{\eta-|\beta|}(\mathbf{y},\mathbf{x})\|({\rm
id}-\mathsf{P})\tau^{(0)}_{\mathbf{z}\mathbf{y}}\|_{\mathsf{T}_{|\beta|}}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{model1}}}}{{\lesssim}}\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}f_{\eta}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}\sum_{|\beta|=\alpha}^{\eta-\alpha}d^{\eta-|\beta|}(\mathbf{y},\mathbf{x})d^{|\beta|}(\mathbf{z},\mathbf{y}),$

which yields (121) in virtue of Lemma 3 and $N_{0}\leq\|u\|_{0}\leq 1$.

Step 4\. We are now in the position to complete the proof of Theorem 1.
Indeed, we may apply Proposition 1 to the jet $\mathbf{y}\mapsto
U(\mathbf{x},\mathbf{y})$, as we have verified (24) in Step 5.3 and (25) in
Step 5.3, where the set $\mathsf{A}_{+}\cap(0,\eta]$ plays the role of
$\mathsf{A}$.

Observe that for $\mathbf{x}\in B_{1}(0)$ and $\mathbf{y}\in
B_{\operatorname{dist}_{\mathbf{x}}}(\mathbf{x})$ we have

$\displaystyle
u(\mathbf{y})-u(\mathbf{x})-f_{\eta}(\mathbf{x}).\tau^{(0)}_{\mathbf{y}\mathbf{x}}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{jc15}}}}{{=}}u(\mathbf{y})-u(\mathbf{x})-f_{\eta}(\mathbf{x}).\Pi_{\mathbf{x}}(\mathbf{y})=U(\mathbf{x},\mathbf{y})$

and thus by (69) and $d\leq\operatorname{dist}_{\mathbf{x}}\leq 1$ and the
definition of $\|U\|_{0}$ in (22)

$\displaystyle\|U\|_{0}\lesssim\|u\|_{B_{1}(0)}+N_{0}+\sup_{\mathbf{x}\in
B_{1}(0)}\operatorname{dist}_{\mathbf{x}}|\nu(\mathbf{x})|\lesssim\|u\|_{0}+\|\nu\|_{w,1}^{\prime}.$

Moreover, the definition of $\gamma(\mathbf{x},\mathbf{y})$ in Step 5.3, (43)
and $f_{\eta}.\mathsf{z}_{x}=\nu$, cf. (45), imply

$\displaystyle\nu(\mathbf{y})-\nu(\mathbf{x})-f_{\eta}(\mathbf{x}).\tau^{(1)}_{\mathbf{y}\mathbf{x}}=-\gamma(\mathbf{x},\mathbf{y}).$

Therefore, as a result of Steps 5.3 and 5.3, the output (26) implies

$\displaystyle[u]_{w,\eta}+[\nu]_{w,\eta}^{\prime}$ $\displaystyle\lesssim
M+\|U\|_{0}$
$\displaystyle\lesssim\|u\|_{0}+(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})\|u\|_{0}^{1-\frac{\alpha}{\eta}}+\|\nu\|_{w,1}^{\prime},$

and thus via
$\|\nu\|_{w,1}^{\prime}\leq\|u\|_{w,\eta}^{\frac{1}{\eta}}\|u\|_{0}^{1-\frac{1}{\eta}}\lesssim\varepsilon\|u\|_{w,\eta}+C_{\varepsilon}\|u\|_{0}$
for every $\varepsilon>0$

$\displaystyle\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime}$
$\displaystyle\lesssim(1+C_{\varepsilon})\|u\|_{0}+(\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime})\|u\|_{0}^{1-\frac{\alpha}{\eta}}+\varepsilon\|u\|_{w,\eta}.$

Choosing $\|u\|_{0}$ and $\varepsilon$ sufficiently small, we may absorb to
arrive at

(123) $\displaystyle\|u\|_{w,\eta}+\|\nu\|_{w,\eta}^{\prime}$
$\displaystyle\lesssim\|u\|_{0}.$

In particular,

(124)
$\displaystyle[u]_{w,\alpha}\lesssim\|u\|_{w,\eta}^{\frac{\alpha}{\eta}}\|u\|_{0}^{1-\frac{\alpha}{\eta}}\lesssim\|u\|_{0}.$

If $\mathbf{x},\mathbf{y}\in B_{\frac{1}{3}}(0)$, then
$d(\mathbf{y},\mathbf{x})\leq\frac{2}{3}\leq\operatorname{dist}_{\mathbf{x}}$,
so that $\mathbf{y}\in B_{\operatorname{dist}_{\mathbf{x}}}(\mathbf{x})$ and
thus (124) implies

$\displaystyle|u(\mathbf{y})-u(\mathbf{x})|$
$\displaystyle\lesssim\operatorname{dist}_{\mathbf{x}}^{\alpha}|u(\mathbf{y})-u(\mathbf{x})|$
$\displaystyle\leq[u]_{w,\alpha}d^{\alpha}(\mathbf{y},\mathbf{x})\lesssim\|u\|_{0}d^{\alpha}(\mathbf{y},\mathbf{x}),$

which is the assertion. By the same argument, using (123) instead of (124), we
obtain (16). ∎

##  6\. Acknowledgements

The third author is very grateful to Joscha Diehl for many useful
conversations on rough paths and regularity structures during his tenure at
the MPI in Leipzig. He also thanks Alexandra Neamţu and Benjamin Seeger for
pointers to the literature on quasi-linear SPDE with colored noise.

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