1+\log_{2}(K)$ is bounded by $K$. We look at
$\operatorname{dist}_{Q}(\mu_{p_{0},t},\mu_{p_{0},t_{k}})$, but since we are
dealing with two flat measures that intersects $B_{Q^{\prime}}$, Lemma A.5 in
[Fen1] shows that

(5.39)
$\operatorname{dist}_{Q}(\mu_{p_{0},t},\mu_{p_{0},t_{k}})\lesssim\operatorname{dist}_{Q^{\prime}}(\mu_{p_{0},t},\mu_{p_{0},t_{k}})$

and then Lemma 5.8 and the fact that $\ell(Q^{\prime})\approx_{K}\ell(Q)$
entail that

(5.40)
$\operatorname{dist}_{Q}(\mu_{p_{0},t},\mu_{p_{0},t_{k}})\lesssim\operatorname{dist}_{Q^{\prime}}(\mu_{p_{0},t},\sigma)+\operatorname{dist}_{Q^{\prime}}(\sigma,\mu_{p_{0},t_{k}})\lesssim\alpha_{\sigma}(Q^{\prime})\leq
C_{K}\alpha_{\sigma}(Q).$

A similar reasoning gives that

(5.41) $\operatorname{dist}_{Q}(\mu_{p_{0},t_{j}},\mu_{p_{0},t_{j+1}})\lesssim
C_{K}\alpha_{\sigma}(Q)$

whenever $0\leq j\leq k-1$. The combination of (5.38), (5.40), and (5.41)
shows the claim (5.37) and thus (5.36).

In the general case, we pick the smallest ancestor $Q^{*}$ of $Q$ such that
$p\in\Pi(\frac{3}{2}B_{Q^{*}})$ and $|t|\leq\ell(Q^{*})/2$, and we apply
(5.36) to get

$\operatorname{dist}_{Q}(\sigma,\mu_{p,t})\leq C_{K}\alpha_{\sigma}(Q^{*}).$

The lemma follows then by simply observing that
$\alpha_{\sigma}(Q^{*})\lesssim\alpha_{\sigma,\beta}(Q)$. $\square$

We need the constant

(5.42) $c_{\beta}:=\int_{\mathbb{R}^{n-1}}(1+|p|^{2})^{-\frac{d+\beta}{2}}dy$

and the unit vector $N_{p,t}$ defined as the vector

(5.43) $N_{p,t}(X):=[\nabla\operatorname{dist}(.,\Lambda(p,t))](X)$

which is of course constant on the two connected components of
$\mathbb{R}^{n}\setminus\Lambda(p,t)$. We are now ready to compare $D_{\beta}$
with the distance to $\Lambda(p,t)$.

######  Lemma 5.44.

Let $Q\in\mathcal{S}$, $X\in W_{\Omega}(Q)$, $p\in\Pi(2^{5}Q)$ and
$2^{-5}\ell(Q)\leq|t|\leq 2^{5}\ell(Q)$. We have

(5.45)
$|D^{-\beta}_{\beta}(X)-c_{\beta}\lambda(p,t)\operatorname{dist}(X,\Lambda(p,t))^{-\beta}|\leq
C\ell(Q)^{-\beta}\alpha_{\sigma,\beta}(Q).$

and

(5.46) $|\nabla[D^{-\beta}_{\beta}](X)+\beta
c_{\beta}\lambda(p,t)\operatorname{dist}(X,\Lambda(p,t))^{-\beta-1}N_{p,t}(X)|\\\
\leq C\ell(Q)^{-\beta-1}\alpha_{\sigma,\beta+1}(Q),$

where the constant $C>0$ depends only $C_{\sigma}$ and $\beta$.

Proof: Denote $r=\left|t\right|$, and $d=n-1$. By the definition of
$W_{\Omega}(Q)$, $\operatorname{dist}(X,\partial\Omega)>\ell(Q)/2$ and $X\in
2B_{Q}$. We show that in addition,

(5.47) $X\in B(\mathfrak{b}^{t}(p),2^{6}\ell(Q)),$

and

(5.48)
$\operatorname{dist}(X,\Lambda(p,t)\cup\partial\Omega)>\frac{\ell(Q)}{20}.$

Since $X\in 2B_{Q}$, $\left|\Pi(X)-p\right|\leq(2^{5}+2)\ell(Q)$. Then
$\left|\mathfrak{b}(\Pi(X))-\mathfrak{b}(p)\right|\leq(1+2\epsilon_{0})(2^{5}+2)\ell(Q)$
because $\mathfrak{b}$ is the graph of a $2\epsilon_{0}$-Lipschitz function.
Write

$\left|X-\mathfrak{b}^{t}(p)\right|\leq\left|X-\mathfrak{b}(\Pi(X))\right|+\left|\mathfrak{b}(\Pi(X))-\mathfrak{b}(p)\right|+\left|\mathfrak{b}(p)-\mathfrak{b}^{t}(p)\right|,$

then use (4.14) and (5.18) to get

$\left|X-\mathfrak{b}^{t}(p)\right|\leq(1+2\epsilon_{0})\left(\delta(X)+(2^{5}+2)\ell(Q)\right)+2\epsilon_{0}r\leq
2^{6}\ell(Q),$

and thus (5.47) follows. To see (5.48), we only need to show that
$\operatorname{dist}(X,\Lambda(p,t))>\frac{\ell(Q)}{20}$. Notice that $(\nabla
b^{t}(p),-1)$ is a normal vector of the plane $\Lambda(p,t)$, and that
$\mathfrak{b}^{t}(p)\in\Lambda(p,t)$. So

(5.49)
$\operatorname{dist}(X,\Lambda(p,t))=\frac{\left|\left(X-\mathfrak{b}^{t}(p)\right)\cdot\left(\nabla
b^{t}(p),-1\right)\right|}{\left|(\nabla
b^{t}(p),1)\right|}=\frac{\left|(\Pi(X)-p)\cdot\nabla
b^{t}(p)+\left(b^{t}(p)-\Pi^{\bot}(X)\right)\right|}{\sqrt{\left|\nabla
b^{t}(p)\right|^{2}+1}}\\\
\geq\frac{1}{2}\left(\left|\Pi^{\bot}(X)-b^{t}(p)\right|-\left|(\Pi(X)-p)\cdot\nabla
b^{t}(p)\right|\right)\geq\frac{1}{2}\left|\Pi^{\bot}(X)-b^{t}(p)\right|-C\,2^{5}\ell(Q)\epsilon_{0},$

by $\left\|\nabla b^{t}\right\|_{\infty}\leq C\epsilon_{0}$ (see (6.2)). We
have that $\left|b(\Pi(X))-b(p)\right|\leq
2\epsilon_{0}\left|\Pi(X)-p\right|\leq 2^{6}\epsilon_{0}$, and that
$\left|\Pi^{\bot}(X)-b(\Pi(X))\right|\geq\operatorname{dist}(X,\Gamma_{S})\geq\frac{\delta(X)}{1+3\epsilon_{0}}\geq\frac{\ell(Q)}{2(1+3\epsilon_{0})}$
by (4.15). So

$\left|\Pi^{\bot}(X)-b^{t}(p)\right|\geq\left|\Pi^{\bot}(X)-b(\Pi(X))\right|-\left|b(\Pi(X))-b(p)\right|-\left|b^{t}(p)-b(p)\right|\geq\frac{\ell(Q)}{5}$

by (5.18). Then $\operatorname{dist}(X,\Lambda(p,t))>\frac{\ell(Q)}{20}$
follows from this and (5.49).

Now we prove (5.45). We intend to cut the integral
$D_{\beta}^{-\beta}=\int_{\partial\Omega}\left|X-y\right|^{-d-\beta}d\sigma(y)$
into pieces. So we introduce a cut-off function $\theta_{0}\in
C_{c}^{\infty}(B(0,r/2))$, which is radial,
${\mathds{1}}_{B(0,r/4)}\leq\theta_{0}\leq{\mathds{1}}_{B(0,r/2)}$, and
$\left|\nabla\theta_{0}\right|\leq 2r$. Then we set
$\theta_{k}(y):=\theta_{0}(2^{-k}y)-\theta_{0}(-2^{-k+1}y)$ for $k\geq 1$ and
$y\in\mathbb{R}^{n}$, and define
$\widetilde{\theta}_{k}(y)=\theta_{k}(y-\mathfrak{b}^{t}(p))$ for
$k\in\mathbb{N}$. Denote $B_{k}=B(\mathfrak{b}^{t}(p),2^{k-1}r)$. We have that
$\operatorname{supp}\widetilde{\theta}_{0}\subset B_{0}$,
$\operatorname{supp}\widetilde{\theta}_{k}\subset B_{k}\setminus B_{k-2}$ for
$k\geq 1$, and that

$\sum_{k\in\mathbb{N}}\widetilde{\theta}_{k}=1.$

Now we can write

$D_{\beta}(X)^{-\beta}=\sum_{k\in\mathbb{N}}\int_{\partial\Omega}\left|X-y\right|^{-d-\beta}\widetilde{\theta}_{k}(y)d\sigma(y)=:\sum_{k\in\mathbb{N}}\int_{\partial\Omega}f_{k}(y)d\sigma(y),$

with $f_{k}(y)=\left|X-y\right|^{-d-\beta}\widetilde{\theta}_{k}(y)$. We
intend to compare $\int f_{k}(y)d\sigma(y)$ and $\int f_{k}(y)d\mu_{p,t}(y)$.
Both integrals are well-defined because of (5.48). Observe that

$\sum_{k\in\mathbb{N}}\int
f_{k}(y)d\mu_{p,t}(y)=\lambda(p,t)\int_{\Lambda(p,t)}\left|X-y\right|^{-d-\beta}d\mu_{\Lambda(p,t)}(y)\\\
=\lambda(p,t)\int_{\mathbb{R}^{d}}\left(\operatorname{dist}(X,\Lambda(p,t))^{2}+\left|y\right|^{2}\right)^{-(d+\beta)/2}dy=\lambda(p,t)c_{\beta}\operatorname{dist}(X,\Lambda(p,t))^{-\beta}$

by a change of variables. So

(5.50)
$D^{-\beta}_{\beta}(X)-c_{\beta}\lambda(p,t)\operatorname{dist}(X,\Lambda(p,t))^{-\beta}=\sum_{k\in\mathbb{N}}\int
f_{k}\,(d\sigma-d\mu_{p,t}).$

We are interested in the Lipschitz properties of $f_{k}$ because we intend to
use Wasserstein distances. We claim that

(5.51) $\left|X-y\right|\geq c2^{k}r\quad\text{when
}y\in\partial\Omega\cup\Lambda(p,t)\text{ is such that
}\widetilde{\theta}_{k}(y)\neq 0,$

where $c=10^{-1}2^{-21}$. In fact, by (5.47) and the support properties of
$\widetilde{\theta}_{k}$, if $k\geq 15$, then

$\left|X-y\right|\geq 2^{k-3}r-2^{6}\ell(Q)\geq(2^{k-3}-2^{11})r\geq
2^{k-4}r\quad\text{for }y\in\operatorname{supp}\widetilde{\theta}_{k}.$

If $0\leq k<15$, then by (5.48), for $y\in\partial\Omega\cup\Lambda(p,t)$,

$\left|X-y\right|\geq\operatorname{dist}(X,\partial\Omega\cup\Lambda(p,t))\geq\frac{\ell(Q)}{20}\geq\frac{2^{-6}r}{10}\geq\frac{2^{-21}}{10}2^{k}r.$

So (5.51) is justified. But $f_{k}$ is not a Lipschitz function in
$\mathbb{R}^{n}$ because $y$ can get arbitrary close to $X$ when $k$ is small.
Set

$\widetilde{f}_{k}(y):=\max\left\\{\left|X-y\right|,c\,2^{k}r\right\\}^{-d-\beta}\widetilde{\theta}_{k}(y).$

Then by (5.51), $\widetilde{f}_{k}(y)=f_{k}(y)$ for
$y\in\partial\Omega\cup\Lambda(p,t)$, and therefore,

(5.52) $\int
f_{k}\,(d\sigma-d\mu_{p,t})=\int\widetilde{f}_{k}\,(d\sigma-d\mu_{p,t}).$

The good thing about $\widetilde{f}_{k}$ is that it is Lipschitz. A direct
computation shows that $\left\|\widetilde{f}_{k}\right\|_{\infty}\leq
C\left(2^{k}r\right)^{-d-\beta}$, and
$\left\|\nabla\widetilde{f}_{k}\right\|_{\infty}\leq C(2^{k}r)^{-d-\beta-1}$.
Moreover, $\widetilde{f}_{k}$ is supported on
$B(\mathfrak{b}^{t}(p),2^{k-1}r)$, which is contained in
$B(x_{Q^{(k)}},10^{3}\ell(Q^{(k)}))$. To see this, one can use (5.18), (5.19),
and $\left|x_{Q}-x_{Q^{(k)}}\right|\leq 2^{k-1}\ell(Q)$ to get that

$\left|\mathfrak{b}^{t}(p)-x_{Q^{(k)}}\right|\leq\left(2\epsilon_{0}+2^{k-1}\right)r+10\,2^{5}\ell(Q)+2^{k-1}\ell(Q)\leq
2^{5}(2^{k}+11)\ell(Q)\leq 10^{3}2^{k}\ell(Q).$

Write

$\int\widetilde{f}_{k}\,(d\sigma-d\mu_{p,t})=\int\widetilde{f}_{k}\,(d\sigma-d\mu_{Q^{(k)}})+\int\widetilde{f}_{k}\,(d\mu_{Q}-d\mu_{p,t})+\sum_{j=1}^{k}\int\widetilde{f}_{k}\,(d\mu_{Q^{(j)}}-d\mu_{Q^{(j-1)}})\\\
=:I+II+\sum_{j=1}^{k}III_{j}.$

By the definition (3.11) of $\mu_{Q^{(k)}}$ and properties of
$\widetilde{f}_{k}$, $\left|I\right|\leq
C\left(2^{k}r\right)^{-\beta}\alpha_{\sigma}(Q,k)$. We then have
$|II|\leq\left(2^{k}r\right)^{-\beta}\operatorname{dist}_{Q^{(k)}}(\mu_{Q},\mu_{p,t})$,
but because we are looking at the Wasserstein distance between two flat
measures whose supports intersect $10B_{Q}$, Lemma A.5 in [Fen1] shows that

$\operatorname{dist}_{Q^{(k)}}(\mu_{Q},\mu_{p,t})\lesssim\operatorname{dist}_{Q}(\mu_{Q},\mu_{p,t})$

and thus

$|II|\lesssim\left(2^{k}r\right)^{-\beta}\operatorname{dist}_{Q}(\mu_{Q},\mu_{p,t})\leq\left(2^{k}r\right)^{-\beta}\Big{(}\operatorname{dist}_{Q}(\mu_{Q},\sigma)+\operatorname{dist}_{Q}(\sigma,\mu_{p,t})\Big{)}\lesssim\left(2^{k}r\right)^{-\beta}\alpha_{\sigma,\beta}(Q)$

by Lemma 5.34. The terms $III_{j}$ can be bounded by a Wasserstein distance
between planes, and similarly to $II$, we get

$|III_{j}|\lesssim\left(2^{k}r\right)^{-\beta}\operatorname{dist}_{Q^{(k)}}(\mu_{Q^{(j)}},\mu_{Q^{(j-1)}})\lesssim\left(2^{k}r\right)^{-\beta}\operatorname{dist}_{Q^{(j)}}(\mu_{Q^{(j)}},\mu_{Q^{(j-1)}})\lesssim\left(2^{k}r\right)^{-\beta}\alpha_{\sigma}(Q,j).$

Altogether, we obtain that

$\left|\int\widetilde{f}_{k}\,(d\sigma-d\mu_{p,t})\right|\leq
C\left(2^{k}r\right)^{-\beta}\left(\alpha_{\sigma,\beta}(Q)+\sum_{j=0}^{k}\alpha_{\sigma}(Q,j)\right).$

Then by (5.52) and (5.50),

$\left|D^{-\beta}_{\beta}(X)-c_{\beta}\lambda(p,t)\operatorname{dist}(X,\Lambda(p,t))^{-\beta}\right|\leq
C\sum_{k\in\mathbb{N}}\left(2^{k}r\right)^{-\beta}\left(\alpha_{\sigma,\beta}(Q)+\sum_{j=0}^{k}\alpha_{\sigma}(Q,j)\right)\\\
\leq C\ell(Q)^{-\beta}\alpha_{\sigma,\beta}(Q),$

which is (5.45).

We claim that (5.46) can be established similarly to (5.45) as long as one
expresses the left-hand side of (5.46) appropriately. A direct computation
shows that

$\nabla(D_{\beta}^{-\beta})(X)=-(d+\beta)\int\left|X-y\right|^{-d-\beta-2}(X-y)d\sigma(y).$

On the other hand,

$\int\left|X-y\right|^{-d-\beta-2}(X-y)d\mu_{\Lambda(p,t)}(y)=N_{p,t}(X)\int\left|X-y\right|^{-d-\beta-2}(X-y)\cdot
N_{p,t}(X)d\mu_{\Lambda(p,t)}(y)\\\
=N_{p,t}(X)\int\left|X-y\right|^{-d-\beta-2}\operatorname{dist}(X,\Lambda(p,t))d\mu_{\Lambda(p,t)}(y)=c_{\beta+2}\operatorname{dist}(X,\Lambda(p,t))^{-\beta-1}N_{p,t}(X).$

By [Fen1] (3.30), $(\beta+d)c_{\beta+2}=\beta c_{\beta}$ for all $\beta>0$.
Hence

$\nabla[D^{-\beta}_{\beta}](X)+\beta
c_{\beta}\lambda(p,t)\operatorname{dist}(X,\Lambda(p,t))^{-\beta-1}N_{p,t}(X)\\\
=-(d+\beta)\int\left|X-y\right|^{-d-\beta-2}(X-y)\left(d\sigma(y)-d\mu_{p,t}(y)\right).$

Now we set $f_{k}^{\prime}(y)=\left|X-y\right|^{-d-\beta-2}(X-y)$. Using
(5.47) and (5.48), we can see that $f_{k}^{\prime}$ is Lipschitz on
$\partial\Omega\cup\Lambda(p,t)$. Then we can play with measures as before to
obtain (5.46). $\square$

######  Corollary 5.53.

Let $Q\in\mathcal{S}$, $X\in W_{\Omega}(Q)$, $p\in\Pi(2^{5}Q)$ and
$2^{-5}\ell(Q)\leq|t|\leq 2^{5}\ell(Q)$. We have

(5.54) $\left|\dfrac{\nabla
D_{\beta}(X)}{D_{\beta}(X)}-\dfrac{N_{p,t}(X)}{\operatorname{dist}(X,\Lambda(p,t))}\right|\leq
C\ell(Q)^{-1}\alpha_{\sigma,\beta}(Q).$

where the constant $C>0$ still depends only $C_{\sigma}$ and $\beta$.

Proof: To lighten the notation, we denote by $\mathcal{O}_{CM}$ any quantity
such that

$|\mathcal{O}_{CM}|\leq C\alpha_{\sigma,\beta}(Q)$

for some constant $C$. Then by (5.46),

$\frac{\nabla
D_{\beta}(X)}{D_{\beta}(X)}=-\frac{1}{\beta}\frac{\nabla[D_{\beta}^{-\beta}](X)}{D_{\beta}^{-\beta}(X)}\\\
=-\frac{1}{\beta}\left(\frac{-\beta
c_{\beta}\lambda(p,t)\operatorname{dist}(X,\Lambda(p,t))^{-\beta-1}N_{p,t}(X)}{D_{\beta}^{-\beta}(X)}+\frac{\ell(Q)^{-\beta-1}\mathcal{O}_{CM}}{D_{\beta}^{-\beta}(X)}\right).$

Using $D_{\beta}(X)\approx\delta(X)\approx\ell(Q)$ and (5.45), we can further
write the above as

$\frac{\nabla
D_{\beta}(X)}{D_{\beta}(X)}=\dfrac{N_{p,t}(X)}{\operatorname{dist}(X,\Lambda(p,t))}+\ell(Q)^{-1}\mathcal{O}_{CM},$

which implies the corollary. $\square$

##  6\. The bi-Lipschitz change of variable $\rho_{\mathcal{S}}$

The results in this section are similar, identical, or often even easier than
the ones found in Sections 2, 3, and 4 of [DFM2]. Many proofs will only be
sketched and we will refer to the corresponding result in [DFM2] for details.

As in the previous sections, we take $0<\epsilon_{0}\ll\epsilon_{1}\ll 1$ and
we use Lemma 3.16 with such $\epsilon_{0},\epsilon_{1}$ to obtain a collection
$\mathfrak{S}$ of coherent regimes. We take then $\mathcal{S}$ that either
belongs to $\mathfrak{S}$, or is a coherent regime included in an element of
$\mathfrak{S}$. We keep the notations introduced in Sections 3, 4, and 5.

###  6.1. Construction of $\rho_{\mathcal{S}}$

In this section, the gradients are column vectors. The other notation is
fairly transparent. A hyperplane $P$ is equipped with an orthonormal basis and
$\nabla_{p}$ correspond to the vector of the derivatives in each coordinate of
$p$ in this basis; $\partial_{t}$ or $\partial_{s}$ are the derivatives with
respect to $t$ or $s$, that are always explicitly written; $\nabla_{p,t}$ or
$\nabla_{p,s}$ are the gradients in $\mathbb{R}^{n}$ seen as $P\times
P^{\bot}$.

######  Lemma 6.1.

The quantities $\nabla_{p,t}b^{t}$ and $t\nabla_{p,t}\nabla_{p}b^{t}$ are
bounded, that is, for any $t\neq 0$ and any $p_{0}\in P$,

(6.2) $|\nabla_{p,t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|\leq C\epsilon_{0}.$

In addition, $|\partial_{t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|\in
CM_{P\times(P^{\bot}\setminus\\{0\\})}$, that is, for any $r>0$ and any
$p_{0}\in P$,

(6.3)
$\iint_{B(p_{0},r)}\Big{(}|\partial_{t}b^{t}(p)|^{2}+|t\nabla_{p,t}\nabla_{p}b^{t}(p)|^{2}\Big{)}\frac{dt}{t}\,dp\leq
C\epsilon_{0}^{2}r^{n-1}.$

In both cases, the constant $C>0$ depends only on $n$ (and $\eta$).

Proof: The result is well known and fairly easy. The boundedness is proven in
Lemma 3.17 of [DFM2], while the Carleson bound is established Lemma 4.11 in
[DFM2] (which is itself a simple application of the Littlewood-Paley theory
found in [Ste, Section I.6.3] to the bounded function $\nabla b$). $\square$

Observe that the convention that we established shows that
$(\nabla_{p}b^{t}(p))^{T}$ is $n-1$ dimensional horizontal vector. We define
the map $\rho:P\times P^{\bot}\to P\times P^{\bot}$ as

(6.4) $\rho_{\mathcal{S}}(p,t):=(p-t(\nabla b^{t}(p))^{T},t+b^{t}(p))$

if $t\neq 0$ and $\rho_{\mathcal{S}}(p,0)=\mathfrak{b}(p)$. Because the
codimension of our boundary in 1 in our paper, contrary to [DFM2] which stands
in the context of domains with higher codimensional boundaries, our map is way
easier than the one found in [DFM2]. However, the present mapping is still
different from the one found in [KP], and has the same weak and strong
features as the change of variable in [DFM2]. Note that the $i^{th}$
coordinate of $\rho_{\mathcal{S}}$, $1\leq i\leq n-1$, is

(6.5) $\rho_{\mathcal{S}}^{i}(p,t):=p_{i}-t\partial_{p_{i}}b^{t}(p).$

Note that $\rho_{\mathcal{S}}$ is continuous on $P\times
P^{\bot}=\mathbb{R}^{n}$, because both $t\nabla b^{t}$ and $b^{t}-b$ converges
(uniformly in $p\in P$) to $0$ as $t\to 0$. The map $\rho_{\mathcal{S}}$ is
$C^{\infty}$ on $\mathbb{R}^{n}\setminus P$, and we compute the Jacobian
$\operatorname{Jac}$ of $\rho_{\mathcal{S}}$ which is

(6.6)
$\operatorname{Jac}(p,t)=\begin{pmatrix}I-t\partial_{p_{i}}\partial_{p_{j}}b^{t}(p)&\partial_{p_{i}}b^{t}(p)\\\
-t\partial_{t}\partial_{p_{j}}b^{t}(p)-\partial_{p_{j}}b^{t}(p)&1+\partial_{t}b^{t}(p)\end{pmatrix},$

where $i$ and $j$ refers to respectively the line and the column of the
matrix. We define the approximation of the Jacobian $\operatorname{Jac}$ as

(6.7) $J=\begin{pmatrix}I&\partial_{p_{i}}b^{t}(p)\\\
-\partial_{p_{j}}b^{t}(p)&1\end{pmatrix}=\begin{pmatrix}I&\nabla_{p}b^{t}(p)\\\
-(\nabla_{p}b^{t}(p))^{T}&1\end{pmatrix}.$

######  Lemma 6.8.

We have the following pointwise bounds:

  1. (i)

$\displaystyle\|J-I\|\lesssim|\nabla_{p}b^{t}|\lesssim\epsilon_{0}$,

  2. (ii)

$\displaystyle\|\operatorname{Jac}-J\|\lesssim|\partial_{t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|\lesssim\epsilon_{0}$,

  3. (iii)

$\displaystyle|\det(J)-1|\lesssim|\nabla_{p}b^{t}|\lesssim\epsilon_{0}$,

  4. (iv)

$\displaystyle|\det(\operatorname{Jac})-\det(J)|\lesssim|\partial_{t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|$,

  5. (v)

$\displaystyle\|(\operatorname{Jac})^{-1}-J^{-1}\|\lesssim|\partial_{t}b|+|t\nabla_{p,t}\nabla_{p}b^{t}|$,

  6. (vi)

$\displaystyle|\nabla_{p,t}\det(J)|+\||\nabla_{p,t}J^{-1}|\|\lesssim|\nabla_{p,t}\nabla_{p}b^{t}|$,

In each estimate, the constants depends only on $n$ and $\eta$.

Proof: Only a rapid proof is provided, and details are carried out in the
proof of Lemmas 3.26, 4.12, 4.13, and 4.15 in [DFM2].

The items (i) and (ii) are direct consequences of (6.2) and the definitions of
$J$ and $\operatorname{Jac}$.

For items $(iii)$ and $(iv)$, we use the fact that the determinant is the sum
of products of coefficients of the matrix. More precisely, the Leibniz formula
states that

(6.9) $\det(M):=\sum_{\sigma\in
S_{n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}M_{i,\sigma(i)},$

where $S_{n}$ is the sets of permutations of $\\{1,\dots,n\\}$ and
$\operatorname{sgn}$ is the signature. So the difference between the
determinant of two matrices $M_{1}$ and $M_{2}$ is the sum of products of
coefficients of $M_{1}$ and $M_{2}-M_{1}$, and each product contains at least
one coefficient of $M_{2}-M_{1}$. With this observation, $(iii)$ and $(iv)$
follow from $(i)$ and $(ii)$.

The items $(iii)$ and $(iv)$ shows that both $\det(J)$ and
$\det(\operatorname{Jac})$ are close to $1$ \- say in $(1/2,2)$ \- as long as
$\epsilon_{0}$ is small enough. This implies that

(6.10)
$\left|\frac{1}{\det(\operatorname{Jac})}-\frac{1}{\det(J)}\right|=\left|\frac{\det(J)-\det(\operatorname{Jac})}{\det(\operatorname{Jac})\det(J)}\right|\lesssim|\partial_{t}b|+|t\nabla_{p,t}\nabla_{p}b^{t}|$

by $(iv)$. Cramer’s rule states that the coefficients of $M^{-1}$ is the
quotient of a linear combination of product of coefficients of $M$ over
$\det(M)$. By using Cramer’s rule to $\operatorname{Jac}$ and $J$, (6.10), and
$(ii)$, we obtain $(v)$.

Finally, the bound on $\nabla\det(J)$ and $\nabla J^{-1}$ are obtained by
taking the gradient respectively in (6.9) and in Cramer’s rule. $\square$

######  Lemma 6.11.

For any $p\in P$ and $t\in P^{\bot}\setminus\\{0\\}$, we have

(6.12)
$(1-C\epsilon_{0})|t|\leq\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Gamma_{\mathcal{S}})\leq|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)|\leq(1+C\epsilon_{0})|t|$

and

(6.13) $|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)-(0,t)|\leq C\epsilon_{0}|t|,$

where $C>0$ depends only on $n$ (and $\eta$).

Proof: The lemma is an analogue of Lemma 3.40 in [DFM2]. But since the lemma
is key to understand why $\rho_{\mathcal{S}}$ is a bi-Lipschitz change of
variable, and since it is much easier in our case, we prove it carefully.

By definition of $\rho_{\mathcal{S}}$,

$\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)-(0,t)=(-t(\nabla
b^{t}(p))^{T},b^{t}(p)-b(p)).$

So the mean value theorem applied to the continuous function $t\mapsto
b^{t}(p)$ [recall that $b$ is Lipschitz and $b^{t}$ is the convolution of $b$
with a mollifier, so we even have a uniform convergence of $b^{t}$ to $b$]
entails that

(6.14)
$\begin{split}|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)-(0,t)|&\leq|t\nabla
b^{t}(p)|+|b^{t}(p)-b(p)|\\\ &\leq|t\nabla
b^{t}(p)|+|t|\sup_{s\in(0,|t|)}|\partial_{s}b^{s}(p)|\lesssim\epsilon_{0}|t|\end{split}$

by (6.2). Therefore, (6.13) is proven and we have

(6.15) $|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)|\leq(1+C\epsilon_{0})|t|,$

is the upper bound in (6.12). The middle bound of (6.12) is immediate, since
$\mathfrak{b}(p)\in\Gamma_{\mathcal{S}}$. It remains thus to prove the lower
bound in (6.12). Let $q\in P$ be such that
$|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(q)|=\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Gamma_{\mathcal{S}})$.
We know that

$|\mathfrak{b}(q)-\mathfrak{b}(p)|\leq|\mathfrak{b}(q)-\rho_{\mathcal{S}}(p,t)|+|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)|\leq
2|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)|\leq 3|t|,$

if $\epsilon_{0}\ll 1$ is small enough, hence $|q-p|\leq 3|t|$ too. So

$\begin{split}\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Gamma_{\mathcal{S}})&=|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(q)|\geq|\mathfrak{b}(p)-\mathfrak{b}(q)+(0,t)|-|\rho_{\mathcal{S}}(p,t)-\mathfrak{b}(p)-(0,t)|\\\
&\geq|b(p)-b(q)+t|-C\epsilon_{0}|t|\end{split}$

by (6.13). But by Lemma 3.40, the function $b$ is $2\epsilon_{0}$-Lipschitz,
so we can continue with

$\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Gamma_{\mathcal{S}})\geq(1-C{\epsilon_{0}})|t|-|b(p)-b(q)|\geq(1-C{\epsilon_{0}})|t|-2\epsilon_{0}|p-q|\geq(1-C^{\prime}{\epsilon_{0}})|t|.$

The lemma follows. $\square$

######  Lemma 6.16.

The map $\rho_{\mathcal{S}}$ is a bi-Lipschitz change of variable that maps
$P$ to $\Gamma_{\mathcal{S}}$.

Proof: See Theorem 3.53 in [DFM2] for more details. We shall show that
$\rho_{\mathcal{S}}$ is a bi-Lipschitz change of variable from
$P\times(0,\infty)$ to

$\Omega^{+}_{\mathcal{S}}:=\\{(p,t)\in P\times P^{\bot},\,t>b(p)\\}$

and a similar argument also give that $\rho_{\mathcal{S}}$ is a bi-Lipschitz
change of variable from $P\times(-\infty,0)$ to

$\Omega^{-}_{\mathcal{S}}:=\\{(p,t)\in P\times P^{\bot},\,t<b(p)\\}.$

The lemma follows because we know that $\rho_{\mathcal{S}}$ is continuous on
$P\times P^{\bot}$.

First, we know by the lower bound in (6.12) that the range of
$\rho_{\mathcal{S}}(P\times(0,\infty))$ never intersects
$\Gamma_{\mathcal{S}}$, so since $\rho_{\mathcal{S}}$ is connected, it means
that $\rho_{\mathcal{S}}(P\times(0,\infty))$ is included in either
$\Omega^{+}_{\mathcal{S}}$ or $\Omega^{-}_{\mathcal{S}}$. A quick analysis of
$\rho_{\mathcal{S}}$, for instance (6.13), shows that
$\rho_{\mathcal{S}}(P\times(0,\infty))\subset\Omega^{+}_{\mathcal{S}}$.

At any point $(p,t)\in P\times(0,\infty)$ the Jacobian of $\rho_{\mathcal{S}}$
is close to the identity, as shown by $(i)$ and $(ii)$ of Lemma 6.8. So
$\rho_{\mathcal{S}}$ is a local diffeomorphism, and the inversion function
theorem shows that there exists a neighborhood $V_{p,t}\subset
P\times(0,\infty)$ of $(p,t)$ such that $\rho_{\mathcal{S}}$ is a bijection
between $V_{p,t}$ and its range $\rho(V_{p,t})$, which is a neighborhood of
$\rho_{\mathcal{S}}(p,t)$. Since the Jacobian is uniformly close to the
identity, all the $\rho_{\mathcal{S}}:\,V_{p,t}\mapsto\rho(V_{p,t})$ are bi-
Lipschitz maps with uniform Lipschitz constant.

If $z\in\Omega^{+}_{\mathcal{S}}$, we define the degree of the map
$\rho_{\mathcal{S}}$ as

$N(z):=\text{``number of points $(p,t)\in P\times(0,\infty)$ such that
$\rho(p,t)=z$''}\in\mathbb{N}\cup\\{+\infty\\}.$

We want to prove that $N(z)$ is constantly equal to 1. If this is true, then
the lemma is proven and we can construct the inverse $\rho^{-1}$ locally by
inversing the appropriate bijection
$\rho_{\mathcal{S}}:\,V_{p,t}\mapsto\rho(V_{p,t})$.

We already know that the number of points that satisfies $\rho(p,t)=z$ is
countable, because we can cover $P\times(0,\infty)$ by a countable union of
the neighborhoods $V_{p,t}$ introduced before. Moreover, if $N(z)\geq v>0$,
then we can find $v$ points $(p_{i},t_{i})\in P\times(0,\infty)$ such that
$\rho_{\mathcal{S}}(p_{i},t_{i})=z$ and so $v$ disjoint neighborhoods
$V_{p_{i},t_{i}}$ of $(p_{i},t_{i})$. Consequently, each point $z^{\prime}$ in
the neighborbood $\bigcap_{i}\rho_{\mathcal{S}}(V_{p_{i},t_{i}})$ of $z$
satisfies $N(z^{\prime})\geq v$. This proves that $N$ is constant on any
connected component, that is

$\text{$N$ is constant on $\Omega^{+}_{\mathcal{S}}$}.$

It remains to prove that $N(z_{0})=1$ for one point $z_{0}$ in
$\Omega^{+}_{\mathcal{S}}$. Take $p_{0}$ far from the support of $b$, for
instance $\operatorname{dist}(p_{0},\Pi(Q(\mathcal{S}))\geq
99\ell(Q(\mathcal{S}))$ and $t_{0}=\ell(Q(\mathcal{S}))$. In this case, we
have $\rho_{\mathcal{S}}(p_{0},t_{0})=(p_{0},t_{0})$ and
$\operatorname{dist}(\rho_{\mathcal{S}}(p_{0},t_{0}),\Gamma_{\mathcal{S}})=t_{0}$.
Let $(p_{1},t_{1})\in P\times(0,\infty)$ be such that
$\rho_{\mathcal{S}}(p_{1},t_{1})=(p_{0},t_{0})$, the bound 6.12 entails that
$|t_{1}-t_{0}|\leq C\epsilon_{1}|t_{0}|\leq\ell(Q(\mathcal{S}))$ and (6.12)
implies that $|p_{1}-p_{0}|\leq C\epsilon_{0}|t_{1}|\leq\ell(Q(\mathcal{S}))$.
Those conditions force $p_{1}$ to stay far away from the support of $b$, which
implies that $\rho_{\mathcal{S}}(p_{1},t_{1})=(p_{1},t_{1})=(p_{0},t_{0})$. We
just proved that $N(p_{0},t_{0})=1$, as desired. $\square$

###  6.2. Properties of the operator $L_{\mathcal{S}}$

######  Lemma 6.17.

Let $L=-\mathop{\operatorname{div}}\mathcal{A}\nabla$ is a uniformly elliptic
operator satisfying (1.6) and (1.7) on $\Omega$. Construct on
$\rho_{\mathcal{S}}^{-1}(\Omega)$ the operator
$L_{\mathcal{S}}=-\mathop{\operatorname{div}}\mathcal{A}_{\mathcal{S}}\nabla$
with

(6.18)
$\mathcal{A}_{\mathcal{S}}(p,t):=\det(\operatorname{Jac}(p,t))\operatorname{Jac}^{-T}(p,t)\mathcal{A}(\rho_{\mathcal{S}}(p,t))\operatorname{Jac}^{-1}(p,t)\qquad\text{
for }(p,t)\in\rho_{\mathcal{S}}^{-1}(\Omega).$

Then $L_{\mathcal{S}}$ is the conjugate operator of $L$ by
$\rho_{\mathcal{S}}$, that is, $u\circ\rho_{\mathcal{S}}$ is a weak solution
to $L_{\mathcal{S}}(u\circ\rho_{\mathcal{S}})=0$ in
$\rho_{\mathcal{S}}^{-1}(\Omega)$ if and only if $u$ is a weak solution to
$Lu=0$ in $\Omega$.

Proof: The maps $\rho_{\mathcal{S}}$ is a bi-Lipschitz change of variable on
$\mathbb{R}^{n}=P\times P^{\bot}$, so the construction (6.18) properly define
a matrix of coefficients in $L^{\infty}(\rho_{\mathcal{S}}^{-1}(\Omega))$.

Let $u$ be a weak solution to $Lu=0$ in $\Omega$. Then, for any $\varphi\in
C^{\infty}_{0}(\rho_{\mathcal{S}}^{-1}(\Omega))$, we have

$\begin{split}\iint_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}}\nabla(u\circ\rho_{\mathcal{S}})\cdot\nabla\varphi\,dp\,dt&=\iint_{\mathbb{R}^{n}}\det(\operatorname{Jac})\operatorname{Jac}^{-T}(\mathcal{A}\circ\rho_{\mathcal{S}})\operatorname{Jac}^{-1}\nabla(u\circ\rho_{\mathcal{S}})\cdot\nabla\varphi\,dp\,dt\\\
&=\iint_{\mathbb{R}^{n}}\det(\operatorname{Jac})(\mathcal{A}\circ\rho_{\mathcal{S}})\operatorname{Jac}^{-1}\nabla(u\circ\rho_{\mathcal{S}})\cdot\operatorname{Jac}^{-1}\nabla\varphi\,dp\,dt\\\
&=\iint_{\mathbb{R}^{n}}\det(\operatorname{Jac})(\mathcal{A}\circ\rho_{\mathcal{S}})(\nabla
u\circ\rho_{\mathcal{S}})\cdot(\nabla[\varphi\circ\rho_{\mathcal{S}}^{-1}]\circ\rho_{\mathcal{S}})\,dp\,dt\end{split}$

because $\nabla(f\circ\rho_{\mathcal{S}})$ is equal to the matrix
multiplication $\operatorname{Jac}(\nabla f\circ\rho_{\mathcal{S}})$ by
definition of the Jacobian. Recall that $\det(\operatorname{Jac})>0$, so doing
the change of variable $X=\rho_{\mathcal{S}}(p,t)$ gives

(6.19)
$\iint_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}}\nabla(u\circ\rho_{\mathcal{S}})\cdot\nabla\varphi\,dp\,dt=\iint_{\mathbb{R}^{n}}\mathcal{A}\nabla
u\cdot\nabla[\varphi\circ\rho_{\mathcal{S}}^{-1}]\,dX.$

The function $\varphi\circ\rho_{\mathcal{S}}^{-1}$ may not be smooth anymore,
but is still compactly supported in $\Omega$ and in
$W^{1,\infty}(\Omega)\subset W^{1,2}_{loc}(\Omega)$, so
$\varphi\circ\rho_{\mathcal{S}}^{-1}$ is a valid test function for the weak
solution $u$, and so the right-hand side of (6.19) is 0. We conclude that

$\iint_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}}\nabla(u\circ\rho_{\mathcal{S}})\cdot\nabla\varphi\,dp\,dt=0$

for any $\varphi\in C^{\infty}_{0}(\rho_{\mathcal{S}}^{-1}(\Omega))$, hence
$u\circ\rho_{\mathcal{S}}$ is a weak solution to
$L_{\mathcal{S}}(u\circ\rho_{\mathcal{S}})=0$ in
$\rho_{\mathcal{S}}^{-1}(\Omega)$.

The same reasoning shows that $u$ is a weak solution to $Lu=0$ in $\Omega$
whenever $u\circ\rho_{\mathcal{S}}$ is a weak solution to
$L_{\mathcal{S}}(u\circ\rho_{\mathcal{S}})=0$ in
$\rho_{\mathcal{S}}^{-1}(\Omega)$. The lemma follows. $\square$

We want to say that $A_{\mathcal{S}}$ satisfies the same Carleson-type
condition as $A\circ\rho_{\mathcal{S}}$. For instance, we want to say that
$\delta\nabla A\in CM_{\Omega}$ \- which implies
$(\delta\circ\rho_{\mathcal{S}})\nabla(A\circ\rho_{\mathcal{S}})\in
CM_{\rho_{\mathcal{S}}^{-1}(\Omega)}$ \- will give that
$(\delta\circ\rho_{\mathcal{S}})\nabla A_{\mathcal{S}}\in
CM_{\rho_{\mathcal{S}}^{-1}(\Omega)}$. However, it is not true, for the simple
reason that the Carleson estimates on $\operatorname{Jac}$ are related to the
set $\mathbb{R}^{n}\setminus P$ while the ones on $A\circ\rho_{\mathcal{S}}$
are linked to the domain $\rho_{\mathcal{S}}^{-1}(\Omega)$. Since
$A_{\mathcal{S}}$ is the product of these two objects, we only have Carleson
estimates for $A_{\mathcal{S}}$ in the areas of $\mathbb{R}^{n}$ where
$\rho_{\mathcal{S}}(\partial\Omega)$ looks like $P$.

######  Lemma 6.20.

Assume that the matrix function $\mathcal{A}$ defined on $\Omega$ satisfies
(1.6) and (1.7), and can be decomposed as
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ where

(6.21) $|\delta\nabla\mathcal{B}|+|\mathcal{C}|\in CM_{\Omega}(M)$

Then the matrix $\mathcal{A}_{\mathcal{S}}$ constructed in (6.18) can also be
decomposed as
$\mathcal{A}_{\mathcal{S}}=\mathcal{B}_{\mathcal{S}}+\mathcal{C}_{\mathcal{S}}$
where $\mathcal{B}_{\mathcal{S}}$ satisfies (1.6) and (1.7) with the constant
$2C_{\mathcal{A}}$, $|t\nabla\mathcal{B}_{\mathcal{S}}|$ is uniformly bounded
by $CC_{\mathcal{A}}$, and

(6.22)
$(|t\nabla\mathcal{B}_{\mathcal{S}}|+|\mathcal{C}_{\mathcal{S}}|){\mathds{1}}_{\rho^{-1}_{\mathcal{S}}(W^{*}_{\Omega}(\mathcal{S}))}\in
CM_{\mathbb{R}^{n}\setminus P}(C(\epsilon_{0}^{2}+M))$

for a constant $C$ that depends only on $n$ and the ellipticity constant
$C_{\mathcal{A}}$.

Proof: Let $\mathcal{A}=\mathcal{B}+\mathcal{C}$ as in the lemma. Without loss
of generality, we can choose $\mathcal{B}$ to be a smooth average of
$\mathcal{A}$ (see Lemma 2.1) and so $\mathcal{B}$ satisfies (1.6) and (1.7)
with the constant $C_{\mathcal{A}}$ and $|\delta\nabla\mathcal{B}|\leq
CC_{\mathcal{A}}$. Define

$\mathcal{B}_{\mathcal{S}}:=\det(J)J^{-T}(\mathcal{B}\circ\rho_{\mathcal{S}})J^{-1}$

and of course
$\mathcal{C}_{\mathcal{S}}:=\mathcal{A}_{\mathcal{S}}-\mathcal{B}_{\mathcal{S}}$.
First, Lemma 6.8 shows that $\det(J)$ is close to $1$ and $J^{-1}$ is close to
the identity, so $\mathcal{B}_{\mathcal{S}}$ satisfies (1.6) and (1.7) with
the constant $(1+C\epsilon_{0})C_{\mathcal{A}}\leq 2C_{\mathcal{A}}$.
Moreover, the same Lemma 6.8 gives that $|\det(J)|+\|J^{-1}\|\leq 3$,
$\|\operatorname{Jac}-I\|\leq 3$, and
$\displaystyle|\nabla_{p,t}\det(J)|+\||\nabla_{p,t}J^{-1}|\|\lesssim|\nabla_{p,t}\nabla_{p}b^{t}|$,
and hence

$|\nabla\mathcal{B}_{\mathcal{S}}|\lesssim|(\nabla\mathcal{B})\circ\rho_{\mathcal{S}}|+|\nabla_{p,t}\nabla_{p}b^{t}|,$

and

$\begin{split}|\mathcal{C}_{\mathcal{S}}|&\lesssim|\det(\operatorname{Jac})-\det(J)|+\|\operatorname{Jac}^{-1}-J^{-1}\|+|\mathcal{C}\circ\rho_{\mathcal{S}}|\\\
&\lesssim|\partial_{t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|+|\mathcal{C}\circ\rho_{\mathcal{S}}|.\end{split}$

Lemma 6.1 entails that $|t\nabla_{p,t}\nabla_{p}b^{t}|\lesssim\epsilon_{0}\leq
1\leq C_{\mathcal{A}}$, so $\mathcal{B}_{\mathcal{S}}$ verifies
$|t\nabla\mathcal{B}_{\mathcal{S}}|\lesssim C_{\mathcal{A}}$, so thus (6.22)
is the only statement we still have to prove. Lemma 6.1 also implies that
$|\partial_{t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|\in
CM_{P\times(0,\infty)}(C\epsilon_{0}^{2})$. Therefore, it suffices to
establish that

(6.23)
$(|t\nabla\mathcal{B}\circ\rho_{\mathcal{S}}|+|\mathcal{C}\circ\rho_{\mathcal{S}}|){\mathds{1}}_{\rho^{-1}_{\mathcal{S}}(W^{*}_{\Omega}(\mathcal{S}))}\in
CM_{P\times(0,\infty)}(CM).$

Take $p_{0}\in P$ and $r_{0}>0$. We want to show that

(6.24)
$\iint_{B(p_{0},r_{0})\cap\rho^{-1}_{\mathcal{S}}(W^{*}_{\Omega}(\mathcal{S}))}(|t\nabla\mathcal{B}\circ\rho_{\mathcal{S}}|^{2}+|\mathcal{C}\circ\rho_{\mathcal{S}}|^{2})\,\frac{dt}{t}\,dp\leq
CM(r_{0})^{n-1}.$

If $\rho_{\mathcal{S}}(B(p_{0},r_{0}))\cap
W^{*}_{\Omega}(\mathcal{S})=\emptyset$, the left hand side above is zero and
there is nothing to prove. Otherwise, pick a point
$X\in\rho_{\mathcal{S}}(B(p_{0},r_{0}))\cap W^{*}_{\Omega}(\mathcal{S})$. The
fact that $X\in\rho_{\mathcal{S}}(B(p_{0},r_{0}))$ means that

(6.25) $|X-\mathfrak{b}(p_{0})|\leq(1+C\epsilon_{0})r_{0}.$

since $\rho_{\mathcal{S}}(p_{0})=\mathfrak{b}(p_{0})$ and
$\|\operatorname{Jac}-I\|\leq C\epsilon_{0}$ by Lemma 6.8. Because $b$ is
$2\epsilon_{0}$-Lipschitz with $\epsilon_{0}\ll 1$, we deduce

(6.26)
$|X-\mathfrak{b}(\Pi(X))|\leq(1+\epsilon_{0})|X-\mathfrak{b}(p_{0})|\leq(1+2C\epsilon_{0})r_{0}.$

The fact that $X\in W^{*}_{\Omega}(\mathcal{S})$ implies by (4.14) that

(6.27) $\delta(X)\leq(1+2\epsilon_{0})|X-\mathfrak{b}(\Pi(X))|\leq 2r_{0}$

thanks to (6.26). Moreover, if $x\in\partial\Omega$ is such that
$|X-x|=\delta(X)$,

(6.28)
$|x-\mathfrak{b}(p_{0})|\leq|x-\mathfrak{b}(\Pi(X))|+|\mathfrak{b}(\Pi(X)-\mathfrak{b}(p_{0})|\\\
\leq
2\epsilon_{0}\delta(X)+(1+\epsilon_{0})|\Pi(X)-p_{0}|\leq\frac{1}{2}r_{0}+(1+\epsilon_{0})|X-\mathfrak{b}(p_{0})|\leq
2r_{0}$

by using in order (4.16), the fact that $b$ is $2\epsilon_{0}$-Lipschitz,
(6.27), and (6.25). Fix $X_{0}\in\rho_{\mathcal{S}}(B(p_{0},r_{0}))\cap
W^{*}_{\Omega}(\mathcal{S})$ and $x_{0}\in\partial\Omega$ such that
$|X_{0}-x_{0}|=\delta(X_{0})$. The inequalities (6.25) and (6.28) show that,

$|X-x_{0}|\leq|X-\mathfrak{b}(p_{0})|+|x_{0}-\mathfrak{b}(p_{0})|\leq
4r_{0}\qquad\text{ for }X\in\rho_{\mathcal{S}}(B(p_{0},r_{0}))\cap
W^{*}_{\Omega}(\mathcal{S}),$

that is

(6.29) $\rho_{\mathcal{S}}(B(p_{0},r_{0}))\cap
W^{*}_{\Omega}(\mathcal{S})\subset B(x_{0},4r_{0}).$

We are now ready to conclude. We make the change of variable
$X=\rho_{\mathcal{S}}(p,s)$ in (6.24), and since $\rho_{\mathcal{S}}$ is a bi-
Lipschitz change of variable that almost preserves the distances (because
$\|\operatorname{Jac}-I\|\leq C\epsilon_{0}\ll 1$), we obtain

$\begin{split}\iint_{B(p_{0},r)\cap\rho^{-1}_{\mathcal{S}}(W^{*}_{\Omega}(\mathcal{S}))}&(|t\nabla\mathcal{B}\circ\rho_{\mathcal{S}}|^{2}+|\mathcal{C}\circ\rho_{\mathcal{S}}|^{2})\,\frac{dt}{t}\,dp\\\
&\leq 2\iint_{B(x_{0},4r)\cap
W^{*}_{\Omega}(\mathcal{S})}(|\operatorname{dist}(X,\Gamma_{\mathcal{S}})\nabla\mathcal{B}|^{2}+|\mathcal{C}|^{2})\,\frac{dX}{\operatorname{dist}(X,\Gamma_{\mathcal{S}})}\\\
&\leq
4\iint_{B(x_{0},4r)}(|\delta\nabla\mathcal{B}|^{2}+|\mathcal{C}|^{2})\,\frac{dX}{\delta(X)}\leq
CM(r_{0})^{n-1}\end{split}$

by using (4.15) and then the fact that
$|\delta\nabla\mathcal{B}|+|\mathcal{C}|\in CM_{\Omega}(M)$. The lemma
follows. $\square$

###  6.3. Properties of the composition of the smooth distance by
$\rho_{\mathcal{S}}$

The change of variable $\rho_{\mathcal{S}}$ maps
$P\times(P^{\bot}\setminus\\{0\\})$ to
$\mathbb{R}^{n}\setminus\Gamma_{\mathcal{S}}$, so for any
$X\in\mathbb{R}^{n}\setminus\Gamma_{\mathcal{S}}$, the quantities
$N_{\rho^{-1}_{\mathcal{S}}(X)}(Y)$ and $\Lambda(\rho^{-1}_{\mathcal{S}}(X))$
make sense as $N_{p,t}(Y)$ and $\Lambda(p,t)$, respectively, where
$(p,t)=\rho^{-1}_{\mathcal{S}}(X)$. With this in mind, we have the following
result.

######  Lemma 6.30.

For any $Q\in\mathcal{S}$, we have

(6.31) $\iint_{W_{\Omega}(Q)}\left|\frac{\nabla
D_{\beta}(X)}{D_{\beta}(X)}-\frac{N_{\rho^{-1}_{\mathcal{S}}(X)}(X)}{\operatorname{dist}(X,\Lambda(\rho^{-1}_{\mathcal{S}}(X))}\right|^{2}\,\delta(X)\,dX\leq
C|\alpha_{\sigma,\beta}(Q)|^{2}\sigma(Q),$

with a constant $C>0$ that depends only on $n$, $C_{\sigma}$, and $\beta$.

Proof: The lemma is a consequence of Corollary 5.53 and the definition of
$\rho_{\mathcal{S}}$.

First , Lemma 4.1 (d) entails that
$W_{\Omega}(Q)\subset\mathbb{R}^{n}\setminus\Gamma_{\mathcal{S}}$, which means
that the quantities $N_{\rho^{-1}_{\mathcal{S}}(X)}$ and
$\Lambda(\rho^{-1}_{\mathcal{S}}(X))$ are well defined in (6.31). Let $X\in
W_{\Omega}(Q)$ and set $(p,t)=\rho^{-1}_{\mathcal{S}}(X)$.

On one hand, Lemma 6.11 gives that

$\operatorname{dist}(X,\Gamma_{\mathcal{S}})\leq|X-\mathfrak{b}(p)|\leq(1+C\epsilon_{0})|t|\leq(1+C^{\prime}\epsilon_{0})\operatorname{dist}(X,\Gamma_{\mathcal{S}})$

and

$|X-\mathfrak{b}(p)-(0,t)|\leq C\epsilon_{0}|t|.$

By projecting the left-hand side on $P$, the latter implies that

$|\Pi(X)-p|\leq C\epsilon_{0}|t|.$

On the other hand, since $X\in W_{\Omega}(Q)$, Lemma 4.13 gives that

$\operatorname{dist}(X,\Gamma_{\mathcal{S}})\leq|X-\mathfrak{b}(\Pi(X))|\leq(1+2\epsilon_{0})\delta(X)\leq(1+C\epsilon_{0})\operatorname{dist}(X,\Gamma_{\mathcal{S}}),$

and, if $x\in Q$ is such that $|X-x|=\delta(X)$, then by (4.16),

$|\mathfrak{b}(\Pi(X))-x|\leq 2\epsilon_{0}\delta(X),$

which implies that

$|\Pi(X)-\Pi(x)|\leq 2\epsilon_{0}\delta(X),$

Altogether, we have

$\delta(X)(1-C\epsilon_{0})\leq|t|\leq(1+C\epsilon_{0})\delta(X)$

and

$\operatorname{dist}(p,\Pi(Q))\leq|p-\Pi(x)|\leq\left|p-\Pi(X)\right|+\left|\Pi(X)-\Pi(x)\right|\leq
C\epsilon_{0}\delta(X).$

If we throw in the fact that $\delta(X)\in[\ell(Q)/2,\ell(Q)]$ by definition
of $W_{\Omega}(Q)$, then we easily observe that $p$ and $t$ satisfy the
assumptions of Corollary 5.53, and so

$\left|\frac{\nabla
D_{\beta}(X)}{D_{\beta}(X)}-\frac{N_{\rho^{-1}_{\mathcal{S}}(X)}(X)}{\operatorname{dist}(X,\Lambda(\rho^{-1}_{\mathcal{S}}(X))}\right|\leq
C\ell(Q)^{-1}\alpha_{\sigma,\beta}(Q)\qquad\text{ for }X\in W_{\Omega}.$

We conclude that

$\begin{split}\iint_{W_{\Omega}(Q)}\left|\frac{\nabla
D_{\beta}(X)}{D_{\beta}(X)}-\frac{N_{\rho^{-1}_{\mathcal{S}}(X)}(X)}{\operatorname{dist}(X,\Lambda(\rho^{-1}_{\mathcal{S}}(X))}\right|^{2}\,\delta(X)\,dX&\leq
C|W_{\Omega}(Q)||\ell(Q)^{-1}\alpha_{\sigma,\beta}(Q)|^{2}\ell(Q)\\\ &\leq
C|\alpha_{\sigma,\beta}(Q)|^{2}\sigma(Q)\end{split}$

because $|W_{\Omega}(Q)|\approx\sigma(Q)\ell(Q)$ by (4.8) and (1.1). The lemma
follows. $\square$

######  Lemma 6.32.

We have

(6.33)
$\iint_{\rho_{\mathcal{S}}^{-1}(W_{\Omega}(\mathcal{S}))}\left|\frac{\nabla
t}{t}-\frac{\operatorname{Jac}(p,t)N_{p,t}(\rho_{\mathcal{S}}(p,t))}{\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Lambda(p,t))}\right|^{2}\left|t\right|\,dt\,dp\leq
C(\epsilon_{0})^{2}\sigma(Q(\mathcal{S}))$

where $C>0$ depends only on $n$ (and $\eta$).

Proof: From the definition, we can see that $\Lambda(p,t)$ is the affine plane
that goes through the point $\mathfrak{b}^{t}(p)$ and whose directions are
given by the vectors $(q,q\nabla b^{t}(p))$, that is $\Lambda(p,t)$ is the
codimension 1 plane that goes through $\mathfrak{b}^{t}(p)$ and with upward
unit normal vector

$N_{p,t}=\frac{1}{|(-(\nabla b^{r}(p))^{T},1)|}\begin{pmatrix}-\nabla
b^{t}(p)\\\ 1\end{pmatrix}=\frac{1}{\sqrt{1+|\nabla
b^{t}(p)|^{2}}}\begin{pmatrix}-\nabla b^{t}(p)\\\ 1\end{pmatrix}$

The vector function $N_{p,t}(X)$ is just $+N_{p,t}$ or $-N_{p,t}$, depending
whether $X$ lies above or below $\Lambda(p,t)$.

Observe that $\rho_{\mathcal{S}}(p,t)-\mathfrak{b}^{t}(p)=t(-(\nabla
b^{t}(p))^{T},1)$, which means that $\mathfrak{b}^{t}(p)$ is the projection of
$\rho_{\mathcal{S}}(p,t)$ onto $\Lambda(p,t)$ and that

$\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Lambda(p,t))=|t||(-(\nabla
b^{t}(p))^{T},1)|=|t|\sqrt{1+|\nabla b^{t}(p)|^{2}}.$

Moreover, $\rho_{\mathcal{S}}(p,t)$ lies above $\Lambda(p,t)$ if $t>0$ and
below otherwise, that is

$N_{p,t}(\rho(p,t))=\operatorname{sgn}(t)N_{p,t}=\frac{\operatorname{sgn}(t)}{\sqrt{1+|\nabla
b^{t}(p)|^{2}}}\begin{pmatrix}-\nabla b^{t}(p)\\\ 1\end{pmatrix}.$

From all this, we deduce

(6.34)
$\begin{split}\frac{J(p,t)N_{p,t}(\rho_{\mathcal{S}}(p,t))}{\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Lambda(p,t))}&=\frac{1}{t(1+|\nabla
b^{t}(p)|^{2})}\begin{pmatrix}I&\nabla b^{t}(p)\\\ -(\nabla
b^{t}(p))^{T}&1\end{pmatrix}\begin{pmatrix}-\nabla b^{t}(p)\\\
1\end{pmatrix}\\\ &=\frac{1}{t}\begin{pmatrix}0_{\mathbb{R}^{n-1}}\\\
1\end{pmatrix}=\frac{\nabla t}{t}.\end{split}$

Recall that
$|\operatorname{Jac}-J|\lesssim|\partial_{t}b^{t}|+|t\nabla_{p,t}\nabla_{p}b^{t}|$.
Together with (6.34), we obtain that the left-hand side of (6.33) is equal to

(6.35)
$I=\iint_{\rho^{-1}_{\mathcal{S}}(W_{\Omega}(\mathcal{S}))}\left|\frac{\operatorname{Jac}(p,t)-J(p,t)}{t(1+|\nabla
b^{t}(p)|^{2})}\begin{pmatrix}-\nabla b^{t}(p)\\\
1\end{pmatrix}\right|^{2}\left|t\right|\,dt\,dp\\\
\lesssim\iint_{\rho_{\mathcal{S}}^{-1}(W_{\Omega}(\mathcal{S}))}(|\partial_{t}b^{t}|^{2}+|t\nabla_{p,t}\nabla_{p}b^{t}|^{2})\,\frac{dt}{\left|t\right|}\,dp.$

Take $X_{0}\in W_{\Omega}(\mathcal{S})$, and notice that the set
$W_{\Omega}(\mathcal{S})$ is included in
$B(\mathfrak{b}(\Pi(X_{0})),4\ell(Q(\mathcal{S})))$ by definition of
$W_{\Omega}(\mathcal{S})$ and by (4.16). Since the Jacobian of
$\rho_{\mathcal{S}}$ is close to the identity, $\rho_{\mathcal{S}}^{-1}$
almost preserves the distance, and hence
$\rho^{-1}_{\mathcal{S}}(W_{\Omega}(\mathcal{S}))\subset
B(\Pi(X_{0}),5\ell(Q(\mathcal{S})))$. We conclude that

$I\lesssim\iint_{B(\Pi(X_{0}),5\ell(Q(\mathcal{S})))}(|\partial_{t}b^{t}|^{2}+|t\nabla_{p,t}\nabla_{p}b^{t}|^{2})\,\frac{dt}{t}\,dp\lesssim(\epsilon_{0})^{2}\ell(Q(\mathcal{S}))^{n-1}\lesssim(\epsilon_{0})^{2}\sigma(Q(\mathcal{S}))$

by Lemma 6.1 and then (1.1). The lemma follows. $\square$

##  7\. The flat case.

In this section, we intend to prove an analogue of Theorem 1.12 when the
boundary is flat, that is when the domain is $\Omega_{0}:=\mathbb{R}^{n}_{+}$.
This is our main argument on the PDE side (contrary to other sections which
are devoted to geometric arguments) and the general case of Chord-Arc Domains
is eventually brought back to this simpler case.

We shall bring a little bit of flexibility in the following manner. We will
allow $\Omega$ to be different from $\mathbb{R}^{n}_{+}$, but we shall stay
away from the parts where $\partial\Omega$ differs from
$\partial\mathbb{R}^{n}_{+}$ with some cut-off functions. More exactly, we
shall use cut-off functions $\phi$ that guarantee that
$\delta(X):=\operatorname{dist}(X,\partial\Omega)\approx t$ whenever
$X=(x,t)\in\operatorname{supp}\phi$. We shall simply use $\mathbb{R}^{n-1}$
for $\partial\mathbb{R}^{n}=\mathbb{R}^{n-1}\times\left\\{0\right\\}$. We
start with the precise definition of the cut-off functions that we are
allowing.

######  Definition 7.1.

We say that $\phi\in L^{\infty}(\Omega)$ is a cut-off function associated to
both $\partial\Omega$ and $\mathbb{R}^{n-1}$ if $0\leq\phi\leq 1$, and there
is a constant $C_{\phi}\geq 1$ such that $|\nabla\phi|\leq
C_{\phi}\delta^{-1}$,

(7.2) $(C_{\phi})^{-1}|t|\leq\delta(X)\leq C_{\phi}|t|\qquad\text{ for all
}X=(x,t)\in\operatorname{supp}\phi,$

and there exists a collection of dyadic cubes $\\{Q_{i}\\}_{i\in I_{\phi}}$ in
$\mathbb{D}_{\partial\Omega}$ such that

(7.3) $\\{Q_{i}\\}_{i\in I_{\phi}}$ is finitely overlapping with an overlap of
at most $C_{\phi}$,

and

(7.4)
$\Omega\cap(\operatorname{supp}\phi)\cap\operatorname{supp}(1-\phi)\subset\bigcup_{i\in
I_{\phi}}W_{\Omega}^{**}(Q_{i}).$

The condition (7.2) allows us to say that

(7.5) $\text{ if, for $x\in\partial\Omega$ and $r>0$,
$B(x,r)\cap\operatorname{supp}\phi\neq\emptyset$,}\\\ \text{ then there exists
$y\in\mathbb{R}^{n-1}$ such that $B(x,r)\subset B(y,Cr)$;}$

so we can pass from Carleson measures in $\Omega$ to Carleson measure in
$\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}$. For instance, we have

(7.6) $\begin{array}[]{c}f\in CM_{\Omega}(M)\implies
f\phi,\,f{\mathds{1}}_{\operatorname{supp}\phi}\in
CM_{\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}}(C^{\prime}_{\phi}M),\\\
\delta\nabla g\in CM_{\Omega}(M)\implies t\phi\nabla g\in
CM_{\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}}(C^{\prime}_{\phi}M).\end{array}$

and vice versa. The conditions (7.3) and (7.4) ensure that
${\mathds{1}}_{(\operatorname{supp}\phi)\cap\operatorname{supp}(1-\phi)}$ (and
hence $\delta\nabla\phi$) satisfies the Carleson measure condition on
$\Omega$. So by (7.6),

(7.7)
$|t\nabla\phi|+{\mathds{1}}_{\operatorname{supp}\nabla\phi}+{\mathds{1}}_{(\operatorname{supp}\phi)\cap\operatorname{supp}(1-\phi)}\in
CM_{\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}}(C^{\prime}_{\phi}).$

And if the support of of $\phi$ is contained in a ball of radius $r$ centered
on $\partial\Omega$, then

(7.8)
$\iint_{\Omega}\big{(}|\nabla\phi|t+|t\nabla\phi|^{2}\big{)}\,\frac{dt}{t}\,dy\lesssim
r^{n-1}.$

We are ready to state the main result of the section.

######  Lemma 7.9.

Let $\Omega$ be a Chord-Arc Domain and let
$L=-\operatorname{div}\mathcal{A}\nabla$ be a uniformly elliptic operator on
$\Omega$, that is $\mathcal{A}$ verifies (1.6) and (1.7). Assume that the
$L^{*}$-elliptic measure $\omega_{L^{*}}\in A_{\infty}(\sigma)$, where $L^{*}$
is the adjoint operator of $L$, and $\sigma$ is an Ahlfors regular measure on
$\partial\Omega$. Let $\phi$ be as in Definition 7.1 and be supported in a
ball $B:=B(x,r)$ centered on the boundary $\partial\Omega$. Assume that the
coefficients $\mathcal{A}$ can be decomposed as
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ where

(7.10)
$(|t\nabla\mathcal{B}|+|\mathcal{C}|){\mathds{1}}_{\operatorname{supp}\phi}\in
CM_{\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}}(M).$

Then for any non-negative nontrivial weak solution $u$ to $Lu=0$ in
$2B\cap\Omega$ with zero trace on $\partial\Omega\cap 2B$, one has

(7.11) $\iint_{\Omega}|t|\left|\frac{\nabla u}{u}-\frac{\nabla
t}{t}\right|^{2}\phi^{2}\,dt\,dy=\iint_{\Omega}|t|\left|\nabla\ln\Big{(}\frac{u}{|t|}\Big{)}\right|^{2}\phi^{2}\,dt\,dy\leq
C(1+M)r^{n-1},$

where $C$ depends only on the dimension $n$, the elliptic constant
$C_{\mathcal{A}}$, the 1-sided CAD constants of $\Omega$, the constant
$C_{\phi}$ in Definition 7.1, and the intrinsic constants in
$\omega_{L^{*}}\in A_{\infty}(\sigma)$.

The above lemma is the analogue of Theorem 2.21 from [DFM4] in our context,
and part of our proof will follow the one from [DFM4] but a new argument is
needed to treat the non-diagonal structure of $\mathcal{A}$.

We need $\omega_{L^{*}}\in A_{\infty}(\sigma)$ for the proof of the following
intermediate lemma. Essentially, we need that the logarithm of the Poisson
kernel lies in $BMO$. Let us state and prove it directly in the form that we
need.

######  Lemma 7.12.

Let $\Omega$, $L$, $\phi$, $B:=B(x,r)$, and $u$ be as in Lemma 7.9. Assume
that $\omega_{L^{*}}\in A_{\infty}(\sigma)$ as in Lemma 7.9. Then there exists
$K:=K(u,B)$ such that

$\iint_{\Omega}|\nabla\phi|\left|\ln\Big{(}\frac{Ku}{|t|}\Big{)}\right|dt\,dy\leq
Cr^{n-1},$

where $C$ depends only on $n$, $C_{\mathcal{A}}$, the 1-sided CAD constants of
$\Omega$, the constant $C_{\phi}$ in Definition 7.1, and the intrinsic
constants in $\omega_{L^{*}}\in A_{\infty}(\sigma)$.

Proof of Lemma 7.12. The first step is to replace $Ku/t$ by the elliptic
measure. Take $X_{0}\in B(x,r)\cap\Omega$ and $X_{1}\in\Omega\setminus
B(x,4r)$ to be two corkscrew points for $x$ at the scale $r$. If $G(Y,X)$ is
the Green function associated to $L$ in $\Omega$ and
$\\{\omega^{X}_{*}\\}_{X\in\Omega}$ is the elliptic measure associated to the
adjoint $L^{*}$, the CFMS estimates (Lemma 2.18) entails, for $Y\in
W_{\Omega}^{**}(Q)\cap B$, that

$\frac{u(Y)}{u(X_{0})}\approx\frac{G(Y,X_{1})}{G(X_{0},X_{1})}\approx\frac{\ell(Q)}{r}\frac{\sigma(\Delta)}{\sigma(Q)}\frac{\omega_{*}^{X_{1}}(Q)}{\omega_{*}^{X_{1}}(\Delta)},$

where $\Delta=B\cap\partial\Omega$. Moreover, if
$Y=(y,t)\in\operatorname{supp}\phi\cap W^{**}_{\Omega}(Q)$, then
$\ell(Q)\approx|t|$ by (7.2). Altogether, we have

(7.13)
$\frac{u(Y)}{|t|}\approx\frac{u(X_{0})}{r}\frac{\sigma(\Delta)}{\sigma(Q)}\frac{\omega_{*}^{X_{1}}(Q)}{\omega_{*}^{X_{1}}(\Delta)}\qquad\text{
for }Y=(y,t)\in\operatorname{supp}\phi\cap W_{\Omega}^{**}(Q).$

Set $K:=r/u(X_{0})$, and $I_{\phi}^{\prime}:=\left\\{i\in
I_{\phi}:W_{\Omega}^{**}(Q_{i})\text{ intersects
}\operatorname{supp}\nabla\phi\right\\}$,

(7.14)
$\begin{split}\iint_{\Omega}|\nabla\phi|\left|\ln\Big{(}\frac{Ku}{|t|}\Big{)}\right|dt\,dy&\lesssim\sum_{i\in
I_{\phi}^{\prime}}\ell(Q_{i})^{-1}\int_{W_{\Omega}^{**}(Q_{i})}\left|\ln\Big{(}\frac{Ku}{|t|}\Big{)}\right|\,dt\,dy\\\
&\lesssim\sum_{i\in
I_{\phi}^{\prime}}\sigma(Q_{i})\left[1+\left|\ln\Big{(}\frac{\sigma(\Delta)}{\sigma(Q_{i})}\frac{\omega_{*}^{X_{1}}(Q_{i})}{\omega_{*}^{X_{1}}(\Delta)}\Big{)}\right|\right]\end{split}$

by (7.4), (7.13), and the fact that
$|W^{**}_{\Omega}(Q_{i})|\approx\ell(Q_{i})\sigma(Q_{i})$.

The second step is to use the fact that $\omega^{X_{1}}_{*}$ is
$A_{\infty}$-absolutely continuous with respect to $\sigma$. To that
objective, we define for $k\in\mathbb{Z}$

$\mathcal{I}_{k}:=\Big{\\{}i\in
I_{\phi}^{\prime},\,2^{k}\leq\frac{\sigma(\Delta)}{\sigma(Q_{i})}\frac{\omega_{*}^{X_{1}}(Q_{i})}{\omega_{*}^{X_{1}}(\Delta)}\leq
2^{k+1}\Big{\\}}$

and then $E_{k}:=\bigcup_{i\in\mathcal{I}_{k}}Q_{i}$. Since the collection
$\\{Q_{i}\\}_{i\in I_{\phi}}$ is finitely overlapping, due to (7.3), the bound
(7.14) becomes

(7.15)
$\begin{split}\iint_{\Omega}|\nabla\phi|\left|\ln\Big{(}\frac{Ku}{|t|}\Big{)}\right|dt\,dy&\lesssim\sum_{k\in\mathbb{Z}}(1+|k|)\sigma(E_{k}).\end{split}$

We want thus to estimate $\sigma(E_{k})$. Observe first that for any $i\in
I_{\phi}^{\prime}$, $W_{\Omega}^{**}(Q_{i})$ intersects
$\operatorname{supp}\phi\subset B$. Therefore $Q_{i}$ and $E_{k}$ have to be
inside $\Delta^{*}:=C\Delta$ for a large $C$ depending only on the constant
$K^{**}$ in (4.6). The finite overlapping (7.3) also implies that

$\frac{\sigma(\Delta^{*})}{\sigma(E_{k})}\frac{\omega_{*}^{X_{1}}(E_{k})}{\omega_{*}^{X_{1}}(\Delta^{*})}\approx
2^{k}$

For $k\geq 0$, we have

(7.16) $\frac{\sigma(E_{k})}{\sigma(\Delta^{*})}\approx
2^{-k}\frac{\omega_{*}^{X_{1}}(E_{k})}{\omega_{*}^{X_{1}}(\Delta^{*})}\lesssim
2^{-k}.$

The elliptic measure $\omega_{*}^{X_{1}}$ is $A_{\infty}$-absolutely
continuous with respect to $\sigma$ by assumption, so for $k\leq 0$, we use
the characterization (iv) from Theorem 1.4.13 in [Ken] to deduce

(7.17)
$\frac{\sigma(E_{k})}{\sigma(\Delta^{*})}\lesssim\left(\frac{\omega_{*}^{X_{1}}(E_{k})}{\omega_{*}^{X_{1}}(\Delta^{*})}\right)^{\theta}\approx
2^{k\theta}\left(\frac{\sigma(E_{k})}{\sigma(\Delta^{*})}\right)^{\theta}\lesssim
2^{k\theta}$

for some $\theta\in(0,1)$ independent of $x$, $r$, and $k$. We reinject (7.16)
and (7.17) in (7.15) to conclude that

$\begin{split}\iint_{\Omega}|\nabla\phi|\left|\ln\Big{(}\frac{Ku}{|t|}\Big{)}\right|dt\,dy&\lesssim\sigma(\Delta^{*})\sum_{k\in\mathbb{Z}}(1+|k|)2^{-|k|\theta}\lesssim\sigma(\Delta^{*})\lesssim
r^{n-1}\end{split}$

because $\sigma$ is Ahlfors regular. The lemma follows. $\square$

Proof of Lemma 7.9. The proof is divided in two parts: the first one treats
the case where $\mathcal{B}_{i,n}=0$ for $i<n$, and the second one shows that
we can come back to the first case by a change of variable, by adapting the
method presented in [Fen2].

Observe that $\phi$ can be decomposed as $\phi=\phi_{+}+\phi_{-}$ where
$\phi_{+}={\mathds{1}}_{t>0}\phi$ and $\phi_{-}={\mathds{1}}_{t<0}\phi$. Both
$\phi_{+}$ and $\phi_{-}$ are as in Definition 7.1 with constant $C_{\phi}$.
So it is enough to prove the lemma while assuming

(7.18) $\operatorname{supp}\phi\subset\\{t\geq
0\\}=\overline{\mathbb{R}^{n}_{+}}.$

The proof of the case $\operatorname{supp}\phi\subset\mathbb{R}^{n}_{-}$ is of
course identical up to obvious changes.

Step 1: Case where $\mathcal{B}_{i,n}=0$ for $i<n$ on
$\operatorname{supp}\phi$ and $\mathcal{B}$ satisfies (1.6) and (1.7) with the
same constant $C_{\mathcal{A}}$ as $\mathcal{A}$. If $b:=\mathcal{B}_{n,n}$,
this assumption on $\mathcal{B}$ implies that

(7.19) $\mathcal{B}\nabla t\cdot\nabla
v\,\phi^{2}=b\,\partial_{t}v\,\phi^{2}.$

whenever $v\in W^{1,1}_{loc}(\Omega)$ and

(7.20) $(C_{\mathcal{A}})^{-1}\leq b\leq C_{\mathcal{A}}.$

We want to prove (7.11) with the assumption (7.18), and for this, we intend to
establish that

(7.21)
$T:=\iint_{\mathbb{R}^{n}_{+}}t\left|\nabla\ln\Big{(}\frac{u}{t}\Big{)}\right|^{2}\phi^{2}\,dt\,dy\lesssim
T^{\frac{1}{2}}r^{\frac{n-1}{2}}+r^{n-1},$

which implies the desired inequality (7.11) provided that $T$ is a priori
finite. However that is not necessary the case, because some problems can
occur when $t$ is close to 0. So we take $\psi\in C^{\infty}(\mathbb{R})$ such
that $\psi(t)=0$ when $t<1$, $\psi(t)=1$ when $t\geq 2$, and $0\leq\psi\leq
1$. We construct then $\psi_{k}(Y)=\psi(2^{k}\delta(Y))$ and
$\phi_{k}=\phi\psi_{k}$. It is not very hard to see that

$\operatorname{supp}\nabla\psi_{k}:=\\{X\in\Omega,\,2^{-k}\leq\delta(X)\leq
2^{1-k}\\}\subset\bigcup_{Q\in\mathbb{D}_{k}}W_{\Omega}^{**}(Q)$

and therefore that $\phi_{k}$ is as in Definition 7.1 (with
$C_{\phi_{k}}=C_{\phi}+1$). The quantity

$T(k):=\iint_{\mathbb{R}^{n}_{+}}t\left|\nabla\ln\Big{(}\frac{u}{t}\Big{)}\right|^{2}\phi_{k}^{2}\,dt\,dy=\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{\nabla
u}{u}-\frac{\nabla t}{t}\right|^{2}\phi_{k}^{2}\,dt\,dy$

is finite, because $\phi_{k}$ is compactly supported in both $\Omega$ and
$\mathbb{R}^{n}_{+}$ (the fact that $\nabla u/u$ is in $L^{2}_{loc}(\Omega)$
for a non-negative nontrivial solution to $Lu=0$ is a consequence of the
Caccioppoli inequality and the Harnack inequality). So, we prove (7.21) for
$T(k)$ instead of $T$, which implies $T(k)\lesssim r^{n-1}$ as we said, and
take $k\to\infty$ to deduce (7.11).

We are now ready for the core part of the proof, which can be seen as an
elaborate integration by parts. Our previous discussion established that we
(only) have to prove (7.21), and that we can assume that $\phi$ is compactly
supported in $\Omega\cap\mathbb{R}^{n}_{+}$. We use the ellipticity of
$\mathcal{A}$ and the boundedness of $b$ to write

$\begin{split}T&=\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{\nabla
u}{u}-\frac{\nabla t}{t}\right|^{2}\phi^{2}\,dt\,dy\leq
C_{\mathcal{A}}^{2}\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{A}}{b}\left(\frac{\nabla
u}{u}-\frac{\nabla t}{t}\right)\cdot\left(\frac{\nabla u}{u}-\frac{\nabla
t}{t}\right)\phi^{2}dtdy\\\
&=C_{\mathcal{A}}^{2}\left(\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{A}\nabla
u}{bu}\cdot\left(\frac{\nabla u}{u}-\frac{\nabla
t}{t}\right)\,t\phi^{2}\,dt\,dy-\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{A}\nabla
t}{bt}\cdot\nabla\ln\Big{(}\frac{u}{t}\Big{)}\,t\phi^{2}\,dt\,dy\right)\\\
&:=C_{\mathcal{A}}^{2}(T_{1}+T_{2}).\end{split}$

We deal first with $T_{2}$. We use the fact that
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ and (7.19) to obtain

$T_{2}=-\iint_{\mathbb{R}^{n}_{+}}\partial_{t}\ln\Big{(}\frac{u}{t}\Big{)}\,\phi^{2}\,dt\,dy-\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{C}}{b}\nabla
t\cdot\nabla\ln\Big{(}\frac{u}{t}\Big{)}\,\phi^{2}\,dt\,dy:=T_{21}+T_{22}.$

The term $T_{22}$ can be then bounded with the help of the Cauchy-Schwarz
inequality as follows

$T_{22}\leq\|b^{-1}\|_{\infty}\left(\iint_{\mathbb{R}^{n}_{+}}|\mathcal{C}|^{2}\phi^{2}\,\frac{dt}{t}\,dy\right)^{\frac{1}{2}}\left(\iint_{\mathbb{R}^{n}_{+}}t\left|\nabla\ln\Big{(}\frac{u}{t}\Big{)}\right|^{2}\,\phi^{2}\,dt\,dy\right)^{\frac{1}{2}}\lesssim
r^{\frac{n-1}{2}}T^{\frac{1}{2}}$

by (7.10). As for $T_{21}$, observe that multiplying by any constant $K$
inside the logarithm will not change the term (because we differentiate the
logarithm). As a consequence, we have

$\begin{split}T_{21}&=-\iint_{\mathbb{R}^{n}_{+}}\partial_{t}\ln\Big{(}\frac{Ku}{t}\Big{)}\,\phi^{2}\,dt\,dy=\iint_{\mathbb{R}^{n}_{+}}\ln\Big{(}\frac{Ku}{t}\Big{)}\,\partial_{n}[\phi^{2}]\,dt\,dy\\\
&\leq\iint_{\mathbb{R}^{n}_{+}}|\nabla\phi|\left|\ln\Big{(}\frac{Ku}{t}\Big{)}\right|\,dt\,dy\lesssim
r^{n-1}\end{split}$

by successively using integration by parts and Lemma 7.12.

We turn to $T_{1}$, and we want now to use the fact that $u$ is a weak
solution to $Lu=0$. So we notice that

$\begin{split}T_{1}&=-\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{A}}{b}\nabla
u\cdot\nabla\Big{(}\frac{t}{u}\Big{)}\,\phi^{2}\,dt\,dy\\\
&=-\iint_{\mathbb{R}^{n}_{+}}\mathcal{A}\nabla
u\cdot\nabla\Big{(}\frac{t\phi^{2}}{bu}\Big{)}\,\,dt\,dy+2\iint_{\mathbb{R}^{n}_{+}}\mathcal{A}\nabla
u\cdot\nabla\phi\,\Big{(}\frac{t\phi}{bu}\Big{)}\,dt\,dy-\iint_{\mathbb{R}^{n}_{+}}\mathcal{A}\nabla
u\cdot\nabla b\,\Big{(}\frac{t\phi^{2}}{b^{2}u}\Big{)}\,dt\,dy\\\
&:=-T_{11}+2T_{12}-T_{13}.\end{split}$

Since $\phi$ is compactly supported, we have that $u>\epsilon_{\phi}$ on
$\operatorname{supp}\phi$ (by the Harnack inequality, see Lemma 2.15) and
$\nabla u\in L^{2}_{loc}(\Omega)$ (by the Caccioppoli inequality, see Lemma
2.14). Therefore $t\phi^{2}/(bu)$ is a valid test function for the solution
$u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, and then $T_{11}=0$. As for $T_{12}$,
we have

$\begin{split}T_{12}&=\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{A}}{b}\left(\frac{\nabla
u}{u}-\frac{\nabla
t}{t}\right)\cdot\nabla\phi\,(t\phi)\,dt\,dy+\iint_{\mathbb{R}^{n}_{+}}\frac{\mathcal{A}}{b}\nabla
t\cdot\nabla\phi\,\phi\,dt\,dy:=T_{121}+T_{122}.\end{split}$

The term $T_{121}$ is similar to $T_{22}$. The boundedness of $\mathcal{A}/b$
and the Cauchy-Schwarz inequality infer that

$T_{121}\leq\left(\iint_{\mathbb{R}^{n}_{+}}t|\nabla\phi|^{2}\,\frac{dt}{t}\,dy\right)^{\frac{1}{2}}\left(\iint_{\mathbb{R}^{n}_{+}}t\left|\nabla\ln\Big{(}\frac{u}{t}\Big{)}\right|^{2}\,\phi^{2}\,dt\,dy\right)^{\frac{1}{2}}\lesssim
r^{\frac{n-1}{2}}T^{\frac{1}{2}}$

by (7.8). The quantity $T_{122}$ is even easier since

$T_{122}\lesssim\iint_{\mathbb{R}^{n}_{+}}|\nabla\phi|\,dt\,dy\lesssim
r^{n-1},$

again by (7.8). It remains to bound $T_{13}$. We start as for $T_{12}$ by
writing

$\begin{split}T_{13}&=\iint_{\mathbb{R}^{n}_{+}}\mathcal{A}\left(\frac{\nabla
u}{u}-\frac{\nabla t}{t}\right)\cdot\nabla
b\,\frac{t\phi^{2}}{b^{2}}\,dt\,dy+\iint_{\mathbb{R}^{n}_{+}}\mathcal{A}\nabla
t\cdot\nabla b\,\frac{\phi^{2}}{b^{2}}\,dt\,dy:=T_{131}+T_{132}.\end{split}$

The term $T_{131}$ is like $T_{121}$, and by using $t\nabla b\in
CM_{\mathbb{R}^{n}_{+}}$ instead of $t\nabla\phi\in CM_{\mathbb{R}^{n}_{+}}$ ,
we obtain that $T_{131}\lesssim r^{(n-1)/2}T^{1/2}$. The term $T_{132}$ does
not contain the solution $u$, but it is a bit harder than $T_{122}$ to deal
with, because $\nabla b$ is not as nice as $\nabla\phi$. We use
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ and (7.19) to get

$\begin{split}T_{132}&=\iint_{\mathbb{R}^{n}_{+}}(\partial_{t}b)\,\frac{\phi^{2}}{b}\,dt\,dy+\iint_{\mathbb{R}^{n}_{+}}\mathcal{C}\nabla
t\cdot\nabla b\,\frac{\phi^{2}}{b^{2}}\,dt\,dy:=T_{1321}+T_{1322}.\end{split}$

We easily deal with $T_{1322}$ by using the Cauchy-Schwarz inequality as
follows:

$T_{1322}\leq\|b^{-1}\|_{\infty}^{2}\left(\iint_{\mathbb{R}^{n}_{+}}|\mathcal{C}|^{2}\phi^{2}\,\frac{dt}{t}\,dy\right)^{\frac{1}{2}}\left(\iint_{\mathbb{R}^{n}_{+}}|t\nabla
b|^{2}\phi^{2}\,\frac{dt}{t}\,dy\right)^{\frac{1}{2}}\lesssim r^{n-1}$

by (7.10). As last, observe that

$T_{1321}=\iint_{\mathbb{R}^{n}_{+}}\partial_{t}[\ln(b)\phi^{2}]\,\,dt\,dy-\iint_{\mathbb{R}^{n}_{+}}\partial_{t}\phi\,\phi\ln(b)\,dt\,dy,$

but the first integral in the right-hand side above is zero, so

$|T_{1321}|\lesssim\|\ln(b)\|_{\infty}\iint_{\mathbb{R}^{n}_{+}}|\partial_{t}\phi|\,dt\,dy\lesssim
r^{n-1},$

by (7.8) and the fact that $b\approx 1$. The inequality (7.11) under the three
assumptions (7.19), (7.20), and (7.10) follows.

Step 2: We can assume that
$\left\|t\left|\nabla_{y}\mathcal{B}\right|\right\|_{\infty}$ is as small as
we want.

We construct

(7.22) $\widetilde{\mathcal{A}}:=\mathcal{A}\phi+(1-\phi)I,$

where $I$ is the identity matrix. Note that $\widetilde{\mathcal{A}}$ is
elliptic with the same elliptic constant $C_{\mathcal{A}}$ as $\mathcal{A}$.
We choose then a bump function $\theta\in C^{\infty}_{0}(\mathbb{R}^{n})$
supported in $B(0,1/10)$, that is $0\leq\theta\leq 1$ and
$\iint_{\mathbb{R}^{n}}\theta\,dX=1$. We construct
$\theta_{y,t}(z,s)=t^{-n}\theta\big{(}\frac{z-y}{t},\frac{s-t}{t}\big{)}$,
which satisfies $\iint_{\mathbb{R}^{n}}\theta_{y,t}=1$, and then

(7.23)
$\widetilde{\mathcal{B}}(y,t):=\iint_{\mathbb{R}^{n}}\widetilde{\mathcal{A}}\,\theta_{y,Nt}\,dz\,ds.$

for a large $N$ to be fixed later to ensure that (7.28) below is invertible.
Since $\widetilde{\mathcal{B}}$ is some average of $\widetilde{\mathcal{A}}$,
then

(7.24) $\widetilde{\mathcal{B}}$ is elliptic and bounded with the same
constant $C_{\mathcal{A}}$ as $\widetilde{\mathcal{A}}$ and $\mathcal{A}$.

The construction is similar to the one done in Lemma 2.1, so we do not write
the details again. Observe also that

(7.25)
$|t\nabla_{y}\widetilde{\mathcal{B}}(y,t)|\lesssim\frac{1}{N}\|\widetilde{\mathcal{A}}\|_{\infty}\quad\text{
and
}\quad|t\,\partial_{t}\widetilde{\mathcal{B}}(y,t)|\lesssim\|\widetilde{\mathcal{A}}\|_{\infty}.$

In addition, we have that

$|\nabla\widetilde{\mathcal{B}}(y,t)|\lesssim
t^{-n}\iint_{B_{Nt/10}(y,Nt)}\Big{(}|\nabla\mathcal{B}|\phi+|\nabla\phi|+\frac{1}{t}|\mathcal{C}|\phi\Big{)}dz\,ds,$

and if $\widetilde{\mathcal{C}}$ denotes
$(\mathcal{A}-\widetilde{\mathcal{B}}){\mathds{1}}_{\operatorname{supp}\phi}$,
the Poincaré inequality entails that

$\int_{\Delta(x,t)}\int_{t}^{3t}|\widetilde{\mathcal{C}}(z,s)|^{2}\frac{ds}{s}\,dz\\\
\lesssim\int_{\Delta(x,2Nt)}\int_{t}^{9Nt}\Big{(}s^{2}|\nabla\mathcal{B}|^{2}\phi^{2}+|\mathcal{C}|^{2}\phi^{2}+s^{2}|\nabla\phi|^{2}+|{\mathds{1}}_{(\operatorname{supp}\phi)\cap\operatorname{supp}(1-\phi)}|^{2}\Big{)}\frac{ds}{s}\,dz,$

which means that
$t|\nabla\widetilde{\mathcal{B}}|+|\widetilde{\mathcal{C}}|\in
CM_{\mathbb{R}^{n}_{+}}$ by (7.10), and (7.7).

Step 3: The change of variable. We write $\widetilde{\mathcal{B}}$ as the
block matrix

(7.26) $\widetilde{\mathcal{B}}=\begin{pmatrix}B_{1}&B_{2}\\\
B_{3}&b\end{pmatrix},$

where $b$ is the scalar function $\widetilde{\mathcal{B}}_{n,n}$, so $B_{1}$
is a matrix of order $n-1$, $B_{2}$ and $B_{3}$ are respectively a vertical
and a horizontal vector of length $n-1$. We use $v$ for the horizontal vector
$v=-(B_{2})^{T}/b$, and we define

(7.27) $\rho(y,t):=(y+tv(y,t),t),$

which is a Lipschitz map from $\mathbb{R}^{n}_{+}$ to $\mathbb{R}^{n}_{+}$
(since $v$ and $t|\nabla v|$ are uniformly bounded, see (7.24) and (7.25)),
and we compute its Jacobian

(7.28) $Jac_{\rho}:=\begin{pmatrix}I+t\nabla_{y}v&0\\\
v+t\partial_{t}v&1\end{pmatrix}.$

We can choose $N$ big enough in (7.25) such that $Jac_{\rho}$ is invertible
and even $\det(Jac_{\rho})\geq 1/2$. Let $J_{\rho}$ be the matrix

(7.29) $J_{\rho}:=\begin{pmatrix}I&0\\\ v&1\end{pmatrix}.$

We easily have that

(7.30) $|Jac_{\rho}-J_{\rho}|+|\det(Jac_{\rho})^{-1}-1|\lesssim|t\nabla
v|\lesssim|t\nabla\widetilde{\mathcal{B}}|.$

We aim to use $\rho$ for a change of variable. If $u$ is a weak solution to
$L=-\operatorname{div}\mathcal{A}\nabla$, then $u\circ\rho^{-1}$ is solution
to $L_{\rho}=-\operatorname{div}(\mathcal{A}_{\rho}\circ\rho^{-1})\nabla$
where

(7.31)
$\mathcal{A}_{\rho}=\det(Jac_{\rho})^{-1}(Jac_{\rho})^{T}\mathcal{A}\,Jac_{\rho}.$

We want to compute $\mathcal{A}_{\rho}$. To lighten the notation, we write
$\mathcal{O}_{CM}$ for a scalar function, a vector, or a matrix which
satisfies the Carleson measure condition with respect to $\mathbb{R}^{n}_{+}$,
i.e. $\mathcal{O}_{CM}$ can change from one line to another as long as
$\mathcal{O}_{CM}\in CM_{\mathbb{R}^{n}_{+}}$. So (7.30) becomes

(7.32) $Jac_{\rho}=J_{\rho}+\mathcal{O}_{CM}\quad\text{ and
}\quad\det(Jac_{\rho})^{-1}=1+\mathcal{O}_{CM}.$

Remember that by construction, the matrix $\mathcal{A}$ equals
$\widetilde{\mathcal{B}}+\widetilde{\mathcal{C}}=\widetilde{\mathcal{B}}+\mathcal{O}_{CM}$
on $\operatorname{supp}\phi$, and that $\operatorname{Jac}_{\rho}$ and
$\mathcal{A}$ are uniformly bounded, so

(7.33)
$\begin{split}({\mathds{1}}_{\operatorname{supp}\phi})\mathcal{A}_{\rho}&={\mathds{1}}_{\operatorname{supp}\phi}\begin{pmatrix}I&v^{T}\\\
0&1\end{pmatrix}\begin{pmatrix}B_{1}&B_{2}\\\
B_{3}&b\end{pmatrix}\begin{pmatrix}I&0\\\ v&1\end{pmatrix}+\mathcal{O}_{CM}\\\
&={\mathds{1}}_{\operatorname{supp}\phi}\begin{pmatrix}B_{1}+v^{T}B_{3}+B_{2}v+bvv^{T}&B_{2}+bv^{T}\\\
B_{3}+bv&b\end{pmatrix}+\mathcal{O}_{CM}\\\
&={\mathds{1}}_{\operatorname{supp}\phi}\underbrace{\begin{pmatrix}b(B_{1}+v^{T}B_{3}+B_{2}v+bvv^{T})&0\\\
B_{3}-(B_{2})^{T}&b\end{pmatrix}}_{:=\mathcal{B}_{\rho}}+\mathcal{O}_{CM}\end{split}$

with our choices of $v$. We write $\mathcal{C}_{\rho}$ for
$(\mathcal{A}_{\rho}-\mathcal{B}_{\rho}){\mathds{1}}_{\operatorname{supp}\phi}=\mathcal{O}_{CM}$.
The matrices $\mathcal{B}_{\rho}\circ\rho^{-1}$ and
$\mathcal{C}_{\rho}\circ\rho^{-1}$ satisfy (7.10) (because the Carleson
measure condition is stable under bi-Lipschitz transformations) and
$\mathcal{B}_{\rho}\circ\rho^{-1}$ has the structure (7.19) as in Step 1. So
Step 1 gives that

(7.34)
$\iint_{\mathbb{R}^{n}_{+}}s\left|\nabla\ln\Big{(}\frac{u\circ\rho^{-1}}{s}\Big{)}\right|^{2}\phi^{2}\circ\rho^{-1}\,ds\,dz\lesssim
r^{n-1}.$

If $s$ (and $t$) is also used, by notation abuse, for the projection on the
last coordinate, then

$\begin{split}\iint_{\mathbb{R}^{n}_{+}}t\left|\nabla\ln\Big{(}\frac{u}{t}\Big{)}\right|^{2}\phi^{2}\,dt\,dy&=\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{\nabla
u}{u}-\frac{\nabla t}{t}\right|^{2}\phi^{2}\,dt\,dy\\\
&=\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{Jac_{\rho}\nabla(u\circ\rho^{-1})\circ\rho}{u}-\frac{\nabla
t}{t}\right|^{2}\phi^{2}\,dt\,dy\\\
&\leq\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{Jac_{\rho}\nabla(u\circ\rho^{-1})\circ\rho}{u}-\frac{Jac_{\rho}(\nabla
s)\circ\rho}{s\circ\rho}\right|^{2}\phi^{2}\,dt\,dy\\\ &\hskip
128.0374pt+\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{Jac_{\rho}(\nabla
s)\circ\rho}{s\circ\rho}-\frac{\nabla t}{t}\right|^{2}\phi^{2}\,dt\,dy\\\
&:=I_{1}+I_{2}.\end{split}$

Yet, $\rho$ is a bi-Lipschitz change of variable, so $Jac_{\rho}$ and
$\det(Jac_{\rho})^{-1}$ are uniformly bounded, and we have

(7.35)
$I_{1}\lesssim\iint_{\mathbb{R}^{n}_{+}}t\left|\frac{\nabla(u\circ\rho^{-1})\circ\rho}{u}-\frac{(\nabla
s)\circ\rho}{s\circ\rho}\right|^{2}\phi^{2}\,dt\,dy\\\
\lesssim\iint_{\mathbb{R}^{n}_{+}}s\left|\frac{\nabla(u\circ\rho^{-1})}{u\circ\rho^{-1}}-\frac{\nabla
s}{s}\right|^{2}\phi^{2}\circ\rho^{-1}\,ds\,dz\\\
=\iint_{\mathbb{R}^{n}_{+}}s\left|\nabla\ln\Big{(}\frac{u\circ\rho^{-1}}{s}\Big{)}\right|^{2}\phi^{2}\circ\rho^{-1}\,ds\,dz\lesssim
r^{n-1}$

by (7.34). As for $I_{2}$, we simply observe that $s\circ\rho=t$ and

$Jac_{\rho}(\nabla s)\circ\rho=\nabla t$

to deduce that $I_{2}=0$. The lemma follows. $\square$

##  8\. Proof of Theorem 1.21

In this section we prove Theorem 1.21, using the same strategy as our proof of
Theorem 1.12. As mentioned in the introduction, we shall explain how to change
the 5-step sketch of proof given in Subsection 1.3 to prove Theorem 1.21.

Fix a bounded solution $u$ of $Lu=0$ in $\Omega$ with
$\left\|u\right\|_{L^{\infty}(\Omega)}\leq 1$ and a ball $B=B(x_{0},r)$
centered on $\partial\Omega$ with radius $r$. By the same argument as Step 1
in in Subection 1.3, it suffices to show that there exists some constant
$C\in(0,\infty)$ depending only on $n$, $M$ and the UR constants of
$\partial\Omega$, such that

(8.1)
$I:=\sum_{Q\in\mathbb{D}_{\partial\Omega}(Q_{0})}\iint_{W_{\Omega}(Q)}\left|\nabla
u(X)\right|^{2}\delta(X)dX\leq C\sigma(Q_{0})$

for any cube $Q_{0}\in\mathbb{D}_{\partial\Omega}$ that satisfies
$Q_{0}\subset\frac{8}{7}B\cap\partial\Omega$ and $\ell(Q_{0})\leq 2^{-8}r$.

Then observe that if $E\subset\Omega$ is a Whitney region, that is,
$E\subset\frac{7}{4}B$ and $\operatorname{diam}(E)\leq K\delta(E)$, then

(8.2) $\iint_{E}\left|\nabla u\right|^{2}\delta\,dX\leq
C_{K}\operatorname{diam}(E)^{-1}\iint_{E^{*}}\left|u\right|^{2}dX\leq
C_{K}\delta(E)^{n-1},$

by the Caccioppoli inequality and $\left\|u\right\|_{L^{\infty}(\Omega)}\leq
1$, where $E^{*}$ is an enlargement of $E$. This bound (8.2) is the analogue
of (1.26), and proves Step 2.

Step 3 is not modified. We pick $0<\epsilon_{1}\ll\epsilon_{1}\ll 1$ and we
use the corona decomposition constructed in Section 3 to decompose $I$ as
follows.

$I=\sum_{Q\in\mathcal{B}(Q_{0})}\iint_{W_{\Omega}(Q)}\left|\nabla
u\right|^{2}\delta\,dX+\sum_{\mathcal{S}\in\mathfrak{S}(Q_{0})}\iint_{W_{\Omega}(\mathcal{S})}\left|\nabla
u\right|^{2}\delta\,dX=:I_{1}+\sum_{\mathcal{S}\in\mathfrak{S}(Q_{0})}I_{\mathcal{S}}.$

By (8.2) and (3.19),

$I_{1}\leq C\sum_{Q\in\mathcal{B}(Q_{0})}\ell(Q)^{n-1}\leq C\sigma(Q_{0}).$

Step 4 is significantly simpler for Theorem 1.21, because we do not need any
estimate on the smooth distance $D_{\beta}$, but the spirit is the same. That
is, by using the bi-Lipschitz map $\rho_{\mathcal{S}}$ constructed in Section
6, $I_{\mathcal{S}}$ can be turned into an integral on
$\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}$, which can be estimated by an
integration by parts argument. More precisely, for any fixed
$\mathcal{S}\in\mathfrak{S}(Q_{0})$,

$I_{\mathcal{S}}=\iint_{\rho_{\mathcal{S}}^{-1}(W_{\Omega}(\mathcal{S}))}\left|(\nabla
u)\circ\rho_{\mathcal{S}}(p,t)\right|^{2}\delta\circ\rho_{\mathcal{S}}(p,t)\det\operatorname{Jac}(p,t)dpdt\\\
\leq
2\iint\left|\nabla(u\circ\rho_{\mathcal{S}}(p,t))\right|^{2}\operatorname{dist}(\rho_{\mathcal{S}}(p,t),\Gamma_{\mathcal{S}})\left(\Psi_{\mathcal{S}}\circ\rho_{\mathcal{S}}(p,t)\right)^{2}dpdt\\\
\leq 3\iint\left|\nabla
v(p,t)\right|^{2}\left|t\right|\phi(p,t)^{2}dpdt,\qquad
v=u\circ\rho_{\mathcal{S}},\quad\phi=\Psi_{\mathcal{S}}\circ\rho_{\mathcal{S}}$

by (4.15), Lemmata 4.1 (d) and 6.8, as well as (6.12), for $\epsilon_{0}$
sufficiently small.

The fifth step consists roughly in proving the result in
$\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}$. The function $\phi$ is the same as
the one used to prove Theorem 1.12, in particular it is a cutoff function
associated to both $\rho_{\mathcal{S}}^{-1}(\partial\Omega)$ and
$\mathbb{R}^{n-1}$ as defined in Definition 7.1, and it satisfies

(8.3)
$\operatorname{supp}\phi\subset\rho_{\mathcal{S}}^{-1}(W_{\Omega}^{*}(\mathcal{S})),$

and

(8.4)
$\iint\left|\nabla\phi\right|dtdp+\iint\left|\nabla\phi\right|^{2}tdtdp\lesssim\sigma(Q(\mathcal{S})),$

where the implicit constant depends on $n$ and the AR constant in (1.1).
Notice that $v=u\circ\rho_{\mathcal{S}}$ is a bounded solution of
$L_{\mathcal{S}}=-\operatorname{div}\mathcal{A}_{\mathcal{S}}\nabla$ that
satisfies $\left\|v\right\|_{L^{\infty}}\leq 1$, where
$\mathcal{A}_{\mathcal{S}}$ is defined in (6.18). By Lemma 6.20,
$I_{\mathcal{S}}\leq C\sigma(Q(\mathcal{S}))$ will follow from the following
lemma, which is essentially a result in
$\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}$.

######  Lemma 8.5.

Let $L=-\operatorname{div}\mathcal{A}\nabla$ be a uniformly elliptic operator
on $\Omega_{\mathcal{S}}:=\rho^{-1}_{\mathcal{S}}(\Omega)$. Assume that the
coefficients $\mathcal{A}$ can be decomposed as
$\mathcal{A}=\mathcal{B}+\mathcal{C}$ where

(8.6)
$(|t\nabla\mathcal{B}|+|\mathcal{C}|){\mathds{1}}_{\operatorname{supp}\phi}\in
CM_{\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}}(M),$

where $\phi=\Psi_{\mathcal{S}}\circ\rho_{\mathcal{S}}$ is as above. Then for
any solution $v$ of $Lv=0$ in $\rho^{-1}_{\mathcal{S}}(\Omega)$ that satisfies
$\left\|v\right\|_{L^{\infty}}\leq 1$, there holds

(8.7) $\iint_{\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}}\left|\nabla
v\right|^{2}\phi^{2}{|t|}\,dtdy\leq C(1+M)\sigma(Q(\mathcal{S})),$

where $C$ depends only on the dimension $n$, the elliptic constant
$C_{\mathcal{A}}$, the AR constant of $\partial\Omega$, and the implicit
constant in (8.4).

The proof of this lemma is similar to the proof of Lemma 7.9, except that
there is no need to invoke the CFMS estimates and $A_{\infty}$ as in Lemma
7.12, essentially because $v$ is bounded and we do not need information of $v$
on the boundary. For the same reason, with the properties of the cutoff
function $\phi$ in mind, we can forget about the domain
$\Omega_{\mathcal{S}}$, and in particular, we do not need the corkscrew and
Harnack chain conditions in the proof.

###### Proof of Lemma 8.5.

We can decompose
$\phi=\phi\,{\mathds{1}}_{t>0}+\phi\,{\mathds{1}}_{t<0}:=\phi_{+}+\phi_{-}$
and prove the result for each of the functions $\phi_{+}$ and $\phi_{-}$, and
since the proof is the same in both cases (up to a sign), we can restrain
ourselves as in the proof of Lemma 7.9 to the case where
$\phi=\phi{\mathds{1}}_{t>0}$. By an approximation argument as in Step 1 of
the proof of Lemma 7.9, we can assume that
$T:=\iint_{\mathbb{R}^{n}_{+}}\left|\nabla v\right|^{2}t\phi^{2}dydt$ is
finite, and that $\phi$ is compactly supported in
$\Omega\cap\mathbb{R}^{n}_{+}$. We first assume that $\mathcal{B}$ has the
special structure that

(8.8) $\mathcal{B}_{ni}=0\qquad\text{for all }1\leq i\leq
n-1,\qquad\mathcal{B}_{nn}=b.$

Then for any $f\in W_{0}^{1,2}(\mathbb{R}^{n}_{+})$,

(8.9) $\iint\frac{\mathcal{B}}{b}\nabla f\cdot\nabla
t\,dydt=\iint\partial_{t}f\,dydt=0.$

Using ellipticity of $\mathcal{A}$ and boundeness of $b$, we write

$T\leq C_{\mathcal{A}}^{2}\iint\frac{\mathcal{A}}{b}\nabla v\cdot\nabla
v\,\phi^{2}t\,dydt\\\ =C_{\mathcal{A}}^{2}\Big{\\{}\iint\mathcal{A}\nabla
v\cdot\nabla\left(v\phi^{2}b^{-1}t\right)dydt-\iint\mathcal{A}\nabla
v\cdot\nabla\left(\phi^{2}b^{-1}\right)vt\,dydt-\iint\mathcal{A}\nabla
v\cdot\nabla t\,v\phi^{2}b^{-1}dydt\Big{\\}}\\\
=-C_{\mathcal{A}}^{2}\Big{\\{}\iint\mathcal{A}\nabla
v\cdot\nabla\left(\phi^{2}b^{-1}\right)vt\,dydt+\iint\mathcal{A}\nabla
v\cdot\nabla
t\,v\phi^{2}b^{-1}dydt\Big{\\}}=:-C_{\mathcal{A}}^{2}\left(T_{1}+T_{2}\right)$

since $Lv=0$. We write $T_{1}$ as

$T_{1}=2\iint\mathcal{A}\nabla v\cdot\nabla\phi\,\phi
b^{-1}vt\,dydt-\iint\mathcal{A}\nabla v\cdot\nabla
b\,\phi^{2}b^{-2}vt\,dydt=:T_{11}-T_{12}.$

By Cauchy-Schwarz and Young’s inequalities, as well as the boundedness of $v$
and $b$,

$\left|T_{11}\right|\leq\frac{C_{\mathcal{A}}^{-2}}{6}T+C\iint\left|\nabla\phi\right|^{2}t\,dydt,\qquad\left|T_{12}\right|\leq\frac{C_{\mathcal{A}}^{-2}}{8}T+C\iint\left|\nabla
b\right|^{2}t\phi^{2}dtdy.$

So (8.4) and (8.6), as well as (8.3) give that

$\left|T_{1}\right|\leq\frac{C_{\mathcal{A}}^{2}}{4}T+C\ell(Q(\mathcal{S}))^{n-1}.$

For $T_{2}$, we write

$T_{2}=\frac{1}{2}\iint\frac{\mathcal{A}}{b}\nabla\left(v^{2}\phi^{2}\right)\cdot\nabla
t\,dydt-\iint\frac{\mathcal{A}}{b}\nabla\phi\cdot\nabla t\,v^{2}\phi\,dydt\\\
=\frac{1}{2}\iint\frac{\mathcal{C}}{b}\nabla\left(v^{2}\phi^{2}\right)\cdot\nabla
t\,dydt-\iint\frac{\mathcal{A}}{b}\nabla\phi\cdot\nabla
t\,v^{2}\phi\,dydt=:T_{21}+T_{22}$

by writing $\mathcal{A}=\mathcal{B}+\mathcal{C}$ and applying (8.9). For
$T_{21}$, we use Cauchy-Schwarz and Young’s inequalities, and get

$\left|T_{21}\right|\leq\left|\iint\mathcal{C}\nabla v\cdot\nabla
t\,v\phi^{2}b^{-1}dydt\right|+2\left|\iint\mathcal{C}\nabla\phi\cdot\nabla
t\,v^{2}\phi b^{-1}dydt\right|\\\
\leq\frac{C_{\mathcal{A}}^{-2}}{4}T+C\iint\left|\mathcal{C}\right|^{2}\phi^{2}t^{-1}dydt+\iint\left|\nabla\phi^{2}\right|tdydt\leq\frac{C_{\mathcal{A}}^{-2}}{4}T+C\ell(Q(\mathcal{S}))^{n-1}$

by the boundedness of $v$, (8.4), (8.6), and (8.3). The boundedness of the
coefficients and $v$ implies that

$\left|T_{22}\right|\leq C\iint\left|\nabla\phi\right|dydt\leq
C\ell(Q(\mathcal{S}))^{n-1}$

by (8.4). Altogether, we have obtained that
$T\leq\frac{1}{2}T+C\ell(Q(\mathcal{S}))^{n-1}$, and thus the desired estimate
(8.7) follows.

We claim that the lemma reduces to the case when (8.8) holds by almost the
same argument as in Steps 2 and 3 in the proof of Lemma 7.9. That is, we can
assume that
$\left\|\left|\nabla_{y}\mathcal{B}\right|t\right\|_{\infty}\lesssim\frac{C_{\mathcal{A}}}{N}$
with $N$ to be chosen to be sufficiently large, and then we do a change of
variables, which produces the structure (8.8) in the conjugate operator. The
only difference from the proof of Lemma 7.9 is that now we need to choose
$v=-B_{3}/b$ in the bi-Lipschitz map $\rho$ defined in (7.27) because we want
$B_{3}+bv=0$ in (7.33). We leave the details to the reader. ∎

##  9\. The converse

In this section, we show that $(v)\implies(i)$ in Theorem 1.12, that is, we
establish that under certain conditions on the domain $\Omega$ and the
operator $L$, the Carleson condition (1.14) on the Green function implies that
$\partial\Omega$ is uniformly rectifiable. More precisely, we prove the
following.

######  Theorem 9.1.

Let $\Omega$ be a 1-sided Chord-Arc Domain (bounded or unbounded) and let
$L=-\mathop{\operatorname{div}}\mathcal{A}\nabla$ be an uniformly elliptic
operator which satisfies the weak DKP condition with constant $M\in(0,\infty)$
on $\Omega$. Let $X_{0}\in\Omega$, and when $\Omega$ is unbounded, $X_{0}$ can
be $\infty$. We write $G^{X_{0}}$ for the Green funtion of $L$ with pole at
$X_{0}$. Suppose that there exists $C\in(0,\infty)$ and $\beta>0$ such that
for all balls $B$ centered at the boundary and such that $X_{0}\notin 2B$, we
have

(9.2) $\iint_{\Omega\cap B}\left|\frac{\nabla
G^{X_{0}}}{G^{X_{0}}}-\frac{\nabla
D_{\beta}}{D_{\beta}}\right|^{2}D_{\beta}\,dX\leq
C\sigma(B\cap\partial\Omega).$

Then $\partial\Omega$ is uniformly rectifiable.

In [DM2] Theorem 7.1, uniform rectifiability is obtained from some weak
condition on the Green function, namely, $G^{\infty}$ being prevalently close
to $D_{\beta}$. Following [DM2], we say that $G^{\infty}$ is prevalently close
to $D_{\beta}$ if for each choice of $\epsilon>0$ and $M\geq 1$, the set
$\mathcal{G}_{GD_{\beta}}(\epsilon,M)$ of pairs
$(x,r)\in\partial\Omega\times(0,\infty)$ such that there exists a positive
constant $c>0$, with

$\left|D_{\beta}(X)-c\,G^{\infty}(X)\right|\leq\epsilon r\quad\text{for
}X\in\Omega\cap B(x,Mr),$

is Carleson-prevalent.

######  Definition 9.3 (Carleson-prevalent).

We say that $\mathcal{G}\subset\partial\Omega\times(0,\infty)$ is a Carleson-
prevalent set if there exists a constant $C\geq 0$ such that for every
$x\in\partial\Omega$ and $r>0$,

$\int_{y\in\partial\Omega\cap
B(x,r)}\int_{0<t<r}{\mathds{1}}_{\mathcal{G}^{c}}(y,t)\frac{d\sigma(y)dt}{t}\leq
C\,r^{n-1}.$

One could say that our condition (9.2) is stronger than $G^{\infty}$ being
prevalently close to $D_{\beta}$, and so the theorem follows from [DM2]. But
actually it is not so easy to link the two conditions directly. Nonetheless,
we can use Chebyshev’s inequality to derive a weak condition from (9.2), which
can be used as a replacement of $G^{\infty}$ being prevalently close to
$D_{\beta}$ in the proof.

We will soon see that the condition on the operator in Theorem 9.1 can be
relaxed. Again following [DM2], given an elliptic operator
$L=-\mathop{\operatorname{div}}{\mathcal{A}\nabla}$, we say that $L$ is
locally sufficiently close to a constant coefficient elliptic operator if for
every choice of $\tau>0$ and $K\geq 1$, $\mathcal{G}_{cc}(\tau,K)$ is a
Carleson prevalent set, where $\mathcal{G}_{cc}(\tau,K)$ is the set of pairs
$(x,r)\in\partial\Omega\times(0,\infty)$ such that there is a constant matrix
$\mathcal{A}_{0}=\mathcal{A}_{0}(x,r)$ such that

$\iint_{X\in W_{K}(x,r)}\left|\mathcal{A}(X)-\mathcal{A}_{0}\right|dX\leq\tau
r^{n},$

where

(9.4) $W_{K}(x,r)=\left\\{X\in\Omega\cap
B(x,Kr):\operatorname{dist}(X,\partial\Omega)\geq K^{-1}r\right\\}.$

We will actually prove Theorem 9.1 for elliptic operators $L$ that are
sufficiently close locally to a constant coefficient elliptic operator.

The first step of deriving weak conditions from the strong conditions on the
operator and $G^{\infty}$ is the observation that for any integrable function
$F$, if there is a constant $C\in(0,\infty)$ such that

$\iint_{B(x,r)\cap\Omega}\left|F(Y)\right|dY\leq C\,r^{n-1}\quad\text{for
}x\in\partial\Omega,r>0,$

then for any $K\geq 1$,

(9.5) $\int_{y\in
B(x,r)\cap\partial\Omega}\int_{0<t<r}\mathchoice{{\vbox{\hbox{$\textstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-6.94159pt}}{{\vbox{\hbox{$\scriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-3.65501pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.82753pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.06253pt}}\\!\int\\!\\!\\!\\!\\!\int_{W_{K}(y,t)}\left|F(Y)\right|dY\,dt\,d\sigma(y)\leq
C\,K^{n-1}r^{n-1}$

for $x\in\partial\Omega$, $r>0$. This follows immediately from Fubini’s
theorem and the fact that $W_{K}(x,r)$ defined in (9.4) is a Whitney region
which is away from the boundary.

######  Lemma 9.6.

  1. (1)

Let $L=-\mathop{\operatorname{div}}\mathcal{A}\nabla$ be a uniformly elliptic
operator which satisfies the weak DKP condition with constant $M\in(0,\infty)$
on $\Omega$. Then $L$ is locally sufficiently close to a constant coefficient
elliptic operator.

  2. (2)

If $G^{X_{0}}$ satisfies (9.2) for all $B$ centered at the boundary and such
that $X_{0}\notin 2B$, then for every choice of $\epsilon>0$ and $K\geq 1$,
the set

(9.7)
$\mathcal{G}^{X_{0}}(\epsilon,K):=\Big{\\{}(x,r)\in\partial\Omega\times(0,\infty):\,X_{0}\notin
B(x,2Kr)\text{ and }\\\
\iint_{W_{K}(x,r)}\left|\nabla\ln\left(\frac{G^{X_{0}}}{D_{\beta}}(X)\right)\right|^{2}D_{\beta}(X)dX\leq\epsilon\,r^{n-1}\Big{\\}}$

is Carleson-prevalent.

###### Proof.

Both results follow from the previous observation (9.5) and Chebyshev’s
inequality. In fact, for (1), we have $\mathcal{A}=\mathcal{B}+\mathcal{C}$
such that for any $x\in\partial\Omega$ and $r>0$,

(9.8) $\int_{y\in
B(x,r)\cap\Omega}\int_{0<t<r}\mathchoice{{\vbox{\hbox{$\textstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-6.94159pt}}{{\vbox{\hbox{$\scriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-3.65501pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.82753pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.06253pt}}\\!\int\\!\\!\\!\\!\\!\int_{W_{K}(y,t)}\left(\left|\nabla\mathcal{B}\right|^{2}\delta+\left|\mathcal{C}\right|^{2}\frac{1}{\delta}\right)dY\,dt\,d\sigma(y)\leq
M\,K^{n-1}r^{n-1}.$

By the Poincaré inequality, the left-hand side is bounded from below by

$c\int_{y\in
B(x,r)\cap\Omega}\int_{0<t<r}\mathchoice{{\vbox{\hbox{$\textstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-6.94159pt}}{{\vbox{\hbox{$\scriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-3.65501pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.82753pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.06253pt}}\\!\int\\!\\!\\!\\!\\!\int_{W_{K}(y,t)}\left(\left|\mathcal{B}-(\mathcal{B})_{W_{K}(y,t)}\right|^{2}+\left|\mathcal{C}\right|^{2}\right)dY(Kt)^{-1}dt\,d\sigma(y)\\\
\geq\frac{c}{2}\int_{y\in
B(x,r)\cap\Omega}\int_{0<t<r}\mathchoice{{\vbox{\hbox{$\textstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-6.94159pt}}{{\vbox{\hbox{$\scriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-3.65501pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.82753pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.06253pt}}\\!\int\\!\\!\\!\\!\\!\int_{W_{K}(y,t)}\left(\left|\mathcal{A}-(\mathcal{B})_{W_{K}(y,t)}\right|^{2}\right)dY(Kt)^{-1}dt\,d\sigma(y)\\\
\geq\frac{c}{2}\frac{\tau}{K^{n+1}}\int_{y\in
B(x,r)\cap\Omega}\int_{0<t<r}{\mathds{1}}_{\mathcal{G}_{cc}(\tau,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t},$

where we have used the fact that $(\mathcal{B})_{W_{K}(y,t)}$ is a constant
matrix and the definition of the set $\mathcal{G}_{cc}(\tau,K)$. Combining
with (9.8), we have that

$\int_{y\in
B(x,r)\cap\Omega}\int_{0<t<r}{\mathds{1}}_{\mathcal{G}_{cc}(\tau,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t}\leq\frac{CMK^{2n}}{\tau}r^{n-1},$

which proves (1).

Now we justify (2). Let $\epsilon>0$, $K\geq 1$ and $X_{0}\in\Omega$ be fixed,
and let $B$ be a ball of radius $r$ centered at the boundary. Our goal is to
show that

$\int_{y\in
B\cap\partial\Omega}\int_{0<t<r}{\mathds{1}}_{\mathcal{G}^{X_{0}}(\epsilon,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t}\leq
C_{\epsilon,K}\sigma(B\cap\partial\Omega).$

We discuss two cases. If $X_{0}\notin 4KB$, then since $G^{X_{0}}$ satisfies
(9.2) for the ball $2KB$, we have that

$\int_{y\in
B\cap\partial\Omega}\int_{0<t<r}\mathchoice{{\vbox{\hbox{$\textstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-6.94159pt}}{{\vbox{\hbox{$\scriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-3.65501pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.82753pt}}{{\vbox{\hbox{$\scriptscriptstyle\mkern-1.5mu{-}\mkern-7.5mu{-}$}}\kern-1.06253pt}}\\!\int\\!\\!\\!\\!\\!\int_{W_{K}(y,t)}\left|\nabla\ln\left(\frac{G^{X_{0}}}{D_{\beta}}\right)(Y)\right|^{2}D_{\beta}(Y)dYdtd\sigma(y)\leq
CK^{n-1}r^{n-1}.$

Notice that the assumption $X_{0}\notin 4KB$ guarantees that $X_{0}\notin
B(y,2Kt)$ for all $y\in B\cap\partial\Omega$ and $0<t<r$. Therefore, if
$(y,t)\in\mathcal{G}^{X_{0}}(\epsilon,K)^{c}$, then

$\iint_{W_{K}(y,t)}\left|\nabla\ln\left(\frac{G^{X_{0}}}{D_{\beta}}\right)(Y)\right|^{2}D_{\beta}(Y)dY>\epsilon\,r^{n-1}.$

From this, it follows that

(9.9) $\int_{y\in
B\cap\partial\Omega}\int_{0<t<r}{\mathds{1}}_{\mathcal{G}^{X_{0}}(\epsilon,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t}\leq\frac{CK^{2n-1}}{\epsilon}\sigma(B\cap\partial\Omega).$

Now let us deal with the case where $X_{0}\in 4KB$. For $x\in
B\cap\partial\Omega$, we define $B_{x}:=B(x,|x-X_{0}|/20K)$. Since
$\\{B_{x}\\}_{x\in B}$ covers $B\cap\partial\Omega$, we can find a non-
overlapping subcollection $\\{B_{i}\\}_{i\in I}$ such that $\\{5B_{i}\\}_{i\in
I}$ covers $B\cap\partial\Omega$. We write $r_{i}>0$ for the radius of $B_{i}$
and we define

$S:=(B\cap\partial\Omega)\times(0,r)\setminus\bigcup_{i\in
I}(5B_{i}\cap\partial\Omega)\times(0,5r_{i})$

We have

$\int_{y\in
B\cap\partial\Omega}\int_{0<t<r}{\mathds{1}}_{\mathcal{G}^{X_{0}}(\epsilon,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t}\leq\sum_{i\in
I}\int_{y\in
5B_{i}\cap\partial\Omega}\int_{0<t<5r_{i}}{\mathds{1}}_{\mathcal{G}^{X_{0}}(\epsilon,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t}\\\
+\iint_{S}{\mathds{1}}_{\mathcal{G}^{X_{0}}(\epsilon,K)^{c}}(y,t)\frac{dt\,d\sigma(y)}{t}=:T_{1}+T_{2}.$

Since $X_{0}\notin 20KB_{i}$, we can apply (9.9), and we have

$T_{1}\leq C_{K,\epsilon}\sum_{i\in
I}\sigma(5B_{i}\cap\partial\Omega)\lesssim\sum_{i\in
I}\sigma(B_{i}\cap\partial\Omega)\leq\sigma(2B\cap\partial\Omega)\lesssim\sigma(B\cap\partial\Omega),$

because $\\{B_{i}\\}$ is a non-overlapping and included in $2B$. It remains to
prove a similar bound on $T_{2}$. Remark first that

$S\subset\\{(y,t)\in\partial\Omega\times(0,r):\,|y-X_{0}|/100K<t\\},$

and therefore

$T_{2}\leq\int_{0}^{r}\int_{y\in
B(X_{0},100Kt)\cap\partial\Omega}\frac{d\sigma(y)\,dt}{t}\leq
CK^{n-1}r^{n-1}\lesssim\sigma(B\cap\partial\Omega).$

The lemma follows. ∎

Before we continue, we need to adapt Theorem 2.19 in [DM2] to our situation,
that is we want to construct a positive solution in a domain which is the
limit of a sequence of domain.

######  Lemma 9.10.

Let $\Omega_{k}$ be a sequence of 1-sided Chord-Arc domains domains in
$\mathbb{R}^{n}$ with uniform 1-sided CAD constants. Let $\partial\Omega_{k}$
be its Ahlfors regular boundary equipped with a Ahlfors regular measure
$\sigma_{k}$ (such that the constant in (1.1) is uniform in $k$).

Assume that $0\in\partial\Omega_{k}$ and $\operatorname{diam}\Omega_{k}\geq
2^{k}$. Moreover, assume that the $\partial\Omega_{k}$ and $\Omega_{k}$
converges to $E_{\infty}$ and $\Omega_{\infty}$ locally in the Hausdorff
distance, that is, for any $j\in\mathbb{N}$, we have

$\lim_{k\to\infty}d_{0,2^{j}}(E_{\infty},\partial\Omega_{k})=0\text{ and
}\lim_{k\to\infty}d_{0,2^{j}}(\Omega_{\infty},\Omega_{k})=0.$

Here, for a couple of sets $(E,F)$, we define the Hausdorff distance

$d_{0,2^{j}}(E,F):=\sup_{x\in E\cap
B(0,2^{j})}\operatorname{dist}(x,F)+\sup_{y\in F\cap
B(0,2^{j})}\operatorname{dist}(y,E).$

Then $E_{\infty}=\partial\Omega_{\infty}$, $E_{\infty}$ is an unbounded
$(n-1)$-Ahlfors regular set, $\Omega_{\infty}$ is a 1-sided Chord-Arc Domain.
Moreover, if the Radon measure $\sigma$ is any weak-* limit of the
$\sigma_{k}$, then $\sigma$ is an Ahlfors regular measure on
$E_{\infty}=\partial\Omega_{\infty}$.

Let $Y_{0}$ be a corkscrew point of $\Omega_{\infty}$ for the boundary point
$0$ at the scale 1. If $L_{k}=-\operatorname{div}A_{k}\nabla$ and
$L_{\infty}=-\operatorname{div}A_{\infty}\nabla$ are operators - in
$\Omega_{k}$ and $\Omega_{\infty}$ respectively - that satisfies

$\lim_{k\to\infty}\|A_{k}-A_{\infty}\|_{L^{1}(B)}=0\qquad\text{ for any ball
$B$ such that $2B\subset\Omega_{\infty}$},$

and if $u_{k}$ are positive solutions in $\Omega_{k}\cap B(0,2^{k+1})$ to
$L_{k}u_{k}=0$ with $\operatorname{Tr}u_{k}=0$ on $\partial\Omega_{k}\cap
B(0,2^{k+1})$, then the sequence of functions $v_{k}:=u_{k}/u_{k}(Y_{0})$
converges, uniformly on every compact subset of $\Omega_{\infty}$, and in
$W^{1,2}_{\operatorname{loc}}(\Omega_{\infty})$, to $G^{\infty}$, the unique
Green function with pole at infinity which verifies $G^{\infty}(Y_{0})=1$.

###### Proof.

The geometric properties of $E_{\infty}$ and $\Omega_{\infty}$ can be derived
verbatim as in the proof of Theorem 2.19 in [DM2]. The uniform convergence of
a subsequence of $v_{k}$ on any compact set $K\Subset\Omega_{\infty}$ follows
from the standard argument of uniform boundedness of $\left\\{v_{k}\right\\}$
on $K$, and Hölder continuity of solutions. The Caccioppoli inequality would
give the weak convergence of another subsequence of $v_{k}$ to some
$v_{\infty}$ in $W_{\operatorname{loc}}^{1,2}(\Omega_{\infty})$. This is
enough to show that $v_{\infty}\in
W_{\operatorname{loc}}^{1,2}(\Omega_{\infty})\cap
C(\overline{\Omega_{\infty}})$ is a weak solution of $L_{\infty}v_{\infty}=0$
in $\Omega_{\infty}$, as we can write

$\iint_{\Omega_{\infty}}A_{\infty}\nabla v_{\infty}\cdot\nabla\varphi
dX=\iint_{\Omega_{\infty}}A_{\infty}(\nabla v_{\infty}-\nabla
v_{k})\cdot\nabla\varphi\,dX+\iint_{\Omega_{\infty}}(A_{\infty}-A_{k})\nabla
v_{k}\cdot\nabla\varphi\,dX$

for every $\varphi\in C_{0}^{\infty}(\Omega_{\infty})$ and any $k$
sufficiently big so that $\operatorname{supp}\varphi\subset\Omega_{k}\cap
B(0,2^{k+1})$. Therefore, $v_{\infty}=G^{\infty}$ is the Green function with
pole at infinity for $L_{\infty}$ in $\Omega_{\infty}$ and normalized so that
$G^{\infty}(Y_{0})=1$.

That $v_{k}$ converges to $G^{\infty}$ (strongly) in
$W_{\operatorname{loc}}^{1,2}(\Omega_{\infty})$ needs more work, but we can
directly copy the proof of Lemma 2.29 in [DM2]. Roughly speaking, for any
fixed ball $B$ with $4B\subset\Omega$, we would need to introduce an
intermediate function $V_{k}$, which satisfies $L_{k}V_{k}=0$ in $B_{\rho}$
for some $\rho\in(r,2r)$, and $V_{k}=v_{k}$ on the sphere $\partial B_{\rho}$.
We refer the readers to [DS2] for the details. ∎

We shall need the following result on compactness of closed sets, which has
been proved in [DS3].

######  Lemma 9.11 ([DS3] Lemma 8.2).

Let $\left\\{E_{j}\right\\}$ be a sequence of non-empty closed subsets of
$\mathbb{R}^{n}$, and suppose that there exists an $r>0$ such that $E_{j}\cap
B(0,r)\neq\emptyset$ for all $j$. Then there is a subsequence of
$\left\\{E_{j}\right\\}$ that converges to a nonempty closed subset $E$ of
$\mathbb{R}^{n}$ locally in the Hausdorff distance.

Now we are ready to prove the main theorem of this section.

###### Proof of Theorem 9.1.

We prove that $\partial\Omega$ is uniformly rectifiable by showing that
$\Omega_{\rm ext}$ satisfies the corkscrew condition (see Lemma 2.13).
Following the proof of Theorem 7.1 in [DM2], it suffices to show that the set
$\mathcal{G}_{CB}(c)$ is Carleson-prevalent for some $c>0$, where
$\mathcal{G}_{CB}(c)$ is the set of pairs
$(x,r)\in\partial\Omega\times(0,\infty)$ such that we can find $Z_{1},Z_{2}\in
B(x,r)$, that lie in different connected components of
$\mathbb{R}^{n}\setminus\partial\Omega$, and such that
$\operatorname{dist}(Z_{i},\partial\Omega)\geq cr$ for $i=1,2$. To do that, we
will rely on the fact that, on 1-sided CAD domains, if the elliptic measure is
comparable to the surface measure, then the complement $\Omega_{\rm ext}$
satisfies the corkscrew condition, which is implied by the main result of
[HMMTZ].

Thanks to Lemma 9.6, for each choice of $\epsilon>0$ and $M\geq 1$, the sets
$\mathcal{G}^{X_{0}}(\epsilon,M)$ and $\mathcal{G}_{cc}(\epsilon,M)$ are
Carleson-prevalent. So it suffices to show that

(9.12)
$\mathcal{G}^{X_{0}}(\epsilon,M)\cap\mathcal{G}_{cc}(\epsilon,M)\subset\mathcal{G}_{CB}(c)\quad\text{for
some }c>0,\epsilon>0,\text{ and }M\geq 1.$

We prove by contradiction. Assume that (9.12) is false, then for
$c_{k}=\epsilon_{k}=M_{k}^{-1}=2^{-k}$, we can find a 1-sided NTA domain
$\Omega_{k}$ bounded by an Ahlfors regular set $\partial\Omega_{k}$, a point
$X_{k}\in\Omega_{k}$ (or $X_{k}\in\Omega_{k}\cup\left\\{\infty\right\\}$ when
$\Omega$ is unbounded), an elliptic operator
$L_{k}=-\mathop{\operatorname{div}}\mathcal{A}_{k}\nabla$ that is locally
sufficiently close to a constant coefficient elliptic operator, and a pair
$(x_{k},r_{k})\in\partial\Omega_{k}\times(0,\infty)$ for which

$(x_{k},r_{k})\in\mathcal{G}^{X_{k}}(\epsilon_{k},M_{k})\cap\mathcal{G}_{cc}(\epsilon_{k},M_{k})\setminus\mathcal{G}_{CB}(c_{k}).$

By translation and dilation invariance, we can assume that $x_{k}=0$ and
$r_{k}=1$. Notice that $(0,1)\in\mathcal{G}^{X_{k}}(\epsilon_{k},M_{k})$
implies that $X_{k}\notin B(0,2^{k})$, and in particular,
$\operatorname{diam}(\Omega_{k})\geq 2^{k}$, and $X_{k}$ tends to infinity as
$k\to\infty$.

By Lemma 9.11, we can extract a subsequence so that $\Omega_{k}$ converges to
a limit $\Omega_{\infty}$. By Lemma 9.10, $\Omega_{\infty}$ is 1-sided NTA,
$\partial\Omega_{k}$ converges to $\partial\Omega_{\infty}$ which is Ahlfors
regular. Moreover, by Lemma 9.11, we can extract a further subsequence so that
the Ahlfors regular measure $\sigma_{k}$ given on $\partial\Omega_{k}$
converges weakly to an Ahlfors regular measure $\sigma$. Since
$(0,1)\in\mathcal{G}_{cc}(2^{-k},2^{k})$, $\mathcal{A}_{k}$ converges to some
constant matrix $\mathcal{A}_{0}$ in
$L^{1}_{\operatorname{loc}}(\Omega_{\infty})$.

Choose a corkscrew point $Y_{0}\in\Omega_{\infty}$ for some ball $B_{0}$
centered on $\partial\Omega_{\infty}$, and let $G_{k}=G_{k}^{X_{k}}$ be the
Green function for $L_{k}$ in $\Omega_{k}$, normalized so that
$G_{k}(Y_{0})=1$. Since $L_{k}G_{k}=0$ in $\Omega_{k}\cap B(0,2^{k})$, Lemma
9.10 asserts that $G^{k}$ converges to the Green function
$G=G_{\infty}^{\infty}$ with pole at infinity for the constant-coefficient
operator $L_{0}=-\mathop{\operatorname{div}}\mathcal{A}_{0}\nabla$, uniformly
on compact sets of $\Omega_{\infty}$, and in
$W_{\operatorname{loc}}^{1,2}(\Omega_{\infty})$. Since
$\sigma_{k}\rightharpoonup\sigma$, $D_{k}=D_{\beta,\sigma_{k}}$ converges to
$D=D_{\beta,\sigma}$ uniformly on compact sets of $\Omega_{\infty}$, and so
does $\nabla D_{k}$ to $\nabla D$. Since
$(0,1)\in\mathcal{G}^{X_{k}}(2^{-k},2^{k})$,

(9.13) $\iint_{W_{2^{k}}(0,1)}\left|\frac{\nabla G_{k}}{G_{k}}-\frac{\nabla
D_{k}}{D_{k}}\right|^{2}D_{k}(X)dX\leq 2^{-k}\qquad\text{for all
}k\in\mathbb{Z}_{+},$

where $W_{2^{k}}(0,1)$ is the Whitney region defined as in (9.4) for
$\Omega_{k}$. Fix any compact set $K\Subset\Omega_{\infty}$. We claim that

(9.14) $\lim_{k\to\infty}\iint_{K}\left|\frac{\nabla
G_{k}}{G_{k}}-\frac{\nabla
D_{k}}{D_{k}}\right|^{2}D_{k}(X)dX=\iint_{K}\left|\frac{\nabla
G}{G}-\frac{\nabla D}{D}\right|^{2}D(X)dX.$

In fact, since $G$ is a positive solution of $L_{0}G=0$ in $\Omega_{\infty}$
with $G(Y_{0})=1$, the Harnack inequality implies that $G\geq c_{0}$ on $K$
for some $c_{0}>0$. Then the uniform convergence of $G_{k}$ to $G$ on $K$
implies that for $k$ large enough, $\left\\{G_{k}^{-1}\right\\}$ is uniformly
bounded on $K$, and so $G_{k}^{-1}$ converges uniformly to $G^{-1}$ on $K$.
Then (9.14) follows from the fact that $\nabla G_{k}$ converges to $\nabla G$
in $L^{2}(K)$, the uniform convergence of $G_{k}^{-1}$ to $G^{-1}$ on $K$, and
the uniform convergences of $\nabla D_{k}$ and $D_{k}^{-1}$ to $\nabla D$ and
$D^{-1}$.

Now by (9.13) and (9.14), we get that

$\iint_{K}\left|\nabla\ln\left(\frac{G}{D}\right)(X)\right|^{2}D(X)dX=0,$

and so $G=CD_{\beta,\sigma}$ in $\Omega_{\infty}$. We can copy the proof of
Theorem 7.1 of [DM2] verbatim from now on to conclude that this leads to a
contradiction. Roughly speaking, $G=CD_{\beta,\sigma}$ would imply that the
elliptic measure $\omega^{\infty}$ for $L_{0}$, with a pole at $\infty$, is
comparable to $\mathcal{H}_{|\partial\Omega_{\infty}}^{n-1}$. Then by [HMMTZ]
Theorem 1.6 one can conclude that $\partial\Omega_{\infty}$ is uniformly
rectifiable, and hence $\mathbb{R}^{n}\setminus\overline{\Omega}_{\infty}$
satisfies the corkscrew condition, which contradicts with the assumption that
$(0,1)=(x_{k},r_{k})\notin\mathcal{G}_{CB}(c_{k})$. ∎

##  10\. Assuming that $\Omega$ is semi-uniform is not sufficient.

In this subsection, we will give an example of domain where the harmonic
measure on $\partial\Omega$ is $A_{\infty}$-absolutely continuous with respect
to the $(n-1)$-dimensional Hausdorff measure, but where Theorem 1.12 fails. It
is known that the harmonic measure is $A_{\infty}$-absolute continuous with
respect to the surface measure whenever the domain $\Omega$ is semi-uniform
and its boundary is $(n-1)$-Ahlfors regular and uniformly rectifiable (see
[Azz2, Theorem III]). The notion of semi-uniform domain is given by the next
definition.

######  Definition 10.1 (Semi-uniform domains).

We say that $\Omega$ is semi-uniform if it satisfies the corkscrew condition
and (see Definition 2.8) if for every $\Lambda\geq 1$, there exists
$C_{\Lambda}>0$ such that for any $\rho>1$ and every pair of points
$(X,x)\in\Omega\times\partial\Omega$ such that $\left|X-x\right|<\Lambda\rho$,
there exists a Harnack chain of length bounded by $C_{\Lambda}$ linking $X$ to
one of the corkscrew points for $x$ at scale $\rho$.

Semi-uniform domains were first introduced by Aikawa and Hirata in [AiHi]
using cigar curves. The two definitions of semi-uniform domains are known to
be equivalent, see for instance, [Azz2] Theorem 2.3.

Our counterexample is constructed in $\mathbb{R}^{2}$ for simplicity but can
easily be extended to any dimension.

Figure 1. The domain $\Omega$

Our domain (see Figure 1) will be

$\Omega:=\mathbb{R}^{2}\setminus\bigcup_{k\in\mathbb{Z}}\Big{\\{}(x,t)\in\mathbb{R}^{2},|x-2k|+|t|<\frac{1}{2}\Big{\\}}$

Note that $\partial\Omega$ is uniformly rectifiable, but the domain contains
two parts ($\Omega\cap\mathbb{R}^{2}_{+}$ and $\Omega\cap\mathbb{R}^{2}_{-}$)
which are not well connected to each other, that is, this domain does not
satisfy the Harnack Chain Condition (see Definition 2.9). We let the reader
check that the domain is still semi-uniform.

Due to the lack of Harnack chains, the space $\Omega$ does not have a unique -
up to constant - Green function with pole at $\infty$. If we take the pole at
$t\to-\infty$, then we can construct a positive function $G$ which will be
bounded on $\Omega\cap\mathbb{R}^{2}_{+}$, and we shall prove that this is
incompatible with our estimate (1.17) that says that $\frac{\partial_{t}G}{G}$
is “close” to $\frac{1}{t}$ when $t$ is large enough.

###  10.1. Construction of $G$

The goal now will be to construct a positive function in $\Omega$, which is
morally the Green function with pole at $t=-\infty$. We could have used the
usual approach, that is taking the limit when $n$ goes to infinity of - for
instance - $G(X,X_{n})/G(X_{0},X_{n})$ in the right sense, where $G$ is the
Green function on $\Omega$ for the Laplacian, and $X_{n}:=(1,n)$. However, the
authors had difficulty proving the 2-periodicity in $x$ of the limit and
didn’t know where to find the right properties in the literature (as our
domains are unbounded). So we decided to make the construction from scratch.

We want to work with the Sobolev space

$W=\Big{\\{}u\in W^{1,2}_{loc}(\overline{\Omega}),\,u(x,t)=u(x+2,t)\text{
and}\,u(x,-t)=u(x,t)\text{ for $(x,t)\in\Omega$},\\\ \,\iint_{S_{0}}|\nabla
u(x,t)|^{2}dx\,dt<+\infty\Big{\\}}.$

Here and in the sequel $S_{k}$ is the strip
$\Omega\cap\Big{(}[k,k+1)\times\mathbb{R}\Big{)}$. Note that due to the
2-periodicity in $x$ and the symmetry, the function $u\in W$ is defined on
$\mathbb{R}^{2}$ as soon as $u$ is defined on any of the sets $S_{k}$. We will
also need

$W^{+}:=\\{u_{|\Omega\cap\mathbb{R}^{2}_{+}},\,u\in W\\}\text{ and
}W_{0}:=\\{u\in W,\,\operatorname{Tr}(u)=0\text{ on }\partial\Omega\\}.$

We let the reader check that the quantity

$\|u\|_{W}:=\left(\iint_{S_{0}}|\nabla u(x,t)|^{2}dx\,dt\right)^{\frac{1}{2}}$

is a norm on the space $W_{0}$, and the couple $(W_{0},\|.\|_{W})$ is a
Hilbert space.

The bilinear form

$a(u,v):=\iint_{S_{0}}\nabla u\cdot\nabla v\,dt\,dx$

is continuous and coercive on $W_{0}$, so for any $k\in\mathbb{N}$, there
exists $\widetilde{G}_{k}\in W_{0}$ such that

(10.2) $a(\widetilde{G}_{k},v)=\iint_{S_{0}}\nabla\widetilde{G}_{k}\cdot\nabla
v\,dx\,dt=2^{-k}\int_{0}^{1}\int_{-2^{k+1}}^{-2^{k}}v(x,t)\,dt\,dx\qquad\text{
for }v\in W_{0}.$

The first key observation is:

######  Proposition 10.3.

$\widetilde{G}_{k}\in W_{0}$ is a positive weak solution to $-\Delta u=0$ in
$\Omega\cap\\{t>-2^{k}\\}$.

Proof: The fact that $\widetilde{G}_{k}$ is nonnegative is a classical result
that relies on the fact that $u\in W_{0}\implies|u|\in W_{0}$ and the bilinear
form $a(u,v)$ is coercive. See for instance [DFM1], (10.18)–(10.20).

In order to prove that $\widetilde{G}_{k}$ is a solution in
$\Omega\cap\\{t>-2^{k}\\}$, take $\phi\in
C^{\infty}_{0}(\Omega\cap\\{t>-2^{k}\\})$. For $j\in\mathbb{Z}$, let
$\phi_{j}$ be the only symmetric and 2-periodic function in $x$ such that
$\phi_{j}=\phi$ on $S_{j}$. Observe that $\phi_{j}$ is necessary continuous,
and so $\phi_{j}$ lies in $W_{0}$. Thus

$\iint_{\Omega}\nabla\widetilde{G}_{k}\cdot\nabla\phi\,dx\,dt=\sum_{j\in\mathbb{Z}}\iint_{S_{j}}\nabla\widetilde{G}_{k}\cdot\nabla\phi\,dx\,dt=\sum_{j\in\mathbb{Z}}\iint_{S_{j}}\nabla\widetilde{G}_{k}\cdot\nabla\phi_{j}\,dx\,dt\\\
=\sum_{j\in\mathbb{Z}}\iint_{S_{0}}\nabla\widetilde{G}_{k}\cdot\nabla\phi_{j}\,dx\,dt=0$

by (10.2), since $\phi_{j}=\phi\equiv 0$ on $\\{t\leq-2^{k}\\}$ for all
$j\in\mathbb{Z}$.

Since $\widetilde{G}_{k}$ is a solution, which is nonnegative and not
identically equal to 0 (otherwise (10.2) would be false), the Harnack
inequality (Lemma 2.15) entails that $\widetilde{G}_{k}$ is positive. The
proposition follows. $\square$

Let $X_{0}:=(1,0)\in\Omega$. From the above proposition,
$\widetilde{G}_{k}(X_{0})>0$ so we can define

(10.4) $G_{k}(X):=\frac{\widetilde{G}_{k}(X)}{\widetilde{G}_{k}(X_{0})}.$

######  Proposition 10.5.

For each $k\in\mathbb{N}$, the function $G_{k}(X)\in W_{0}$ is a positive weak
solution to $-\Delta u=0$ in $\Omega\cap\\{t>-2^{k}\\}$. Moreover, we have the
following properties:

  1. (i)

for any compact set $K\Subset\overline{\Omega}$, there exists $k:=k(K)$ and
$C:=C(K)$ such that $G_{j}(X)\leq C_{K}$ for all $j\geq k$ and $X\in K$ and
$\\{G_{j}\\}_{j\geq k}$ is equicontinuous on $K$;

  2. (ii)

there exists $C>0$ such that

$\iint_{\Omega\cap([-2,2]\times[-1,1])}|\nabla G_{k}(x,t)|^{2}dx\,dt\leq
C\qquad\text{ for all }k\in\mathbb{N};$

  3. (iii)

there exists $C>0$ such that

$\|G_{k}\|_{W^{+}}^{2}:=\iint_{S_{0}\cap\mathbb{R}^{2}_{+}}|\nabla
G_{k}|^{2}dx\,dt\leq C\qquad\text{ for all }k\in\mathbb{N}.$

Proof: The fact that $G_{k}$ is a positive weak solution is given by
Proposition 10.3. So it remains to prove (i), (ii) and (iii).

We start with (i). Since $G_{k}$ is a weak solution in
$\Omega_{0}:=\Omega\cap[(-4,4)\times(-2,2)]$ when $k\geq 1$, and since
$\Omega_{0}$ is a Chord Arc Domain, we can invoke the classical elliptic
theory and we can show that there exists $C>0$ such that

$\sup_{\Omega\cap([-2,2]\times[-1,1])}G_{k}\leq CG_{k}(1,0)=C\quad\text{for
all }k\geq 1,$

see for instance Lemma 15.14 in [DFM3]. By the 2-periodicity of $G_{k}$, it
means that

$\sup_{k\geq 1}\sup_{\Omega\cap(\mathbb{R}\times[-1,1])}G_{k}\leq C,$

and then since we can link any point of a compact $K\Subset\overline{\Omega}$
back to $\Omega\cap(\mathbb{R}\times[-1,1])$ with a Harnack chain (the length
of the chain depends on $K$), we have

$\sup_{j\geq k}\sup_{K}G_{j}\leq C_{K},$

whenever $G_{j}$ is a solution in the interior of $K$, which is bound to
happen if $j\geq k(K)$ is large enough.

The functions $G_{k}$ are also Hölder continuous up to the boundary in the
areas where they are solutions, so $\\{G_{j}\\}_{j\geq k}$ is equicontinuous
on $K$ as long as $k$ is large enough so that
$K\subset\overline{\Omega}\cap\left\\{t>-2^{k}\right\\}$.

Point (ii) is a consequence of the Caccioppoli inequality at the boundary. We
only need to prove the bound when $k\geq 2$, since all the $G_{k}$ are already
in $W_{0}$ by construction. We have by the Caccioppoli inequality at the
boundary (see for instance Lemma 11.15 in [DFM3]) that

$\iint_{\Omega\cap([-2,2]\times[-1,1])}|\nabla
G_{k}(x,t)|^{2}dx\,dt\lesssim\iint_{\Omega\cap([-4,4]\times[-2,2])}|G_{k}(x,t)|^{2}dx\,dt\\\
\lesssim\sup_{\Omega\cap([-4,4]\times[-2,2])}|G_{k}(x,t)|^{2}\lesssim 1.$

Point (iii) is one of our key arguments. We define $W_{0}^{+}$ as the subspace
of $W^{+}$ that contained the functions with zero trace on
$\partial(\Omega\cap\mathbb{R}^{2}_{+})$.

Since $G_{k}\in W_{0}$, its restriction
$(G_{k})_{|\Omega\cap\mathbb{R}^{2}_{+}}$ is of course in $W^{+}$. Moreover,
$G_{k}$ is a solution to $-\Delta u=0$ in $\Omega\cap\mathbb{R}^{2}_{+}$. We
can invoke the uniqueness in Lax-Milgram theorem (see Lemma 12.2 in [DFM3],
but adapted to our periodic function spaces $W_{0}^{+}$ and $W^{+}$) to get
that $G_{k}$ is the only weak solution to $-\Delta u=0$ in
$\Omega\cap\mathbb{R}^{2}_{+}$ for which the trace on
$\partial(\Omega\cap\mathbb{R}^{2}_{+})$ is
$(G_{k})_{|\partial(\Omega\cap\mathbb{R}^{2}_{+})}$. Moreover,

$\|G_{k}\|_{W^{+}}\leq
C\|(G_{k})_{|\partial(\Omega\cap\mathbb{R}^{2}_{+})}\|_{H^{1/2}_{\partial\Omega_{+}}},$

where $H^{1/2}_{\partial\Omega_{+}}$ is the space of traces on
$\partial\Omega_{+}:=\partial(\Omega\cap\mathbb{R}^{2}_{+})$ for the symmetric
2-periodic functions defined as

$H^{1/2}_{\partial\Omega_{+}}:=\Big{\\{}f:\,\partial\Omega_{+}\mapsto\mathbb{R}\text{
measurable such that $f$ is symmetric and $2$-periodic in $x$,}\\\ \text{ and
}\|f\|_{H^{1/2}_{\partial\Omega_{+}}}:=\left(\int_{\partial\Omega_{+}\cap
S_{0}}\int_{\partial\Omega_{+}\cap
S_{0}}\frac{|f(x)-f(y)|^{2}}{|x-y|^{3/2}}d\mathcal{H}^{1}(x)\,d\mathcal{H}^{1}(y)\right)^{\frac{1}{2}}<+\infty\Big{\\}}.$

So in particular, we have by a classical argument that

$\|(G_{k})_{|\partial(\Omega\cap\mathbb{R}^{2}_{+})}\|^{2}_{H^{1/2}_{\partial\Omega_{+}}}\leq
C\iint_{\Omega\cap([-2,2]\times[-1,1])}|\nabla G_{k}(x,t)|^{2}dx\,dt.$

We conclude that

$\iint_{S_{0}\cap\mathbb{R}^{2}_{+}}|\nabla
G_{k}|^{2}dx\,dt\lesssim\iint_{\Omega\cap([-2,2]\times[-1,1])}|\nabla
G_{k}(x,t)|^{2}dx\,dt\lesssim 1$

by (ii). Point (iii) follows. $\square$

######  Proposition 10.6.

There exists a symmetric (in $x$), 2-periodic (in $x$), positive weak solution
$G\in W_{\operatorname{loc}}^{1,2}(\Omega)\cap C(\overline{\Omega})$ to
$-\Delta G=0$ in $\Omega$ such that $G=0$ on $\partial\Omega$ and $G(X_{0})=1$
and

(10.7) $\iint_{S_{0}\cap\mathbb{R}^{2}_{+}}|\nabla G|^{2}dx\,dt<+\infty.$

Proof: We invoke the Arzelà-Ascoli theorem - whose conditions are satisfied
thanks to Proposition 10.5 (i) - to extract a subsequence of $G_{k}$ that
converges uniformly on any compact to a continuous function $G$. The fact $G$
is non-negative, symmetric, 2-periodic, and satisfies $G(X_{0})=1$ is
immediate from the fact that all the $G_{k}$ are already like this. The
functions $G_{k}$ converges to $G$ in $W^{1,2}_{loc}(\overline{\Omega})$
thanks to the Caccioppoli inequality, and then by using the weak convergence
of $G_{k}$ to $G$ in $W^{1,2}_{loc}(\overline{\Omega})$, we can easily prove
that $G$ is a solution to $-\Delta u=0$ in $\Omega$ (hence $G$ is positive by
the Harnack inequality, since it was already non-negative). The convergence of
$G_{k}$ to $G$ in $W^{1,2}_{loc}(\overline{\Omega})$ also allow the uniform
bound on $\|G_{k}\|_{W^{+}}$ given by Proposition 10.5 (iii) to be transmitted
to $G$, hence (10.7) holds. The proposition follows. $\square$

###  10.2. $G$ fails the estimate given in Theorem 1.12

######  Lemma 10.8.

$\partial_{t}G$ is harmonic in $\Omega$, that is, it is a solution of $-\Delta
u=0$ in $\Omega$, and we have

$\int_{1}^{\infty}\int_{0}^{1}|\nabla\partial_{t}G|^{2}dx\,dt<+\infty.$

Proof: Morally, we want to prove that if $G$ is a solution (to $-\Delta u=0$),
then $\nabla G\in W^{1,2}$, which is a fairly classical regularity result. The
difficulty in our case is that the domain in consideration is unbounded.

Since $G$ is a harmonic function (solution of the Laplacian), the function
$g(x):=G(x,1)$ is smooth. We can prove the bound

$\int_{1}^{\infty}\int_{0}^{1}|\nabla\partial_{x}G|^{2}dx\,dt\lesssim\int_{0}^{1}|g^{\prime}(x)|^{2}dx+\int_{0}^{1}|g^{\prime\prime}(x)|^{2}dx+\int_{1}^{\infty}\int_{0}^{1}|\nabla
G|^{2}dx\,dt<+\infty$

by adapting the proof of Proposition 7.3 in [DFM5] to our simpler context (and
invoking (10.7) and $g\in C^{\infty}(\mathbb{R})$ to have the finiteness of
the considered quantities). In order to have the derivative on the
$t$-derivative, it is then enough to observe

$\int_{1}^{\infty}\int_{0}^{1}|\nabla\partial_{t}G|^{2}dx\,dt\lesssim\int_{1}^{\infty}\int_{0}^{1}|\partial_{x}\partial_{t}G|^{2}dx\,dt+\int_{1}^{\infty}\int_{0}^{1}|\partial_{t}\partial_{t}G|^{2}dx\,dt\\\
=\int_{1}^{\infty}\int_{0}^{1}|\partial_{t}\partial_{x}G|^{2}dx\,dt+\int_{1}^{\infty}\int_{0}^{1}|\partial_{x}\partial_{x}G|^{2}dx\,dt\\\
\lesssim\int_{1}^{\infty}\int_{0}^{1}|\nabla\partial_{x}G|^{2}dx\,dt<+\infty,$

where we use the fact that $G$ is a solution to $-\Delta u=0$ \- i.e.
$\partial_{t}\partial_{t}G=-\partial_{x}\partial_{x}G$ \- for the second line.
The lemma follows. $\square$

We will also need a maximum principle, given by

######  Lemma 10.9.

If $u$ is a symmetric (in $x$), 2-periodic (in $x$) harmonic function in
$\mathbb{R}\times(t_{0},\infty)$ that satisfies

(10.10) $\int_{t_{0}}^{\infty}\int_{0}^{1}|\nabla u|^{2}dx\,dt<+\infty,$

then $u$ has a trace - denoted by $\operatorname{Tr}_{t_{0}}u$ \- on
$\mathbb{R}\times\\{t_{0}\\}$ and

$\inf_{y\in(0,1)}(\operatorname{Tr}_{t_{0}}u)(y)\leq
u(x,t)\leq\sup_{y\in(0,1)}(\operatorname{Tr}_{t_{0}}u)(y)\qquad\text{ for all
}x\in\mathbb{R},\,t>t_{0}.$

Proof: The existence if the trace - in the space
$W^{2,\frac{1}{2}}(\mathbb{R}\times\\{t_{0}\\})$ \- is common knowledge. The
proof of Lemma 12.8 in [DFM3] (for instance) can be easily adapted to prove
our case. $\square$

######  Lemma 10.11.

There exists $C\geq 1$ such that

$C^{-1}\leq G(x,t)\leq C\qquad\text{ for }x\in\mathbb{R},\,t\geq 1.$

Proof: Since $G(1,0)=G(X_{0})=1$ and $G$ is a positive solution, the Harnack
inequality implies that $C^{-1}\leq G(x,1)\leq C$ for $x\in[0,1]$. Since $G$
is symmetric and 2-periodic in $x$, we have $C^{-1}\leq G(x,1)\leq C$ for
$x\in\mathbb{R}$. We conclude with the maximum principle (Lemma 10.9), since
the bound (10.10) is given by (10.7). $\square$

######  Lemma 10.12.

For every $c>0$, there exists $t_{0}\geq 1$ such that

$\partial_{t}G(x,t)\leq\frac{c}{t}\text{ for all $x\in\mathbb{R}$, $t\geq
t_{0}$}.$

Proof: Let $x$ be fixed. Since $G$ is symmetric and 2-periodic in $x$, we can
assume without loss of generality that $x\in(0,1)$. Then recall that
$\partial_{t}G$ is a weak solution in $\Omega$, so in particular, we have the
Moser estimate and the Caccioppoli inequality, which give

(10.13)
$\sup_{y\in\mathbb{R},\,s>4}s|\partial_{t}G(y,s)|\lesssim\sup_{y\in\mathbb{R},\,s>4}s\left(\fint_{s/2}^{2s}\fint_{x-s}^{x+s}|\nabla
G(z,r)|^{2}dz\,dr\right)^{\frac{1}{2}}\lesssim\sup_{y\in\mathbb{R},\,s>1}G\lesssim
1.$

by Lemma 10.11. Moreover, $\partial_{t}G$ is Hölder continuous, that is,

(10.14) $\sup_{y\in(0,1)}|\partial_{t}G(x,t)-\partial_{t}G(y,t)|\leq
Ct^{-\alpha}\left(\fint_{(1-t)/2}^{(1+t)/2}\fint_{t/2}^{3t/2}\left|\partial_{t}G(y,s)\right|^{2}dsdy\right)^{1/2}\\\
\leq Ct^{-\alpha}\sup_{y\in\mathbb{R},\,s>t/2}|\partial_{t}G(y,s)|\leq
C^{\prime}t^{-\alpha-1}\qquad\text{ for }t\geq 8$

by (10.13).

We pick $t_{0}\geq 8$ such that $2C^{\prime}(t_{0})^{-\alpha}\leq c$. Assume
by contradiction that there exist $x\in(0,1)$ and $t\geq t_{0}$ are such that
$\partial_{t}G(x,t)\geq c/t$, then

$\inf_{y\in\mathbb{R}}\partial_{t}G(y,t)=\inf_{y\in(0,1)}\partial_{t}G(y,t)\geq\partial_{t}G(x,t)-\sup_{y\in(0,1)}|\partial_{t}G(x,t)-\partial_{t}G(y,t)|\\\
\geq\frac{c-C^{\prime}t^{-\alpha}}{t}\geq\frac{c}{2t}$

by our choice of $t_{0}$. Since $\partial_{t}G$ is a solution that satisfies
(10.10) - see Lemma 10.8 \- the maximum principle given by Lemma 10.9 entails
that

$\partial_{t}G(y,s)\geq\frac{c}{2t}\qquad\text{ for }y\in\mathbb{R},\,s>t,$

which implies

$\int_{0}^{1}\int_{t}^{\infty}|\nabla G(y,s)|^{2}ds\,dy=+\infty,$

which is in contradiction with (10.7). We conclude that for every $x\in(0,1)$
and $t\geq t_{0}$, we necessary have $\partial_{t}G\leq c/t$. The lemma
follows. $\square$

######  Lemma 10.15.

For any $\beta>0$, there exists a $t_{0}\geq 1$ and $\epsilon>0$ such that

(10.16)
$\left|\frac{\partial_{t}G(x,t)}{G(x,t)}-\frac{\partial_{t}D_{\beta}(x,t)}{D_{\beta}(x,t)}\right|\geq\frac{\epsilon}{t}\qquad\text{
for }x\in\mathbb{R},\,t\geq t_{0}.$

Proof: The set $\partial\Omega$ is $(n-1)$-Ahlfors regular, so (1.4) gives the
equivalence $D_{\beta}(X)\approx\operatorname{dist}(X,\partial\Omega)$ for
$X\in\Omega$, and hence the existence of $C_{1}>0$ (depending on $\beta$ and
$n$) such that

(10.17) $(C_{1})^{-1}t\leq D_{\beta}(x,t)\leq
C_{1}D_{\beta+2}(x,t)\leq(C_{1})^{2}t\qquad\text{ for }x\in\mathbb{R},\,t\geq
1.$

Check then that

$\partial_{t}D_{\beta}(x,t)=\frac{d+\beta}{\beta}D_{\beta}^{1+\beta}(x,t)\int_{(y,s)\in\partial\Omega}|(x,t)-(y,s)|^{-d-\beta-2}(t-s)\,d\sigma(y,s)$

In particular, since $s\leq\frac{1}{2}$ whenever $(y,s)\in\partial\Omega$, we
have, for $(x,t)\in\mathbb{R}\times[1,\infty)$, that

$\partial_{t}D_{\beta}(x,t)\geq\Big{(}t-\frac{1}{2}\Big{)}\frac{n+\beta-1}{\beta}D^{1+\beta}_{\beta}(x,t)\int_{(y,s)\in\partial\Omega}|(x,t)-(y,s)|^{-n-\beta-1}\,d\sigma(y,s)\\\
\geq\frac{t}{2}\frac{n+\beta-1}{\beta}D^{1+\beta}_{\beta}(x,t)D^{-\beta-2}_{\beta+2}(x,t)\geq
c_{\beta,n}$

for some $c_{\beta,n}>0$, by (10.17). In conclusion, using (10.17) again, we
have the existence of $c_{1}>0$ such that

(10.18)
$\frac{\partial_{t}D_{\beta}(x,t)}{D_{\beta}(x,t)}\geq\frac{c_{1}}{t}\qquad\text{
for }x\in\mathbb{R},\,t\geq 1.$

Let $C_{2}$ be the constant in Lemma 10.11. Thanks to Lemma 10.12, there
exists $t_{0}\geq 1$ such that $\partial_{t}G(x,t)\leq c_{1}/(2C_{2}t)$ for
any $x\in\mathbb{R}$ and $t\geq t_{0}$, which means that

(10.19) $\frac{\partial_{t}G(x,t)}{G(x,t)}\leq\frac{c_{1}}{2t}\qquad\text{ for
}x\in\mathbb{R},\,t\geq t_{0}.$

The combination of (10.18) and (10.19) gives (10.16) for $\epsilon=c_{1}/2$.
$\square$

######  Lemma 10.20.

The positive solution $G$ does not satisfies (1.13), proving that assuming
that $\Omega$ is semi-uniform is not sufficient for Theorem 1.12.

Proof: Let $B_{r}$ be the ball of radius $r$ centered at
$(0,\frac{1}{2})\in\partial\Omega$, and take $r\geq 2t_{0}$, where $t_{0}\geq
1$ is the value from Lemma 10.15. We have

$\iint_{\Omega\cap B_{r}}\left|\frac{\nabla G}{G}-\frac{\nabla
D_{\beta}}{D_{\beta}}\right|^{2}D_{\beta}\,dx\,dt\geq\iint_{B_{r}\cap\\{t\geq
t_{0}\\}}\left|\frac{\nabla G}{G}-\frac{\nabla
D_{\beta}}{D_{\beta}}\right|^{2}D_{\beta}\,dx\,dt\\\ \geq
C^{-1}\epsilon^{2}\iint_{B_{r}\cap\\{t\geq t_{0}\\}}\frac{dx\,dt}{t}$

by (10.16) and (1.4). We conclude that

$\frac{1}{\sigma(B_{r})}\iint_{\Omega\cap B_{r}}\left|\frac{\nabla
G}{G}-\frac{\nabla
D_{\beta}}{D_{\beta}}\right|^{2}D_{\beta}\,dx\,dt\gtrsim\ln\Big{(}\frac{r}{t_{0}}\Big{)}\rightarrow+\infty\text{
as }r\to\infty,$

which means that $G$ does not satisfies (1.13). The lemma follows. $\square$

## References

  * [Azz1] J. Azzam. Harmonic measure and the analyst’s traveling salesman theorem. Preprint, arXiv: 1905.09057.
  * [Azz2] J. Azzam. Semi-uniform domains and the $A_{\infty}$ property of harmonic measure. Int. Math. Res. Not. (2021) no. 9, 6717–6771.
  * [AGMT] J. Azzam, J. Garnett, M. Mourgoglou, and X. Tolsa. Uniform rectifiability, elliptic measure, square functions, $\epsilon$-approximability via an ACF monotonicity formula. Int. Math. Res. Not. (to appear).
  * [AHMMT] J. Azzam, S. Hofmann, J. M. Martell, M. Mourgoglou, and X. Tolsa. Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$-solvability of the Dirichlet problem. Invent. Math. 222 (2020), no. 3, 881–993.
  * [AHMNT] J. Azzam, S. Hofmann, J.M. Martell, K. Nyström, T. Toro. A new characterization of chord-arc domains, J. Eur. Math. Soc. 19 (2017), no. 4, 967–981.
  * [AHM3TV] J. Azzam, S. Hoffman, M. Mourgoglou, J. M. Martell, S. Mayboroda, X. Tolsa, A. Volberg. Rectifiability of harmonic measure, Geom. Funct. Anal., 26 (2016), no. 3, 703–728.
  * [AiHi] Aikawa, Hiroaki, and Kentaro Hirata. Doubling conditions for harmonic measure in John domains, In Annales de l’institut Fourier, 58 (2008) no. 2, 429–445.
  * [BH] S. Bortz, S. Hofmann. Harmonic measure and approximation of uniformly rectifiable sets. Rev. Mat. Iberoam. 33 (2017), no. 1, 351–373.
  * [BHHLN] S. Bortz, J. Hoffman, S. Hofmann, J.-L. Luna-Garcia, and K. Nyström. Coronizations and big pieces in metric spaces. Ann. Inst. Fourier (Grenoble) 72 (2022), no. 5, 2037–2078.
  * [CFK] L. Caffarelli, E. Fabes, C. Kenig. Completely singular elliptic-harmonic measures. Indiana Univ. Math. J., 30 (1981), no. 6, 917–924.
  * [CFMS] L. Caffarelli, E. Fabes, S. Mortola, S. Salsa. Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J., 30 (1981), no. 4, 621–640.
  * [CHM] M. Cao, P. Hidalgo-Palencia, J. M. Martell. Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity. Preprint, arXiv:2202.06363.
  * [CHMT] J. Cavero, S. Hofmann, J.M. Martell, and T. Toro. Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates. Trans. of the AMS. (11) 373 (2020), 7901–7935.
  * [Chr] M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math., LX/LXI (1990), 601–628.
  * [D1] G. David. Morceaux de graphes lipschitziens et intégrales singulières sur une surface. (French) [Pieces of Lipschitz graphs and singular integrals on a surface]. Rev. Mat. Iberoamericana 4 (1988), no. 1, 73–114.
  * [D2] G. David. Wavelets and singular integrals on curves and surfaces. Lecture Notes in Mathematics, 1465. Springer-Verlag, Berlin, 1991.
  * [DJ] G. David, D. Jerison. Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals. Indiana Univ. Math. J., 39 (1990), no. 3, 831–845.
  * [DS1] G. David, S. Semmes. Singular integrals and rectifiable sets in $\mathbb{R}^{n}$: Beyond Lipschitz graphs. Asterisque, 193 (1991).
  * [DS2] G. David, S. Semmes. Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, 38. American Mathematical Society, Providence, RI, 1993.
  * [DS3] G. David, S. Semmes. Fractured fractals and broken dreams: self-similar geometry through metric and measure. Vol. 7. Oxford University Press, 1997.
  * [DEM] G. David, M. Engelstein, S. Mayboroda. Square functions, non-tangential limits and harmonic measure in co-dimensions larger than 1. Duke Math. J. 170 (2021), no 3, 455–501.
  * [DFM1] G. David, J. Feneuil, S. Mayboroda. Elliptic theory for sets with higher co-dimensional boundaries. Mem. Am. Math. Soc. 273 (2021), no 1346, iii+123 pp.
  * [DFM2] G. David, J. Feneuil, S. Mayboroda. Dahlberg’s theorem in higher co-dimension. Journal of Functional Analysis, 276 (2019), no. 9 2731–2820.
  * [DFM3] G. David, J. Feneuil, S. Mayboroda. Elliptic theory in domains with boundaries of mixed dimension. Preprint, arXiv:2003.09037.
  * [DFM4] G. David, J. Feneuil, S. Mayboroda. Green function estimates on complements of low-dimensional uniformly rectifiable sets. Math. Ann. (accepted).
  * [DFM5] Z. Dai, J. Feneuil, S. Mayboroda. The regularity problem in domains with lower dimensional boundaries. Preprint, arXiv:2208.00628.
  * [DLM1] G. David, L. Li, S. Mayboroda. Carleson measure estimates for the Green function. Arch. Rational. Mech. Anal. 243 (2022), 1525–1563.
  * [DLM2] G. David, L. Li, S. Mayboroda. Carleson estimates for the Green function on domains with lower dimensional boundaries. J. Funct. Anal. 283 (2022), no. 5, https://doi.org/10.1016/j.jfa.2022.109553.
  * [DM1] G. David, S. Mayboroda. Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets. Int. Math. Res. Not. (2022), rnac109, https://doi.org/10.1093/imrn/rnac109.
  * [DM2] G. David, S. Mayboroda. Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions. Adv. Math. 410 (to appear).
  * [Fen1] J. Feneuil. Absolute continuity of the harmonic measure on low dimensional rectifiable sets. J. Geom. Anal. 32 (2022), no. 10, Paper No. 247, 36 pp.
  * [Fen2] J. Feneuil. A change of variable for Dahlberg-Kenig-Pipher operators. Proc. Am. Math. Soc. 150 (2022), no. 8, 3565–3579.
  * [GMT] J. Garnett, M. Mourgoglou, X. Tolsa. Uniform rectifiability from Carleson measure estimates and $\epsilon$-approximability of bounded harmonic functions. Duke Math. J. 167 (2018), no. 8, 1473–1524.
  * [HM1] S. Hofmann, J.M. Martell. Uniform rectifiability and harmonic measure I: uniform rectifiability implies Poisson kernels in $L^{p}$. Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), no. 3, 577–654.
  * [HM2] S. Hofmann, J.M. Martell. Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in Lp implies uniform rectifiability. Preprint, arXiv:1505.06499.
  * [HMM1] S. Hofmann, J.M. Martell, S. Mayboroda. Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions. Duke Math. J. 165 (2016), no. 12, 2331–2389.
  * [HMM2] S. Hofmann, J.M. Martell, S. Mayboroda. Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains. Preprint, arXiv:1904.13116.
  * [HMMTZ] S. Hofmann, J.M. Martell, S. Mayboroda, T. Toro, Z. Zihui. Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Geom. Funct. Anal. 31 (2021), no. 2, 325–401.
  * [HMU] S. Hofmann, J.M. Martell, I. Uriarte-Tuero. Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^{p}$ imply uniform rectifiability. Duke Math. J., 163 (2014), no. 8, 1601–1654.
  * [HMT1] S. Hofmann, J.M. Martell, T. Toro. $A_{\infty}$ implies NTA for a class of variable coefficient elliptic operators. J. Differential Equations 263 (2017), no. 10, 6147–6188.
  * [HMT2] S. Hofmann, J.M. Martell and T. Toro, General divergence form elliptic operators on domains with ADR boundaries, and on 1-sided NTA domains. In progress.
  * [JK] D. Jerison and C. Kenig. The Dirichlet problem in nonsmooth domains. Ann. of Math. (2) 113 (1981), no. 2, 367–382.
  * [Ken] C. E. Kenig. Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics, 83. Amer. Math. Soc., Providence, RI, 1994.
  * [KP] C. Kenig, J. Pipher. The Dirichlet problem for elliptic equations with drift terms. Publ. Mat., 45 (2001), no. 1, 199–217.
  * [MM] L. Modica, S. Mortola. Construction of a singular elliptic-harmonic measure. Manuscripta Math. 33 (1980/81), no. 1, 81–98.
  * [MT] M. Mourgoglou, X. Tolsa. The regularity problem for the Laplace equation in rough domains. Preprint, arXiv:2110.02205.
  * [MPT] M. Mourgoglou, B. Poggi, X. Tolsa. $L^{p}$-solvability of the Poisson-Dirichlet problem and its applications to the regularity problem. Preprint, arXiv:2207.10554.
  * [NTV] F. Nazarov, X. Tolsa and A. Volberg, _On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1_. Acta Math. 213 (2014), no. 2, 237–321.
  * [Ste] E. M. Stein.Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, N.J., 1993.
  * [Tol] X. Tolsa. Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality. Proc. Lond. Math. Soc. (3), 98 (2009), no. 2, 393–426.