satisfies $2^{-k-1}\leq|x-y|<2^{-k}$. We use (6.5) and decompose the integral
as

$\begin{split}\|\operatorname{Tr}u\|_{H}&:=\int_{\Gamma}\int_{\Gamma}\dfrac{\rho(x,|x-y|)^{2}|\operatorname{Tr}u(x)-\operatorname{Tr}u(y)|^{2}}{m(B(x,|x-y|)\cap\Omega)}\,d\mu(y)\,d\mu(x)\\\
&\lesssim\int_{\Gamma}\int_{\Gamma}\dfrac{\rho(x,|x-y|)^{2}|\operatorname{Tr}u(x)-\operatorname{Tr}_{k(x,y)}u(x)|^{2}}{m(B(x,|x-y|)\cap\Omega)}\,d\mu(y)\,d\mu(x)\\\
&\hskip
28.45274pt+\int_{\Gamma}\int_{\Gamma}\dfrac{\rho(x,|x-y|)^{2}|\operatorname{Tr}u(y)-\operatorname{Tr}_{k(x,y)}u(y)|^{2}}{m(B(x,|x-y|)\cap\Omega)}\,d\mu(y)\,d\mu(x)\\\
&\hskip
56.9055pt+\int_{\Gamma}\int_{\Gamma}\dfrac{\rho(x,|x-y|)^{2}|\operatorname{Tr}_{k(x,y)}u(x)-\operatorname{Tr}_{k(x,y)}u(y)|^{2}}{m(B(x,|x-y|)\cap\Omega)}\,d\mu(y)\,d\mu(x)\\\
&\hskip 28.45274pt:=I_{1}+I_{2}+I_{3}.\end{split}$

Let us first treat the term $I_{3}$. More precisely, we start with the
difference $|\operatorname{Tr}_{k(x,y)}u(x)-\operatorname{Tr}_{k(x,y)}u(y)|$.
To lighten the notation, write $k$ for $k(x,y)$. As before, denote by
$B_{x}^{k}$ and $B_{y}^{k}$ the balls used to define
$\operatorname{Tr}_{k}u(x)$ and $\operatorname{Tr}_{k}u(y)$. That is,
$B_{x}^{k}$ and $B_{y}^{k}$ are such that

$\operatorname{Tr}_{k}u(x)=\fint_{B_{x}^{k}}u\quad\text{ and
}\quad\operatorname{Tr}_{k}u(y)=\fint_{B_{y}^{k}}u.$

Since the balls $B_{x}^{k},B_{y}^{k}$ lie at distances at least $c2^{-k}$ from
the boundary $\Gamma$ and at most $C2^{-k}$ from each other, the Harnack chain
condition (Proposition 2.18) says that we can find a chain of balls joining
$B_{x}^{k}$ to $B_{y}^{k}$, with uniformly bounded length, staying at a
distance at least $c2^{-k}$ from the boundary and at distance at most
$C2^{-k}$ from both $x$ and $y$. We define $P_{x,y}^{k}$ as the union of the
cubes in $\mathcal{W}$ that meet one of the balls of the chain. From what we
just said, $P_{x,y}^{k}$ is a Whitney region associated to both $(x,2^{-k})$
and $(y,2^{-k})$, and so it is the union of a bounded number of adjacent cubes
in ${\mathcal{W}}$. Therefore, similarly to the sets $U^{*}_{Q}$,
$P_{x,y}^{k}$ satisfies the $C(\kappa,M)$ chain condition for some uniform
$\kappa,M$ and is thus fitted for the Poincaré inequality. These observations
allow us to use Theorem 5.24 and write

$\begin{split}|\operatorname{Tr}_{k}u(x)-\operatorname{Tr}_{k}u(y)|&:=\left|\fint_{B_{x}^{k}}u\,dm-\fint_{B_{y}^{k}}u\,dm\right|\lesssim\fint_{B_{x}^{k}}\Big{|}u-\fint_{B_{y}^{k}}u\,dm\Big{|}\,dm\\\
&\lesssim\frac{1}{m(B_{x}^{k})}\int_{P_{x,y}^{k}}\Big{|}u-\fint_{B_{y}^{r}}u\,dm\Big{|}dm\\\
&\lesssim\frac{2^{-k}m(P_{x,y}^{k})^{\frac{1}{2}}}{m(B_{x}^{k})}\left(\int_{P_{x,y}^{k}}\left|\nabla
u\right|^{2}dm\right)^{\frac{1}{2}}.\end{split}$

Since both $P^{k}_{x,y}$ and $B_{x}^{k}$ are Whitney region associated to $y$
and $2^{-k}$ (i.e. there exists a large constant $C$ such that both sets are
contained in $B(y,C2^{-k})$, contain a ball $B$ of radius $C^{-1}2^{-k}$, and
are at distance at least $C^{-1}2^{-k}$ of $\Gamma$), the doubling measure
condition (H4) implies that $m(B_{x}^{k})\approx m(P_{x,y}^{k})\approx
m(B(y,2^{-k-1})\cap\Omega)$. Therefore,

(6.14)
$|\operatorname{Tr}_{k}u(x)-\operatorname{Tr}_{k}u(y)|^{2}\lesssim\frac{2^{-2k}}{m(B(y,2^{-k-1})\cap\Omega)}\int_{P_{x,y}^{k}}\left|\nabla
u\right|^{2}dm.$

We inject (6.14) in $I_{3}$ and observe that $m(B(x,|x-y|)\cap\Omega)\approx
m(B(x,2^{-k(x,y)})\cap\Omega)$ and
$\rho(x,|x-y|)\approx\rho(x,2^{-k(x,y)})\approx\rho(y,2^{-k(x,y)})$ by
(H3)–(H4). Therefore,

(6.15)
$\begin{split}I_{3}&\lesssim\int_{\Gamma}\int_{\Gamma}\frac{2^{-k(x,y)}\rho(x,2^{-k(x,y)})}{m(B(x,2^{-k(x,y)})\cap\Omega)}\frac{2^{-k(x,y)}\rho(y,2^{-k(x,y)})}{m(B(y,2^{-k(x,y)})\cap\Omega)}\int_{P_{x,y}^{k(x,y)}}\left|\nabla
u\right|^{2}dm\,d\mu(x)\,d\mu(y)\\\
&\quad=\int_{\Gamma}\int_{\Gamma}\frac{1}{\mu(B(x,2^{-k(x,y)}))}\frac{1}{\mu(B(y,2^{-k(x,y)}))}\int_{P_{x,y}^{k(x,y)}}\left|\nabla
u\right|^{2}dm\,d\mu(x)\,d\mu(y),\end{split}$

by the definition (2.6) of $\rho$. Since $P_{x,y}^{k(x,y)}$ is a ‘Whitney
region’ for both $x$ and $y$, we have that $x,y\in B(Z,C\delta(Z))$ for $Z\in
P^{k(x,y)}_{x,y}$ where the constant $C\geq 2$ depends only on
$n,C_{1},C_{2}$, and moreover $2^{-k(x,y)}\approx\delta(Z)$. Then by Fubini’s
lemma

(6.16) $\begin{split}I_{3}&\lesssim\int_{\Omega}|\nabla
u(Z)|^{2}dm(Z)\,\int_{x\in
B(Z,C\delta(Z))}\frac{d\mu(x)}{\mu(B(x,c\delta(Z)))}\int_{y\in
B(Z,C\delta(Z))}\frac{d\mu(y)}{\mu(B(y,c\delta(Z)))}.\end{split}$

Yet the doubling property (H3) implies that for $z\in\Gamma\cap
B(Z,C\delta(Z))$,

$\mu(B(z,c\delta(Z)))\gtrsim\mu(B(z,C^{2}\delta(Z)))\geq\mu(B(Z,C\delta(Z))),$

hence we can simply bound $I_{3}$ by

(6.17) $I_{3}\lesssim\int_{\Omega}|\nabla u(Z)|^{2}\,dm(Z)$

as desired.

We turn now to the bound on $I_{1}$. Notice that the estimate for $I_{2}$ is
the same as for $I_{1}$, either by symmetry or since by (H3)–(H4),
$m(B(x,|x-y|)\cap\Omega)\approx\mu(B(y,|x-y|)\cap\Omega)$ and
$\rho(x,|x-y|)\approx\rho(y,|x-y|)$.

Notice that $I_{1}$ depends on $y$ only via $|x-y|$, so, by the doubling
property (H3) again and then (2.6)

(6.18)
$\begin{split}I_{1}&\lesssim\int_{x\in\Gamma}\sum_{k\in\mathbb{Z}}\frac{\rho(x,2^{-k})^{2}|\operatorname{Tr}u(x)-\operatorname{Tr}_{k}u(x)|^{2}}{m(B(x,2^{-k})\cap\Omega)}\int_{y\in
B(x,2^{-k})\setminus B(x,2^{-k-1})}d\mu(y)\,d\mu(x)\\\
&\lesssim\int_{x\in\Gamma}\sum_{k\in\mathbb{Z}}2^{k}\rho(x,2^{-k})|\operatorname{Tr}u(x)-\operatorname{Tr}_{k}u(x)|^{2}\,d\mu(x).\end{split}$

The trace operator is defined for $\mu$-almost every $x\in\Gamma$ by (6.7).
For such $x$, one get by letting $j$ tend to $+\infty$ in (6.9) that

(6.19)
$\left|\operatorname{Tr}_{k}u(x)-\operatorname{Tr}u(x)\right|\lesssim\sum_{i>k}2^{-i}\left(\frac{1}{m(U^{*}_{Q^{i}(x)})}\int_{U^{*}_{Q^{i}(x)}}|\nabla
u|^{p}dm\right)^{\frac{1}{p}}.$

We use the above estimate in (6.18) to obtain that

$\begin{split}I_{1}&\lesssim\int_{x\in\Gamma}\sum_{k\in\mathbb{Z}}2^{k}\rho(x,2^{-k})\left(\sum_{i>k}2^{-i}\left(\frac{1}{m(U^{*}_{Q^{i}(x)})}\int_{U^{*}_{Q^{i}(x)}}|\nabla
u|^{p}dm\right)^{\frac{1}{p}}\right)^{2}\,d\mu(x)\\\
&\qquad=\int_{x\in\Gamma}\sum_{k\in\mathbb{Z}}2^{k}\rho(x,2^{-k})\left(\sum_{i>k}\frac{1}{2^{i}\rho(x,2^{-i})}\,g_{i}(x)\right)^{2}\,d\mu(x),\end{split}$

where

$g_{i}(x)=\rho(x,2^{-i})\left(\frac{1}{m(U^{*}_{Q^{i}(x)})}\int_{U^{*}_{Q^{i}(x)}}|\nabla
u|^{p}dm\right)^{\frac{1}{p}}.$

Thanks to (H5), the sequence $\\{s_{i}\\}_{i\geq k_{0}}$ defined as
$s_{i}:=[2^{i}\rho(x,2^{-i})]^{-1}$ satisfies (6.3). As a consequence, Lemma
6.2 (with $q=2$) gives that for each $x\in\Gamma$,

$\sum_{k\in\mathbb{Z}}2^{k}\rho(x,2^{-k})\left(\sum_{i>k}\frac{1}{2^{i}\rho(x,2^{-i})}\,g_{i}(x)\right)^{2}\lesssim\sum_{i\in\mathbb{Z}}\frac{1}{2^{i}\rho(x,2^{-i})}\,|g_{i}(x)|^{2}.$

Thus the bound on $I_{1}$ becomes

$\begin{split}I_{1}&\lesssim\int_{\Gamma}\sum_{i\in\mathbb{Z}}\frac{1}{2^{i}\rho(x,2^{-i})}|g_{i}(x)|^{2}\,d\mu(x)\\\
&\qquad=\int_{\Gamma}\sum_{i\in\mathbb{Z}}2^{-i}\rho(x,2^{-i})\left(\frac{1}{m(U^{*}_{Q^{i}(x)})}\int_{U^{*}_{Q^{i}(x)}}|\nabla
u|^{p}dm\right)^{\frac{2}{p}}d\mu(x).\end{split}$

We use (6.11) to get rid of the function $\rho$, and then write the bound we
obtained as a sum over $\mathbb{D}$, which gives

$\begin{split}I_{1}&\lesssim\int_{\Gamma}\sum_{i\in\mathbb{Z}}\frac{m(U^{*}_{Q^{i}(x)})^{1-2/p}}{\mu(Q^{i}(x))}\left(\int_{U^{*}_{Q^{i}(x)}}|\nabla
u|^{p}dm\right)^{\frac{2}{p}}d\mu(x)=\sum_{Q\in\mathbb{D}}m(U^{*}_{Q})^{1-2/p}\left(\int_{U^{*}_{Q}}|\nabla
u|^{p}dm\right)^{\frac{2}{p}}.\end{split}$

We can now apply Lemma 6.1 with $q=2/p$ and $g=|\nabla u|^{p}$. Recall recall
that $q>1$ because $p$ comes from Theorem 5.24 and was chosen above (6.9),
with $1<p<2$. We get that

$I_{1}\lesssim\int_{\Omega}|\nabla u|^{2}\,dm;$

Theorem 6.6 follows. $\Box$

We end this section with a useful result concerning the trace of a product.

######  Lemma 6.20.

Let $(\Omega,m,\mu)$ satisfy (H1)–(H6). Suppose $u\in W$ and $\varphi\in
C_{0}^{\infty}(\mathbb{R}^{n})$. Then $u\varphi\in W$, with the product rule

(6.21) $\nabla(u\varphi)=\varphi\nabla u+u\nabla\varphi$

for the gradient, and

(6.22)
$\operatorname{Tr}(u\varphi)(x)=\varphi(x)\operatorname{Tr}u(x)\quad\hbox{for
every point $x\in\Gamma$ satisfying \eqref{defLebesgue}.}$

Proof. This result is the analogue of [DFM2, Lemma 5.4]. The proof is similar,
so we only sketch it.

We start with the simplest case is when we can see $u$ as a distribution on
$\Omega$; this is the case when the stronger form (H6’) of our assumption (H6)
holds. Then $\nabla(u\varphi)=\varphi\nabla u+u\nabla\varphi$ in the sense of
distributions, and we are about to check that $\nabla(u\varphi)\in
L^{2}(\Omega,dm)$.

Choose $Q\in\mathbb{D}$ so large that
$\operatorname{supp}\,\varphi\cap\Omega\subset T_{2Q}$. Then, setting
$\bar{u}=\fint_{U^{*}_{Q}}u\,dm$,

(6.23)
$\|\nabla(u\varphi)\|_{L^{2}(\Omega,dm)}\leq\|\varphi\|_{\infty}\|u\|_{W}+\|\nabla\varphi\|_{\infty}\left(\int_{T_{2Q}}|u-\bar{u}|^{2}\,dm\right)^{\frac{1}{2}}+\|\nabla\varphi\|_{\infty}m(T_{2Q})^{\frac{1}{2}}|\bar{u}|.$

All three terms in the right hand side are finite, since $\varphi$ is smooth,
$u\in W\subset L^{1}(U^{*}_{Q},m)$, and by Theorem 5.24 (Poincaré’s
inequality). Consequently, $u\varphi\in W$ as desired.

As for the trace, observe that if $x$ satisfies (6.8), and if $B_{x}^{k}$
denotes $B(X,\delta(X)/2)$ for some $X\in U_{Q_{k}(x)}$, where as usual
$Q_{k}(x)$ is the only cube of $\mathbb{D}_{k}$ that contains $x$, then

$\fint_{B_{x}^{k}}|\varphi
u-\varphi(x)\operatorname{Tr}u(x)|\,dm\leq\|\varphi\|_{\infty}\fint_{B_{x}^{k}}|u-\operatorname{Tr}u(x)|\,dm+|Tu(x)|\fint_{B_{x}^{k}}|\varphi-\varphi(x)|\,dm.$

The first term converges to 0 as $k\to+\infty$ since $x$ is a Lebesgue point
for $u$, and the second term also tends to 0 since $|Tu(x)|<+\infty$ ($x$ is a
Lebesgue point) and $\varphi$ is continuous. Therefore

$\lim_{\delta(X)\to 0,\atop X\in\gamma(x)}\fint_{B_{x}^{k}}|\varphi
u-\varphi(x)\operatorname{Tr}u(x)|\,dm=0,$

which easily implies
$\operatorname{Tr}(u\varphi)(x)=\varphi(x)\operatorname{Tr}u(x)$ by the
definition of $\operatorname{Tr}$.

This takes care of the lemma when $u$ and $\nabla u$ are taken as
distributions. In general, we used Definition 4.1 to define the space $W$ and
the gradient $\nabla u$. Recall that we wrote $u$ as a limit in
$L^{1}_{loc}(\Omega,m)$ of smooth functions $\varphi_{i}\in
C^{\infty}(\overline{\Omega})$, as in (4.3), and required that (4.2) and (4.4)
hold for a suitable $v\in L^{2}(\Omega,dm)$ which is unique by (H6) and which
we also called $\nabla u$.

Now we want to show that $u\varphi\in W$, so we approximate it by the smooth
functions $\varphi_{i}\varphi$. It is easy to see that the
$\varphi_{i}\varphi$ converge to $u\varphi$ in $L^{1}_{loc}(\Omega,m)$ as in
(4.3), and that $\int_{\Omega}|\nabla(\varphi\varphi_{i})|^{2}<+\infty$ for
every $i$, as in (4.2) (recall that $\varphi\in
C_{0}^{\infty}(\mathbb{R}^{n})$). We try the gradient $w=\varphi
v+u\nabla\varphi=\varphi\nabla u+u\nabla\varphi$ in the definition (4.4).
First observe that $w\in L^{2}(\Omega,dm)$ by the proof of (6.23) (and where
Theorem 5.24 is applied in the general context of $W$). We claim that

$\lim_{i\to\infty}\int_{\Omega}|\nabla(\varphi\varphi_{i})-w|^{2}dm=0,$

as needed for (4.4). The first part of $\nabla(\varphi\varphi_{i})$ is
$\varphi\nabla\varphi_{i}$, which converges to $\varphi v$ in
$L^{2}(\Omega,dm)$, by (4.4). Thus we are left with showing that
$\varphi_{i}\nabla\varphi$ converges to $u\nabla\varphi$ in
$L^{2}(\Omega,dm)$. Or, since $\varphi$ is bounded and
$\operatorname{supp}\,\varphi\cap\Omega\subset T_{2Q}$, that

(6.24) $\lim_{i\to\infty}\int_{T_{2Q}}|\varphi_{i}-u|^{2}dm=0.$

Denote by $c_{i}$ the average of $\varphi_{i}-u$ on $2Q$. Then by Poincaré’
inequality (Theorem 5.24) and (4.4),
$\int_{T_{2Q}}|\varphi_{i}-u-c_{i}|^{2}dm$ tends to $0$. But also (4.3) says
that $\int_{B}|\varphi_{i}-u|dm$ tends to $0$ for some small ball $B\subset
T_{2Q}$, so in fact $c_{i}$ tends to $0$, (6.24) holds, and $u\varphi\in W$
with a derivative equal to $w$. The remaining estimates are as in the easier
case, and Lemma 6.20 follows. $\Box$

##  7\. Poincaré inequalities on the boundary

We are interested in a version of the Poincaré inequality for functions that
have a vanishing trace at the boundary. The proofs shall use the tent sets
$T_{2Q}$ that were constructed in Section 5, where we assumed $\Gamma$ and
$\Omega$ unbounded. But as explained in Section 13, the same construction
works for bounded $\Gamma$ and/or $\Omega$ (with maybe a restriction on the
size of possible $Q$), and the proofs in the section are directly adaptable to
this case. The Poincaré inequalities that we prove here are a local results,
so it makes sense anyway that they don’t depend on the boundedness of
$\Omega$.

######  Theorem 7.1.

Let $(\Omega,m,\mu)$ satisfy (H1)–(H6). There exists $p_{1}\in[1,2)$ and
$k:=k(C_{4})>1$ such that for $p\in[p_{1},2]$, $Q\in\mathbb{D}$, and $u\in W$
such that $\operatorname{Tr}u=0$ on a set $E\subset 2Q$ such that $\mu(E)\geq
c\mu(2Q)$, we have

(7.2) $\left(\frac{1}{m(T_{2Q})}\int_{T_{2Q}}|u|^{kp}\,dm\right)^{1/kp}\leq
C\ell(Q)\left(\frac{1}{m(T_{2Q})}\int_{T_{2Q}}|\nabla
u|^{p}\,dm\right)^{1/p},$

where $T_{2Q}$ is the same tent set over $2Q$ as in (5.12), and $C>0$ depends
only on $n$, the constants $C_{1}$-$C_{6}$, and $c$.

Proof. Take $z\in 2Q$. We start as in the proof of Lemma 5.23, and for each
$j\in\mathbb{N}$ we define $Q_{j}^{z}$ as the dyadic cube in
$\mathbb{D}_{j+k(Q)}$ that contains $z$. Let $X_{j}^{z}$ be a Corkscrew point
associated to $z$ and the scale $\ell(Q^{z}_{j})=2^{-j}\ell(Q)$; by
construction of $\mathcal{W}_{Q_{j}^{z}}$, we can find a cube
$I^{j,z}\in\mathcal{W}_{Q_{j}^{z}}$ containing $X_{j}^{z}$, and we denote by
$Y_{j}^{z}$ the center of $I^{j,z}$. By construction of
$\mathcal{W}_{Q_{j}^{z}}^{*}$,

$B_{j}^{z}:=B(Y_{j}^{z},\delta(Y_{j}^{z})/2)\subset U^{*}_{Q}.$

By Proposition 2.18, we can find a uniform integer $N=N(n,C_{1},C_{2})$ such
that we can link $B_{j}^{z}$ to $B^{z}_{j+1}$ by a Harnack chain of length
$N$. We construct a chain of balls $\\{\mathcal{B}_{i}^{z}\\}$ as follows: for
$j\in\mathbb{N}$, $\\{\mathcal{B}_{i}^{z}\\}_{jN\leq i\leq(j+1)N}$ is the
chain linking $B_{j}^{z}$ to $B^{z}_{j+1}$ given by Proposition 2.18 and used
to built $\mathcal{W}^{*}_{Q^{z}_{j+1}}$. The collection of balls
$(\mathcal{B}_{i}^{z})_{i\geq 0}$ that we just constructed has bounded
overlap, is included in $\gamma^{*}_{Q}(z)$, and is such that
$\operatorname{diam}\,\mathcal{B}_{i}^{z}\approx 2^{-i/N}\ell(Q)$; observe
also that by construction of ${\mathcal{W}}_{Q}$, we can choose $I^{0,z}$ (and
thus $B_{0}^{z}=\mathcal{B}_{0}^{z}$) independent of $z$, hence we write
$B_{0}$ for $B_{0}^{z}$.

For any subset $S\subset T_{2Q}$, we write as before $u_{S}$ for
$\fint_{S}u\,dm$. Theorem 7.1 will follow from Theorem 5.24 as soon as we
prove that for some $p_{1}\in[1,2)$,

(7.3) $|u_{B_{0}}|\lesssim\ell(Q)\left(\frac{1}{m(T_{2Q})}\int_{T_{2Q}}|\nabla
u|^{p}\,dm\right)^{1/p}$

holds for all $p\in[p_{1},2]$.

Let $q\in[p_{0},2]$, where $p_{0}$ is the value provided by Lemma 4.7. Thanks
to Theorem 6.6, for $\mu$-almost every $z\in E$, we have

$\lim_{j\to\infty}|u_{\mathcal{B}^{z}_{jN}}|=0.$

In particular,

$\begin{split}|u_{B_{0}}|&\leq\lim_{j\to+\infty}\big{\\{}|u_{\mathcal{B}^{z}_{jN}}|+|u_{B_{0}}-u_{\mathcal{B}^{z}_{jN}}|\big{\\}}\leq\lim_{j\to\infty}|u_{\mathcal{B}^{z}_{jN}}|+\sum_{0\leq
i<jN}|u_{\mathcal{B}^{z}_{i}}-u_{\mathcal{B}^{z}_{i+1}}|\leq\sum_{i\in\mathbb{N}}|u_{\mathcal{B}^{z}_{i}}-u_{\mathcal{B}^{z}_{i+1}}|.\end{split}$

Since $\mathcal{B}^{z}_{i}\cap\mathcal{B}^{z}_{i+1}$, $\mathcal{B}^{z}_{i}$,
and $\mathcal{B}^{z}_{i+1}$ have comparable sizes, the Poincaré inequality
(Lemma 4.7) gives that

$\begin{split}|u_{\mathcal{B}^{z}_{i}}-u_{\mathcal{B}^{z}_{i+1}}|&\leq|u_{\mathcal{B}^{z}_{i}}-u_{\mathcal{B}^{z}_{i+1}\cap\mathcal{B}^{z}_{i}}|+|u_{\mathcal{B}^{z}_{i+1}}-u_{\mathcal{B}^{z}_{i+1}\cap\mathcal{B}^{z}_{i}}|\\\
&\leq\fint_{\mathcal{B}^{z}_{i+1}\cap\mathcal{B}^{z}_{i}}\left(|u-u_{B^{z}_{i}}|+|u-u_{\mathcal{B}^{z}_{i+1}}|\right)\,dm\\\
&\lesssim\fint_{\mathcal{B}^{z}_{i}}|u-u_{\mathcal{B}^{z}_{i}}|\,dm+\fint_{\mathcal{B}^{z}_{i+1}}|u-u_{\mathcal{B}^{z}_{i+1}}|\,dm\\\
&\lesssim 2^{-i/N}\ell(Q)\left(\fint_{\mathcal{B}^{z}_{i}}|\nabla
u|^{q}\,dm+\fint_{\mathcal{B}^{z}_{i+1}}|\nabla
u|^{q}\,dm\right)^{\frac{1}{q}}.\\\ \end{split}$

So the last two computations yield that, for $\mu$-almost every $z\in E$

$\begin{split}|u_{B_{0}}|&\lesssim\sum_{i\in\mathbb{N}}2^{-i/N}\ell(Q)\left(\fint_{\mathcal{B}^{z}_{i}}|\nabla
u|^{q}dm\right)^{\frac{1}{q}}\lesssim\ell(Q)\left(\sum_{i\in\mathbb{N}}2^{-iq\alpha/N}\fint_{\mathcal{B}^{z}_{i}}|\nabla
u|^{q}dm\right)^{\frac{1}{q}}\end{split}$

where we applied Hölder’s inequality for the last part, and the price to pay
is that we need to introduce $\alpha\in(0,1)$, close to 1, that will be fixed
later on. Now observe that, for $Z\in\mathcal{B}^{z}_{i}$, we have
$\delta(Z)\approx 2^{-i/N}\ell(Q)$ and hence
$2^{-iq\alpha/N}\approx\delta(Z)^{q\alpha}\ell(Q)^{-q\alpha}$. So, by (H4),

$m(\mathcal{B}_{i}^{z})\approx m(B(Z,2\delta(Z)))\quad\text{ for
}Z\in\mathcal{B}_{i}^{z}.$

Hence

$\begin{split}|u_{B_{0}}|&\lesssim\ell(Q)^{1-\alpha}\left(\sum_{i\in\mathbb{N}}\int_{\mathcal{B}^{z}_{i}}|\nabla
u(Z)|^{q}\,\frac{\delta(Z)^{q\alpha}}{m(B(Z,2\delta(Z)))}\,dm(Z)\right)^{\frac{1}{q}}\\\
&\lesssim\ell(Q)^{1-\alpha}\left(\int_{\gamma_{Q}^{*}(z)}|\nabla
u(Z)|^{q}\,\frac{\delta(Z)^{q\alpha}}{m(B(Z,2\delta(Z)))}\,dm(Z)\right)^{\frac{1}{q}}\\\
\end{split}$

since the balls $\mathcal{B}_{i}^{z}$ have bounded overlap and are contained
in $\gamma_{Q}^{*}(z)$ (see the beginning of the proof, slightly above (7.3))
We average over $z\in E$ this estimate for $|u_{B_{0}}|^{q}$ to obtain

$\begin{split}|u_{B_{0}}|&=\left(\fint_{z\in
E}|u_{B_{0}}|^{q}d\mu(z)\right)^{\frac{1}{q}}\\\
&\lesssim\ell(Q)^{1-\alpha}\left(\fint_{E}\int_{\gamma_{Q}^{*}(z)}|\nabla
u(Z)|^{q}\,\frac{\delta(Z)^{q\alpha}}{m(B(Z,2\delta(Z)))}\,dm(Z)\,d\mu(z)\right)^{\frac{1}{q}}\\\
&\lesssim\ell(Q)^{1-\alpha}\left(\fint_{z\in
2Q}\int_{\gamma_{Q}^{*}(z)}|\nabla
u(Z)|^{q}\,\frac{\delta(Z)^{q\alpha}}{m(B(Z,2\delta(Z)))}\,dm(Z)\,d\mu(z)\right)^{\frac{1}{q}}\end{split}$

because we assume $\mu(E)\geq c\mu(2Q)$. Notice that $Z\in\gamma_{Q}^{*}(z)$
implies that $z\in B(Z,C\delta(Z))$. Therefore, by Fubini’s lemma and (5.12),

$\begin{split}|u_{B_{0}}|&\lesssim\ell(Q)^{1-\alpha}\left(\frac{1}{\mu(2Q)}\int_{Z\in
T_{2Q}}|\nabla
u(Z)|^{q}\,\frac{\delta(Z)^{q\alpha}}{m(B(Z,C\delta(Z)))}\,dm(Z)\int_{z\in
B(Z,C\delta(Z))}d\mu(z)\right)^{\frac{1}{q}}.\end{split}$

If $Z\in\Omega$, we pick $z_{0}\in\Gamma$ such that
$|Z-z_{0}|=\,\mathrm{dist}(Z,\Gamma)$, and define $\rho(Z)$ by
$\rho(z_{0},\delta(Z))$. The point $z_{0}$, and then $\rho(Z)$, are not
uniquely defined, but it is easy to check that two choice of $z_{0}$ will be
equivalent (up to constants independent of $Z$), or if the reader prefers, the
estimates below do not depend on our choice of $z_{0}$. The doubling
conditions (H3) and (H4) and the definition (2.6) imply that

$\int_{z\in
B(Z,C\delta(Z))}d\mu(z)\approx\mu(B(z_{0},\delta(Z)))\approx\frac{m(B(z_{0},\delta(Z))\cap\Omega)}{\delta(Z)\rho(z_{0},\delta(Z))}\approx\frac{m(B(Z,2\delta(Z)))}{\delta(Z)\rho(Z)},$

and the estimate on $u_{B_{0}}$ becomes

$|u_{B_{0}}|\lesssim\ell(Q)^{1-\alpha}\left(\frac{1}{\mu(2Q)}\int_{Z\in
T_{2Q}}|\nabla
u(Z)|^{q}\,\frac{\delta(Z)^{q\alpha-1}}{\rho(Z)}\,dm(Z)\right)^{\frac{1}{q}}.$

By (5.3), (5.15), (2.6), and (H4), one can show that
$m(T_{2Q})\approx\ell(Q)\rho(z_{Q},\ell(Q))\mu(2Q)$, where $z_{Q}$ is a fixed
point in $Q$. Therefore, for $p\in(q,2]$,

$\begin{split}|u_{B_{0}}|&\lesssim\ell(Q)\left(\ell(Q)^{1-q\alpha}\rho(z_{Q},\ell(Q))\fint_{Z\in
T_{2Q}}|\nabla
u(Z)|^{q}\frac{\delta(Z)^{q\alpha-1}}{\rho(Z)}dm(Z)\right)^{\frac{1}{q}}\\\
&\lesssim\ell(Q)\Big{(}\fint_{T_{2Q}}|\nabla
u|^{p}dm\Big{)}^{\frac{1}{p}}\left(\big{[}\ell(Q)^{1-\alpha
q}\rho(z_{Q},\ell(Q))\big{]}^{p/(p-q)}\fint_{Z\in
T_{2Q}}\left(\frac{\delta(Z)^{[q\alpha-1]}}{\rho(Z)}\right)^{\frac{p}{p-q}}dm(Z)\right)^{\frac{p-q}{qp}}\end{split}$

by the Hölder inequality. The claim (7.3) will thus be proven if we can
establish that, for some $\alpha\in(0,1)$ close to $1$,

(7.4) $I_{0}:=\fint_{Z\in
T_{2Q}}\left(\frac{\delta(Z)^{[q\alpha-1]}}{\rho(Z)}\right)^{\frac{p}{p-q}}dm(Z)\lesssim[\ell(Q)^{1-\alpha
q}\rho(z_{Q},\ell(Q))]^{-p/(p-q)}$

when $p<2$ is close enough to $2$.

By construction of $T_{2Q}$ (see (5.12) and (5.11)),

$T_{2Q}\subset\bigcup_{R\subset 100Q}U_{R}^{*}$

and the covering has a uniformly finite overlap. Notice also that

$\rho(Z)\approx\rho(R)\qquad\text{ for $Z\in U^{*}_{R}$ and
$R\in\mathbb{D}$},$

where $\rho(R)$ is defined in (5.17) and where the constants are independent
of $Z$ and $R$. We call $\rho(R)$ the value of $\rho(Z)$ for a $Z\in
U^{*}_{R}$. The two last observation allow us to write

(7.5) $I_{0}\lesssim\frac{1}{m(T_{2Q})}\sum_{R\in\mathbb{D}\atop R\subset
100Q}m(U^{*}_{R})\left(\frac{\ell(R)^{[q\alpha-1]}}{\rho(R)}\right)^{\frac{p}{p-q}}.$

We let the reader check that by definition of $\rho$, and by arguments similar
to the ones used to prove Lemma 5.16,

$m(T_{2Q})\approx\ell(Q)\rho(Q)\mu(Q)$

and

$m(U^{*}_{R})\approx\ell(R)\rho(R)\mu(R).$

The bound (7.5) becomes

(7.6) $I_{0}\lesssim\frac{1}{\ell(Q)\rho(Q)\mu(Q)}\sum_{R\in\mathbb{D}\atop
R\subset
100Q}\ell(R)^{[q\alpha-1]\frac{p}{p-q}+1}\rho(R)^{1-\frac{p}{p-q}}\mu(R).$

By (H5),

$\rho(R)\gtrsim\rho(Q)\left(\frac{\delta(R)}{\delta(Q)}\right)^{1-\epsilon}\approx\rho(Q)\left(\frac{\ell(R)}{\ell(Q)}\right)^{1-\epsilon},$

where $\epsilon=C_{5}^{-1}$ is the one given in (H5). We use this to replace
$\rho(R)$ in (7.6) by $\rho(Q)$; notice that the inequality goes in the right
direction because the exponent $1-\frac{p}{p-q}=\frac{-q}{p-q}$ is negative
(recall that $p\in(q,2]$). We get that

(7.7)
$I_{0}\lesssim\rho(Q)^{-p/(p-q)}\ell(Q)^{a}\mu(Q)^{-1}\sum_{R\in\mathbb{D}\atop
R\subset 100Q}\ell(R)^{b}\mu(R),$

with the exponents $a=-1+(\varepsilon-1)\big{(}1-\frac{p}{p-q}\big{)}$ and
$b=[q\alpha-1]\frac{p}{p-q}+1+(1-\varepsilon)\big{(}1-\frac{p}{p-q}\big{)}=[q\alpha-2+\epsilon]\frac{p}{p-q}+2-\epsilon$.

If $2-p\leq\min\\{\epsilon/2,(2-p_{0})/2\\}$, we can pick $\alpha\in(0,1)$
(small) and $q\in[p_{0},2)$ such that $q\alpha-2+\epsilon\geq 0$. With these
values, we can still pick $p\in(q,2]$ as above, and since the power for
$\ell(R)$ is $b>0$, we can bound $\ell(R)$ brutally by $\ell(Q)$, which gives

(7.8)
$I_{0}\lesssim\rho(Q)^{-p/(p-q)}\ell(Q)^{a+b}\mu(Q)^{-1}\sum_{R\in\mathbb{D}\atop
R\subset 100Q}\mu(R)\lesssim\rho(Q)^{-p/(p-q)}\ell(Q)^{a+b}.$

Notice that $a+b=[q\alpha-1]\frac{p}{p-q}$; the claim (7.4) follows from the
observation that $\rho(Q)\approx\rho(z_{Q},\ell(Q))$, and we have seen before
that (7.3) and Theorem 7.1 follow. $\Box$

In the following corollary of Theorem 7.1 we replace the tents $T_{2Q}$ by
balls.

######  Corollary 7.9.

Let $(\Omega,m,\mu)$ satisfy (H1)–(H6). There exists $p_{1}\in[1,2]$ and
$k:=k(C_{4})>1$ such that the following happens for $p\in[p_{1},2]$. Let
$\lambda>1$ be given, and let $x\in\Gamma$, $r>0$, and $u\in W$ be such that
$\operatorname{Tr}u=0$ on $B(x,\lambda r)\cap\Gamma$; then

(7.10) $\left(\fint_{B(x,r)\cap\Omega}|u|^{kp}\,dm\right)^{1/kp}\leq
C_{\lambda}r\left(\fint_{B(x,\lambda r)\cap\Omega}|\nabla
u|^{p}\,dm\right)^{1/p},$

where $C>0$ depends only on $n$, $C_{1}$ to $C_{6}$, and $\lambda$.

Proof. Let $x^{\prime}\in\Gamma$ and $r^{\prime}>0$ be given. Let
$Q^{\prime}\in\mathbb{D}$ be the only dyadic cube such that $x^{\prime}\in
Q^{\prime}$ and $r^{\prime}\leq k(Q^{\prime})<2r^{\prime}$. Then
$B(x^{\prime},r^{\prime})\subset 2Q^{\prime}$, and by Theorem 7.1 and (5.15),
there exists $K>1$ that depends only on $n$, $C_{1}$, and $C_{2}$ such that

(7.11)
$\left(\fint_{B(x^{\prime},r^{\prime})\cap\Omega}|u|^{kp}\,dm\right)^{1/kp}\leq
Cr^{\prime}\left(\fint_{B(x^{\prime},Kr^{\prime})\cap\Omega}|\nabla
u|^{p}\,dm\right)^{1/p},$

provided that $\operatorname{Tr}u\equiv 0$ on $Q^{\prime}\subset
B(x^{\prime},Kr^{\prime})\cap\Gamma$.

This looks like the desired estimate, but the constant $K$ is too large; we
will fix this with a covering argument. Set $\tau=(\lambda-1)r/100K$, with
$\lambda$ as in the statement and $K$ as above. Then let $x\in\Gamma$ and
$r>0$ be given. Denote by $(x_{i})_{i\in I}$ a maximal collection of points of
$\Gamma\cap B(x,(1+2\tau)r)$ such that $|x_{i}-x_{j}|\geq\tau r$ for $i\neq
j$. Thus the balls $B_{i}=B(x_{i},2\tau r)$, cover $\Gamma\cap
B(x,(1+2\tau)r)$, and the sets $D_{i}=\Omega\cap B(x_{i},4\tau r)$ cover

$H:=\big{\\{}X\in\Omega:\,\,\mathrm{dist}(X,\Gamma\cap B(x,(1+2\tau)r)\leq
2\tau r\big{\\}}$

Notice that $I$ has at most $C$ elements, with a constant that depends also on
$\lambda$ and $K$ through $\tau$, but this is all right. We can apply (7.11)
to each $B(x_{i},4\tau r)$, and we get that

(7.12) $\left(\fint_{D_{i}}|u|^{kp}\,dm\right)^{1/kp}\leq C\tau
r\left(\fint_{B(x,\lambda r)\cap\Omega}|\nabla u|^{p}\,dm\right)^{1/p},$

because $B(x_{i},4K\tau r)\subset B(x,\lambda r)$ by choice of $\tau$. We may
sum over $i$ and get that

(7.13) $\left(\fint_{H}|u|^{kp}\,dm\right)^{1/kp}\leq C\tau
r\left(\fint_{B(x,\lambda r)\cap\Omega}|\nabla u|^{p}\,dm\right)^{1/p},$

and now we just need to take care of $H_{1}=\Omega\cap B(x,r)\setminus H$. Let
$(y_{j})_{j\in J}$ be a maximal collection of points of $H_{1}$, with
$|y_{i}-y_{j}|\geq\tau r$ for $i\neq j$. Thus $J$ has at most $C=C(\tau)$
points, and the set $B_{j}=B(y_{j},2\tau r)$, $j\in J$, cover $H$. We want to
control each $\fint_{B_{j}}|u|^{kp}\,dm$, and then we’ll sum.

Fix $j\in J$, and call $z_{j}$ the first point of $[y_{j},x]$ (starting from
$y_{j}$) that lies within $\tau r$ from $\Gamma$. Obviously $z_{j}\in B(x,r)$,
and $B(z_{j},\tau r)\subset H$ because $\Gamma\setminus B(x,(1+2\tau)r)$ is
too far. Now denote by $W_{j}$ the convex hull of $B(y_{j},\tau r)$ and
$B(z_{j},\tau r)$ (a nice tube contained in $\Omega$) and set
$\widetilde{W}_{j}=W_{j}\cup B_{j}$ (with a larger head around $y_{j}$, and
still contained in $\Omega$). It is easy to see that $\widetilde{W}_{j}$
satisfies the chain condition $C(\kappa,M)$ of Definition 5.21, with any small
$\kappa$ chosen in advance, and with an $M$ that depends only on $\kappa$ and
$\tau$; we can take $B(z_{j},\tau r/2)$ as the distinguished ball. This allows
us to apply Theorem 5.24, and prove that

(7.14) $\left(\fint_{B_{j}}|u-\bar{u}|^{kp}\,dm\right)^{1/kp}\leq
C\left(\fint_{\widetilde{W}_{j}}|u-\bar{u}|^{kp}\,dm\right)^{1/kp}\leq
Cr\left(\fint_{\widetilde{W}_{j}}|\nabla u|^{p}\,dm\right)^{1/p},$

where $\bar{u}$ denotes the average of $u$ on $B(z_{j},\tau r/2)$. Now
$|\bar{u}|\leq C\left(\fint_{B(x,\lambda r)\cap\Omega}|\nabla
u|^{p}\,dm\right)^{1/p}$, by (7.13) and because $B(z_{j},\tau r)\subset H$,
and $\widetilde{W}_{j}\subset B(x,(1+2\tau)r)\subset B(x,\lambda r)$ by
definition of $\tau$, we we may sum (7.14) over $j$ and get that

(7.15) $\left(\fint_{H_{1}}|u|^{kp}\,dm\right)^{1/kp}\leq C\tau
r\left(\fint_{B(x,\lambda r)\cap\Omega}|\nabla u|^{p}\,dm\right)^{1/p}.$

We combine this with (7.13) and get (7.11), as needed for Corollary 7.9.
$\Box$

##  8\. The Extension Theorem

The aim of this section is the construction of an extension operator
$\operatorname{Ext}:\,H\to W$ such that the composition
$\operatorname{Tr}\circ\operatorname{Ext}$ is the identity on $H$. The section
can be seen as the dual of Section 6. As in Section Section 6, the results
will be only proven when assuming $\Gamma$ and $\Omega$ unbounded, and the
proof in the bounded case is very similar and is discussed in Section 13.

We assume that $\Gamma$ and $\Omega$ are unbounded. The beginning of this
section is similar to [DFM2, Section 7], but the proof of the density result
Lemma 8.12 is different.

We shall construct $\operatorname{Ext}$ with the help of a Whitney extension.
But first, it is crucial to observe that for any $g\in
L^{1}_{loc}(\Gamma,\mu)$ and $\mu$-almost every $x\in\Gamma$, one has

(8.1) $\lim_{r\to 0}\fint_{B(x,r)}|g(y)-g(x)|\,d\mu(y)=0.$

This is a consequence of the Lebesgue differentiation theorem in doubling
spaces (see for instance [Fed, Sections 2.8-2.9]). It is easy to verify that
$H\subset L^{1}_{loc}(\Gamma,\mu)$ (see (6.5)) and so (8.1) holds for any
function $g\in H$.

Our construction will rely on the family ${\mathcal{W}}$ of dyadic Whitney
cubes already used in Section 5. We associate to $\mathcal{W}$ a partition of
unity $\\{\varphi_{I}\\}_{I\in{\mathcal{W}}}$ where the $\varphi_{I}$ are
smooth functions supported in $2I$ that satisfy $0\leq\varphi_{I}\leq 1$,
$|\nabla\varphi_{I}|\leq C/\ell(I)$ and
$\sum_{I\in{\mathcal{W}}}\varphi_{I}={\mathds{1}}_{\Omega}$.

We record a few properties of ${\mathcal{W}}$, that can be found in [Ste,
Chapter VI]. If two dyadic cubes $I$ and $I^{\prime}$ are such that $2I\cap
2I^{\prime}\neq\emptyset$, then $\ell(I)/\ell(I^{\prime})\in\\{1/2,1,2\\}$,
and also $I^{\prime}\subset 6I$.

Hence, for a given $I$,

(8.2) the number of cubes $I^{\prime}\in{\mathcal{W}}$ such that
$2I^{\prime}\cap 2I\neq 0$ is at most $2\cdot 12^{n}$,

because each such $I^{\prime}$ needs to be a dyadic cube in $6I$ such that
$\ell(I)\geq\ell(I)/2$.

For each $I\in\mathcal{W}$, we write $\delta(I)=\,\mathrm{dist}(I,\Gamma)$,
pick a point $\xi_{I}\in\Gamma$ such that $\,\mathrm{dist}(\xi_{I},I)\leq
2\delta(I)$, and set $B_{I}=B(\xi_{I},\ell(I))$.

We define the extension operator $\operatorname{Ext}$ on functions $g\in
L^{1}_{loc}(\Gamma,\mu)$ by

(8.3) $\operatorname{Ext}g(X):=\sum_{I\in{\mathcal{W}}}\varphi_{I}(X)y_{I},$

where

(8.4) $y_{I}:=\fint_{B_{I}}g(z)\,d\mu(z).$

If we wanted to have an extension operator on - for instance - Lipschitz
function, we could take $y_{I}=g(\xi_{I})$. However, since the function $g$ is
not smooth (and maybe not even defined everywhere), we need this extra
average; a good way to see this is to notice that otherwise we would only use
the values of $g$ on the countable set $\\{\xi_{I}\\}_{I\in{\mathcal{W}}}$,
which does not make sense for functions in $L^{1}_{loc}(\Gamma,\mu)$.

Notice that $\operatorname{Ext}g$ lies in $C^{\infty}(\Omega)$ because (8.2)
yields that the sum in (8.3) is locally finite. Moreover, if $g$ is continuous
on $\Gamma$, then $\operatorname{Ext}g$ is continuous on $\overline{\Omega}$
(see [Ste, Proposition VI.2.2]).

######  Theorem 8.5.

Let $(\Omega,m,\mu)$ satisfies (H1)–(H6). For any $g\in
L^{1}_{loc}(\Gamma,\mu)$

(8.6) $\operatorname{Tr}\circ\operatorname{Ext}g=g\qquad\text{ $\mu$-a.e. in
$\Gamma$}.$

Moreover, $\operatorname{Ext}$ is a bounded linear operator from $H$ to $W$,
i.e. there exists $C:=C(C_{3},C_{4},C_{5})>0$ such that for any $g\in H$,

(8.7) $\int_{\Omega}|\nabla\operatorname{Ext}g|^{2}\,dm\leq
C\|g\|_{H}^{2}:=C\int_{\Gamma}\int_{\Gamma}\frac{\rho(x,|x-y|)^{2}|g(x)-g(y)|^{2}}{m(B(x,|x-y|))}\,d\mu(x)\,d\mu(y).$

Proof. Let $g\in L^{1}_{loc}(\Gamma,\mu)$ be given. We write $u$ for
$\operatorname{Ext}g$ and we want to show that $\operatorname{Tr}u=g$, in the
sense that (6.7) holds with $Tru(x)=g(x)$ for $\mu$-almost every $x\in\Gamma$,
regardless of whether $g\in H$ or $u\in W$. We will actually prove the
following stronger result, anlogous to (6.8): for $\mu$-a.e. $x\in\Gamma$, one
has

(8.8) $\lim_{X\in\gamma(x)\atop{\delta(X)\to
0}}\fint_{B(X,\delta(X)/2)}|u(Z)-g(x)|\,dm(Z)=0.$

Since we only want to prove (8.8) for $\mu$-a.e. point, we can restrict to the
case when $x$ is a Lebesgue point of $g$, that is, when (8.1) is satisfied.

Fix such an $x\in\Gamma$ and $X\in\gamma(x)$. We write $B$ for
$B(X,\delta(X)/2)$. Then

$\fint_{B}|u(Z)-g(x)|\,dm(Z)\leq\frac{1}{m(B)}\sum_{R\in{\mathcal{W}}(B)}\int_{R}|u(Z)-g(x)|\,dm(Z),$

where ${\mathcal{W}}(B)$ is the set of dyadic cubes
$I^{\prime}\in{\mathcal{W}}$ that meet $B$. It is easy to check that
${\mathcal{W}}(B)$ contains a finite number of cubes $I^{\prime}$ (the number
is bounded uniformly in $X\in\Omega$),

for which $\ell(I^{\prime})\approx\delta(X)$, and then, by (H4),
$m(I^{\prime})\approx m(B)$. So (8.8) will be proven if we can establish that

(8.9) $\fint_{I^{\prime}}|u(Z)-g(x)|\,dm(Z)\longrightarrow 0\quad\text{ as
$\delta(I^{\prime})\to 0$},$

where we restrict to dyadic cubes $I^{\prime}\in{\mathcal{W}}$ such that $x\in
KI^{\prime}$ for some large enough constant $K:=K(n)$. Recall from the
definition (8.3) of $u=\operatorname{Ext}g$ that
$u(Z)=\sum_{I\in{\mathcal{W}}}\varphi_{I}(Z)y_{I}$. We observed earlier that
the sum is locally finite on $I^{\prime}$, and the cubes $I$ for which
$\varphi_{I}$ does not vanish identically on $I$ are such that
$I^{\prime}\subset 6I$ and $\ell(I)/\ell(I^{\prime})\in\\{1/2,1,2\\}$. We
deduce that any such $I$ satisfies $B_{I}\subset
K^{\prime}B_{I^{\prime}}\subset B(x,K^{\prime\prime}\delta(I^{\prime}))$ and
$B_{I^{\prime}}\subset K^{\prime}B_{I}\subset
B(x,K^{\prime\prime}\delta(I^{\prime}))$, and then by (H3) that
$\mu(B_{I})\approx\mu(B(x,K^{\prime\prime}\delta(I^{\prime})))$. The
conclusion is that

$\begin{split}\fint_{I^{\prime}}|u(Z)-g(x)|\,dm(Z)&=\fint_{I^{\prime}}\bigg{|}\sum_{I\in{\mathcal{W}}:\,2I\cap
2I^{\prime}\neq\emptyset}\varphi_{I}(Z)\fint_{B_{I}}[g(z)-g(x)]\,d\mu(z)\bigg{|}\,dm(Z)\\\
&\lesssim\sum_{I\in{\mathcal{W}}:\,2I\cap
2I^{\prime}\neq\emptyset}\fint_{B_{I}}|g(z)-g(x)|\,d\mu(z)\\\
&\lesssim\fint_{B(x,K^{\prime\prime}\delta(I^{\prime}))}|g(z)-g(x)|\,d\mu(z)\end{split}$

because the number of $I\in{\mathcal{W}}$ that verify $2I\cap
2I^{\prime}\neq\emptyset$ is uniformly bounded. Thanks to (8.1), the right-
hand side above converges to 0 as $\delta(I^{\prime})\to 0$. The claims (8.9),
(8.8), and then (8.6), follow.

Now, we want to show that for $g\in H$, $u\in W$ and even
$\|u\|_{W}\lesssim\|g\|_{H}$. Recall that $u$ is smooth on $\Omega$ because
the sum in (8.3) is locally finite, so $u$ is locally integrable in $\Omega$,
and its distribution derivative is locally integrable too, and given by

(8.10) $\nabla
u(X)=\sum_{I\in{\mathcal{W}}}y_{I}\nabla\varphi_{I}(X)=\sum_{I\in{\mathcal{W}}}[y_{I}-y_{I^{\prime}}]\nabla\varphi_{I}(X),$

where $I^{\prime}$ is any cube (that may depend on $X$ but not on $I$), and
the identity holds because
$\sum_{I}\nabla\varphi_{I}=\nabla(\sum_{I}\varphi_{I})=\nabla 1=0$. So we only
need to show that $\|u\|_{W}\leq C\|g\|_{H}<+\infty$. First decompose
$\|u\|^{2}_{W}$ as

(8.11)
$\|u\|^{2}_{W}=\sum_{I^{\prime}\in{\mathcal{W}}}\int_{I^{\prime}}|\nabla
u|^{2}\,dm.$

For the moment, we fix $I^{\prime}\in{\mathcal{W}}$ and $X\in I^{\prime}$, and
get a bound on $|\nabla u(X)|$. If ${\mathcal{W}}(I^{\prime})$ denotes the
sets of dyadic cubes $I\in{\mathcal{W}}$ such that $2I$ meets $I^{\prime}$,
then

$|\nabla
u(X)|\leq\sum_{I\in{\mathcal{W}}(I^{\prime})}|y_{I}-y_{I^{\prime}}||\nabla\varphi_{I}(X)|\lesssim\ell(I^{\prime})^{-1}\sum_{I\in{\mathcal{W}}(I^{\prime})}|y_{I}-y_{I^{\prime}}|$

because
$\nabla\varphi_{I}\lesssim\delta(I)^{-1}\approx\delta(I^{\prime})^{-1}$. We
use the definition of $y_{I},y_{I^{\prime}}$, the facts that $I\subset
6I^{\prime}$ and $\delta(I)\approx\delta(I^{\prime})$ to obtain that

$\begin{split}|y_{I}-y_{I^{\prime}}|&\leq\fint_{B_{I}}\fint_{B_{I^{\prime}}}|g(x)-g(y)|d\mu(x)\,d\mu(y)\\\
&\leq\left(\fint_{B_{I}}\fint_{B_{I^{\prime}}}|g(x)-g(y)|^{2}d\mu(x)\,d\mu(y)\right)^{\frac{1}{2}}\\\
&\lesssim\mu(B_{I^{\prime}})^{-1}\left(\int_{100B_{I^{\prime}}}\int_{B_{I^{\prime}}}|g(x)-g(y)|^{2}d\mu(x)\,d\mu(y)\right)^{\frac{1}{2}}.\end{split}$

The combination of the last two computations gives

$|\nabla
u(X)|\lesssim\mu(B_{I^{\prime}})^{-1}\ell(I^{\prime})^{-1}\left(\int_{100B_{I^{\prime}}}\int_{B_{I^{\prime}}}|g(x)-g(y)|^{2}d\mu(x)\,d\mu(y)\right)^{\frac{1}{2}}$

since ${\mathcal{W}}(I^{\prime})$ contains at most $2\cdot 12^{n}$ elements,
and then

$\int_{I^{\prime}}|\nabla
u|^{2}\,dm\lesssim\ell(I^{\prime})^{-2}\mu(B_{I^{\prime}})^{-2}m(I^{\prime})\int_{100B_{I^{\prime}}}\int_{B_{I^{\prime}}}|g(x)-g(y)|^{2}d\mu(x)\,d\mu(y).$

We inject the above estimate in (8.11) and obtain that

$\begin{split}\|u\|_{W}^{2}&\lesssim\sum_{I^{\prime}\in{\mathcal{W}}}\ell(I^{\prime})^{-2}\mu(B_{I^{\prime}})^{-2}m(I^{\prime})\int_{100B_{I^{\prime}}}\int_{B_{I^{\prime}}}|g(x)-g(y)|^{2}d\mu(x)\,d\mu(y)\\\
&\lesssim\int_{\Gamma}\int_{\Gamma}|g(x)-g(y)|^{2}h(x,y)d\mu(x)\,d\mu(y),\end{split}$

where

$h(x,y):=\sum_{I^{\prime}\in{\mathcal{W}}}\ell(I^{\prime})^{-2}\mu(B_{I^{\prime}})^{-2}m(I^{\prime}){\mathds{1}}_{100B_{I^{\prime}}}(x){\mathds{1}}_{B_{I^{\prime}}}(y).$

Fix $x,y\in\Gamma$. Observe that if $I^{\prime}$ satisfies $(x,y)\in
100B_{I^{\prime}}\times B_{I^{\prime}}$, then by (H3),
$\mu(B_{I^{\prime}})\approx\mu(B(x,\ell(I^{\prime})))$ and by (H4),
$m(I^{\prime})\approx m(B(x,\ell(I^{\prime}))\cap\Omega)$. Hence by (2.6)

$\frac{m(I^{\prime})}{\ell(I^{\prime})^{2}\mu(B_{I^{\prime}})^{2}}\approx\frac{\rho(x,\ell(I^{\prime}))}{\ell(I^{\prime})\mu(B(x,\ell(I^{\prime})))}\,.$

Under the same assumption on $I^{\prime}$, we also have $|x-y|\leq
101\ell(I^{\prime})$, so by (H4) again,
$\mu(B(x,\ell(I^{\prime})))^{-1}\lesssim\mu(B(x,|x-y|))^{-1}$. In addition,
(H5) gives that

$\rho(x,\ell(I^{\prime}))\lesssim\rho(x,|x-y|)\left(\frac{\ell(I^{\prime})}{|x-y|}\right)^{1-\epsilon}$

where $\epsilon:=C_{5}^{-1}>0$ (notice that if $\ell(I^{\prime})\leq|x-y|\leq
101\ell(I^{\prime})$, we don’t need to use (H5), just the doubling
properties). All this yields

$h(x,y)\lesssim\frac{\rho(x,|x-y|)}{\mu(B(x,|x-y|))|x-y|^{1-\epsilon}}\sum_{I^{\prime}\in{\mathcal{W}}\atop\delta(I^{\prime})\geq|x-y|/101}\ell(I^{\prime})^{-\epsilon}\,{\mathds{1}}_{B_{I^{\prime}}}(y).$

Since $B_{I^{\prime}}\subset\kappa I^{\prime}$ for some constant
$\kappa:=\kappa(n)>1$ that does not depend on $I^{\prime}$, we see that for
each $j\in\mathbb{Z}$, the number of dyadic cubes $I^{\prime}$ such that
$\ell(I^{\prime})=2^{j}$ and $y\in B_{I^{\prime}}$ is uniformly bounded.
Together with the fact that $\delta(I^{\prime})\approx\ell(I^{\prime})$, this
yields

$\sum_{I^{\prime}\in{\mathcal{W}}\atop\ell(I^{\prime})\geq|x-y|/101}\ell(I^{\prime})^{-\epsilon}\,{\mathds{1}}_{B_{I^{\prime}}}(y)\lesssim\sum_{k\in\mathbb{N}}(2^{k}|x-y|)^{-\epsilon}\lesssim|x-y|^{-\epsilon}.$

Altogether,

$h(x,y)\lesssim\frac{\rho(x,|x-y|)}{\mu(B(x,|x-y|))|x-y|}\approx\frac{\rho(x,|x-y|)^{2}}{m(B(x,|x-y|)\cap\Omega)}$

by (2.6) and thus

$\|u\|_{W}^{2}\lesssim\int_{\Gamma}\int_{\Gamma}\frac{\rho(x,|x-y|)^{2}|g(x)-g(y)|^{2}}{m(B(x,|x-y|)\cap\Omega)}d\mu(x)\,d\mu(y):=\|g\|_{H}^{2}$

as desired (see the definition (6.5)). Theorem 8.5 follows. $\Box$

######  Lemma 8.12.

For every $g\in H$, we can find a sequence $(g_{k})_{k\in\mathbb{N}}$ of
functions in $C^{\infty}(\mathbb{R}^{n})$ whose restrictions to $\Gamma$ (we
still call them $g_{k}$) belong to $H$ and such that $(g_{k})_{k}$ converges
to $g$ in $H$, $L^{2}_{loc}(\Gamma,\mu)$ and $\mu$-a.e. pointwise.

######  Remark 8.13.

The above density result (whose proof doesn’t use Theorem 8.5) actually
entails the Lebesgue density result given as (8.1). The proof of this
implication uses maximal functions, is classical, and is left to the reader.

Proof. For the density of smooth functions, we are given $g\in H$ and we want
to approximate it with smooth functions. The simplest way for us to construct
functions $g_{k}$ will be to use our dyadic decompositions $\mathbb{D}_{k}$ of
$\Gamma$, but coverings of $\Gamma$ with balls of radius $2^{-k}$ would work
as well. We associate to $\mathbb{D}_{k}$ a collection of smooth functions
$\\{\varphi_{Q}\\}_{Q\in\mathbb{D}_{k}}$ such that $\varphi_{Q}$ is supported
in $2Q$, $\sum_{Q\in\mathbb{D}_{k}}\varphi_{Q}=1$ near $\Gamma$, and
$\|\nabla\varphi_{Q}\|_{\infty}\leq C2^{k}$. Finally we set

(8.14) $g_{k}(x)=\sum_{Q\in\mathbb{D}_{k}}\varphi_{Q}(x)y_{Q}$

for $x\in\Gamma$, where we take $y_{Q}=\fint_{2Q}g(y)d\mu(y)$. It is obvious
that $g_{k}$ is a smooth function on $\mathbb{R}^{n}$ (the sum in (8.14) is
locally finite). We shall prove now that

(8.15) $\|g-g_{k}\|^{2}_{H}\leq CJ(k),\ \text{ where
}J(k)=\iint_{x,y\in\Gamma\,;\,|x-y|\leq
2^{3-k}}{\rho(x,|x-y|)^{2}|g(x)-g(y)|^{2}\over
m(B(x,|x-y|)\cap\Omega)}d\mu(x)d\mu(y).$

Notice that $J_{k}$ is a subintegral of $\|g\|_{H}^{2}$, where the domain of
integration decreases to the empty set when $k$ tends to $+\infty$; thus
$\lim_{k\to+\infty}J(k)=0$, and as soon as we prove (8.15), we will get that
$g_{k}\in H$ and $g_{k}$ tends to $g$ in $H$; the density of smooth functions
in $H$ will follow.

We need some notation. Set $h_{k}=g-g_{k}$ and for $Q\in\mathbb{D}$

(8.16) ${\mathcal{R}}_{Q}=\big{\\{}(x,y)\in Q\times
2Q\,;|x-y|\geq\ell(Q)/2\big{\\}}.$

Every pair of points $(x,y)\in\Gamma^{2}$ lies in at least one
${\mathcal{R}}_{Q}$: choose $j$ such that $2^{-j-1}\leq|x-y|<2^{-j}$, let
$Q=Q^{j}(x)$ be the element of ${\mathcal{D}}_{j}$, and observe that $y\in
B(x,2^{-j})\subset 2Q$, hence $(x,y)\in{\mathcal{R}}_{Q}$. Because of this,

(8.17) $\|g-g_{k}\|^{2}_{\mathcal{H}}\leq\sum_{Q\in\mathbb{D}}T_{Q}^{k},$

where

$\displaystyle T_{Q}^{k}$ $\displaystyle:=$
$\displaystyle\iint_{{\mathcal{R}}_{Q}}{\rho(x,|x-y|)^{2}|h_{k}(x)-h_{k}(y)|^{2}\over
m(B(x,|x-y|)\cap\Omega)}d\mu(x)d\mu(y)$ $\displaystyle=$
$\displaystyle\iint_{{\mathcal{R}}_{Q}}{|h_{k}(x)-h_{k}(y)|^{2}\over\mu(B(x,|x-y|))}\frac{\rho(x,|x-y|)}{|x-y|}d\mu(x)d\mu(y)$
$\displaystyle\lesssim$
$\displaystyle\frac{\rho(Q)}{\ell(Q)}\iint_{{\mathcal{R}}_{Q}}\frac{|h_{k}(x)-h_{k}(y)|^{2}}{\mu(2Q)}d\mu(x)d\mu(y)$

because $|x-y|\approx\ell(Q)$, and by (H3)–(H4) and the definitions(2.6) and
(5.17).

We start the estimate of $T_{Q}^{k}$ with the large scales, where we shall
merely estimate the size of $h_{k}$ on $\Gamma$. Let us check that for any
cube $Q^{*}$ such that $\ell(Q^{*})\geq 2^{-k}$,

(8.19) $\int_{2Q^{*}}|h_{k}(x)|^{2}d\mu(x)\leq C\int_{x\in
8Q^{*}}\frac{1}{\mu(B(x,2^{2-k}))}\int_{z\in
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x).$

We shall first estimate the contribution of a a given cube
$Q_{0}\in\mathbb{D}_{k}(Q^{*})$, where

(8.20) $\mathbb{D}_{k}(Q^{*}):=\\{Q_{0}\in\mathbb{D}_{k},\,2Q_{0}\cap
2Q^{*}\neq\emptyset\\},$

and then sum. So let $Q_{0}\in\mathbb{D}_{k}(Q^{*})$ be given. We estimate

(8.21) $\displaystyle a(Q_{0})$ $\displaystyle:=$ $\displaystyle\int_{x\in
2Q_{0}}|h_{k}(x)|^{2}d\mu(x)=\int_{2Q_{0}}|g(x)-g_{k}(x)|^{2}d\mu(x)$
$\displaystyle=$ $\displaystyle\int_{x\in
2Q_{0}}|g(x)-\sum_{Q\in\mathbb{D}_{k}}\varphi_{Q}(x)y_{Q}|^{2}d\mu(x)=\int_{x\in
2Q_{0}}\big{|}\sum_{Q\in\mathbb{D}_{k}}\varphi_{Q}(x)[g(x)-y_{Q}]\big{|}^{2}d\mu(x)$
$\displaystyle\leq$ $\displaystyle\int_{x\in
2Q_{0}}\sum_{Q\in\mathbb{D}_{k}}\varphi_{Q}(x)|g(x)-y_{Q}|^{2}d\mu(x)\leq\int_{x\in
2Q_{0}}\sum_{Q\in\mathbb{D}_{k}(Q_{0})}|g(x)-y_{Q}|^{2}d\mu(x)$

by (8.14), the fact that $\sum_{Q}\varphi_{Q}(x)=1$, Cauchy-Schwarz for a
finite average, and the fact that $\varphi_{Q}(x)=0$ outside of $2Q$. Notice
that when $Q\in\mathbb{D}_{k}(Q_{0})$ and $x\in 2Q_{0}$,

(8.22) $\displaystyle|g(x)-y_{Q}|^{2}$ $\displaystyle=$
$\displaystyle\left|\fint_{z\in
2Q}[g(x)-g(z)]d\mu(z)\right|^{2}\leq\fint_{z\in 2Q}|g(x)-g(z)|^{2}d\mu(z)$
$\displaystyle\leq$ $\displaystyle C\fint_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)$

because for $x\in 2Q_{0}$, the fact that $2Q_{0}\cap 2Q\neq\emptyset$ implies
that $2Q\subset B(x,2^{2-k})$, and $\mu$ is doubling by (H3). Hence, since the
number of element in $\mathbb{D}_{k}(Q_{0})$ is bounded,

(8.23) $\displaystyle a(Q_{0})$ $\displaystyle\leq$ $\displaystyle C\int_{x\in
2Q_{0}}\sum_{Q\in\mathbb{D}_{k}(Q_{0})}\fint_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x)$ $\displaystyle\leq$ $\displaystyle
C\int_{x\in 2Q_{0}}\fint_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x)$ $\displaystyle\leq$ $\displaystyle
C\int_{x\in 2Q_{0}}\frac{1}{\mu(B(x,2^{2-k}))}\int_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x).$

Now, we sum on $Q_{0}\in\mathbb{D}_{k}(Q^{*})$ and get that

(8.24) $\displaystyle\int_{2Q^{*}}|h_{k}(x)|^{2}d\mu(x)$ $\displaystyle\leq$
$\displaystyle\sum_{Q_{0}\in\mathbb{D}_{k}(Q^{*})}\int_{2Q_{0}}|h_{k}(x)|^{2}d\mu(x)\leq\sum_{Q_{0}\in\mathbb{D}_{k}}a(Q_{0})$
$\displaystyle\lesssim$
$\displaystyle\sum_{Q_{0}\in\mathbb{D}_{k}(Q^{*})}\int_{x\in
2Q_{0}}\frac{1}{\mu(B(x,2^{2-k}))}\int_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x)$ $\displaystyle\lesssim$
$\displaystyle\int_{x\in
8Q^{*}}\frac{1}{\mu(B(x,2^{2-k}))}\int_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x)$

because the $2Q_{0}$ cover $Q^{*}$ (actually, they cover $2Q^{*}$), are
contained in $8Q^{\ast}$, and have bounded covering; the estimate (8.19)
follows.

Recall that since $\ell(Q^{\ast})\geq 2^{-k}$, (H5) and Lemma 5.16 imply that

$\frac{2^{2-k}}{\rho(x,2^{2-k})}\lesssim\frac{\ell(Q^{*})}{\rho(Q^{*})}\left(\frac{\ell(Q^{*})}{2^{-k}}\right)^{-\epsilon}$

(with $\epsilon=C_{5}^{-1}$ as usual) and, for $z\in B(x,2^{2-k})\cap\Gamma$,

$\frac{\rho(x,2^{2-k})}{2^{2-k}}\lesssim\frac{\rho(x,|x-z|)}{|x-z|}\left(\frac{2^{-k}}{|x-z|}\right)^{-\epsilon}\lesssim\frac{\rho(x,|x-z|)}{|x-z|}.$

We return to (8.19), use the two estimates above and the fact that
$\mu(B(x,|x-z|))\leq C\mu(B(x,2^{2-k}))$ when $B(x,2^{2-k})$, and get that

(8.25) $\begin{split}\int_{2Q^{*}}|h_{k}|^{2}d\mu&\lesssim\int_{x\in
8Q^{*}}\frac{\rho(x,2^{2-k})}{2^{2-k}\mu(B(x,2^{2-k}))}\frac{2^{2-k}}{\rho(x,2^{2-k})}\int_{z\in\Gamma\cap
B(x,2^{2-k})}|g(x)-g(z)|^{2}d\mu(z)d\mu(x)\\\
&\lesssim\frac{\ell(Q^{*})}{\rho(Q^{*})}\left(\frac{\ell(Q^{*})}{2^{-k}}\right)^{-\epsilon}\int_{x\in
8Q^{*}}\int_{z\in
B(x,2^{2-k})}\frac{\rho(x,|x-z|)|g(x)-g(z)|^{2}}{\mu(B(x,|x-z|))|x-z|}d\mu(z)d\mu(x)\\\
&=\frac{\ell(Q^{*})}{\rho(Q^{*})}\left(\frac{\ell(Q^{*})}{2^{-k}}\right)^{-\epsilon}\int_{x\in
8Q^{*}}\int_{z\in
B(x,2^{2-k})}\frac{\rho(x,|x-z|)^{2}|g(x)-g(z)|^{2}}{m(B(x,|x-z|)\cap\Omega)}d\mu(z)d\mu(x)\end{split}$

where the last estimate comes from the definition of $\rho$. The right-hand
side tends to $0$ (for any fixed $Q^{\ast}$) when $k$ tends to $+\infty$, so
(8.25) means that $(g_{k})$ converges to $g$ in $L^{2}_{loc}(\Gamma,\mu)$.

Let us return to $T^{k}_{Q^{*}}$, starting with the case when $\ell(Q^{*})\geq
2^{-k}$. Observe that by (8) and (8.25)

$\displaystyle T^{k}_{Q^{*}}$ $\displaystyle\lesssim$
$\displaystyle\frac{\rho(Q^{*})}{\ell(Q^{*})}\iint_{{\mathcal{R}}_{Q^{*}}}\frac{|h_{k}(x)-h_{k}(y)|^{2}}{\mu(2Q^{*})}d\mu(x)d\mu(y)$
$\displaystyle\lesssim$
$\displaystyle\frac{\rho(Q^{*})}{\ell(Q^{*})}\iint_{{\mathcal{R}}_{Q^{*}}}\frac{|h_{k}(x)|^{2}+|h_{k}(y)|^{2}}{\mu(2Q^{*})}d\mu(x)d\mu(y)$
$\displaystyle\lesssim$
$\displaystyle\frac{\rho(Q^{*})}{\ell(Q^{*})}\int_{2Q^{*}}|h_{k}(x)|^{2}d\mu(x)$
$\displaystyle\lesssim$
$\displaystyle\left(\frac{\ell(Q^{*})}{2^{-k}}\right)^{-\epsilon}\int_{x\in
8Q^{*}}\int_{z\in
B(x,2^{2-k})}\frac{\rho(x,|x-z|)^{2}|g(x)-g(z)|^{2}}{m(B(x,|x-z|)\cap\Omega)}d\mu(z)d\mu(x).$

We now sum this over $Q^{\ast}$ such that $\ell(Q^{*})\geq 2^{-k}$. Fix
$x,z\in\Gamma$; for each generation $j$, $j\leq k$, there are at most $C$
cubes $Q^{\ast}$, $Q^{*}\in\mathbb{D}_{j}$, such that $x\in 8Q^{*}$.
Therefore,

(8.27) $\sum_{j\leq
k}\sum_{Q^{*}\in\mathbb{D}_{j}}T_{Q^{*}}^{k}\lesssim\sum_{j\geq
k}2^{(j-k)\epsilon}J(k)\lesssim J(k),$

where $J(k)$ is as in (8.15). This part fits with (8.15) (see (8.17)).

For the small scales we shall use the regularity of $g_{k}$. That is, for
$\ell(Q)\leq 2^{-k}$, we recall that $h_{k}=g-g_{k}$, hence, by the first part
of (8), $T_{Q}^{k}\leq 2U_{Q}^{k}+2V_{Q}^{k}$, where

(8.28)
$V_{Q}^{k}:=\iint_{{\mathcal{R}}_{Q}}{|g_{k}(x)-g_{k}(y)|^{2}\over\mu(B(x,|x-y|))}\frac{\rho(x,|x-y|)}{|x-y|}d\mu(x)d\mu(y)$

and

(8.29)
$W_{Q}^{k}:=\iint_{{\mathcal{R}}_{Q}}{|g(x)-g(y)|^{2}\over\mu(B(x,|x-y|))}\frac{\rho(x,|x-y|)}{|x-y|}d\mu(x)d\mu(y)$

are the analogues of $T_{Q}^{k}$ for $g_{k}$ and $g$.

We deduce from the definition (8.16) that $\ell(Q)/2\leq|x-y|\leq 2\ell(Q)$
when $(x,y)\in{\mathcal{R}}_{Q}$, so a given pair $(x,y)$ cannot lie in more
than $C$ sets ${\mathcal{R}}_{Q}$, $Q\in\mathbb{D}$, so (8.29) yields

(8.30) $\sum_{j>k}\sum_{Q\in\mathbb{D}_{j}}W_{Q}^{k}\lesssim J(k).$

As for the $V_{Q}^{k}$, we decide to estimate $|g_{k}(x)-g_{k}(y)|$ rather
brutally. Again we localize at the scale $2^{-k}$. Let
$Q_{0}\in\mathbb{D}_{k}$ be given, and then pick $x\in Q_{0}$ and $y\in
2Q_{0}$. We want to estimate $|g_{k}(x)-g_{k}(y)|$ in terms of

(8.31) $b(Q_{0}):=\fint_{y\in 2Q_{0}}\fint_{z\in
4Q_{0}}|g(y)-g(z)|^{2}d\mu(y)d\mu(z).$

By (8.14),

(8.32) $\displaystyle|g_{k}(x)-g_{k}(y)|$ $\displaystyle=$
$\displaystyle\Big{|}\sum_{Q\in\mathbb{D}_{k}}[\varphi_{Q}(x)-\varphi_{Q}(y)]y_{Q}\Big{|}=\Big{|}\sum_{Q\in\mathbb{D}_{k}}[\varphi_{Q}(x)-\varphi_{Q}(y)][y_{Q}-y_{Q_{0}}]\Big{|}$
$\displaystyle\leq$
$\displaystyle\sum_{Q\in\mathbb{D}_{k}}|\varphi_{Q}(x)-\varphi_{Q}(y)|\,|y_{Q}-y_{Q_{0}}|$

because $\sum_{Q}\varphi_{Q}(x)=\sum_{Q}\varphi_{Q}(y)=1$. Notice that if
$|\varphi_{Q}(x)-\varphi_{Q}(y)|>0$, then $x$ or $y$ lies in $2Q$, hence $Q$
lies in the collection $\mathbb{D}_{k}(Q_{0})$ of (8.20). For such balls,

(8.33) $\displaystyle|y_{Q}-y_{Q_{0}}|$ $\displaystyle=$
$\displaystyle\Big{|}\fint_{y\in 2Q_{0}}\fint_{z\in
2Q}[g(y)-g(z)]\,d\mu(z)\,d\mu(y)\Big{|}\leq\fint_{y\in 2Q_{0}}\fint_{z\in
2Q}|g(y)-g(z)|\,d\mu(z)\,d\mu(y)$ $\displaystyle\leq$
$\displaystyle\Big{\\{}\fint_{y\in 2Q_{0}}\fint_{z\in
2Q}|g(y)-g(z)|^{2}d\mu(y)d\mu(z)\Big{\\}}^{1/2}$ $\displaystyle\leq$
$\displaystyle C\Big{\\{}\fint_{y\in 2Q_{0}}\fint_{z\in
4Q_{0}}|g(y)-g(z)|^{2}d\mu(y)d\mu(z)\Big{\\}}^{1/2}=Cb(Q_{0})^{1/2},$

where $b(Q_{0})$ is as in (8.31), and because $2Q\subset 4Q_{0}$ for
$Q\in\mathbb{D}_{k}(Q_{0})$, and by (H3). Recall that $|\nabla\varphi_{Q}|\leq
C{2^{k}}$. Since there are no more than $C$ cubes $Q\in\mathbb{D}_{k}(Q_{0})$,
(8.32) yields

(8.34)
$|g_{k}(x)-g_{k}(y)|^{2}\lesssim|x-y|^{2}2^{2k}\sup_{Q\in\mathbb{D}_{k}(Q_{0})}|y_{Q}-y_{Q_{0}}|^{2}\lesssim|x-y|^{2}2^{2k}b(Q_{0}).$

We shall use this soon, but for the moment let us estimate a given
$V_{Q}^{k}$, $\ell(Q)\leq 2^{-k}$. Observe that by (H3)–(H4) and the
definitions(2.6) and (5.17), and as in the last part of (8),

$V_{Q}^{k}\lesssim\frac{\rho(Q)}{\ell(Q)}\iint_{{\mathcal{R}}_{Q}}\frac{|g_{k}(x)-g_{k}(y)|^{2}}{\mu(2Q)}d\mu(x)d\mu(y).$

Then let $Q_{0}$ denote the cube of $\mathbb{D}_{k}$ that contains $x$, then
$x\in Q_{0}$ and $y\in 2Q_{0}$ when $(x,y)\in{\mathcal{R}}_{Q}$, so we can use
(8.34) and get that

(8.35) $\displaystyle V_{Q}^{k}$ $\displaystyle\lesssim$
$\displaystyle\frac{\rho(Q)}{\ell(Q)}\iint_{{\mathcal{R}}_{Q}}\frac{|x-y|^{2}2^{2k}b(Q_{0})}{\mu(2Q)}\,d\mu(y)\,d\mu(x).$

Since $|x-y|\approx\ell(Q)$ when $(x,y)\in{\mathcal{R}}_{Q}$,

(8.36) $\displaystyle V_{Q}^{k}$ $\displaystyle\lesssim$ $\displaystyle
2^{2k}\ell(Q)\rho(Q)\int_{x\in Q}\int_{y\in
2Q}\frac{b(Q_{0})}{\mu(2Q)}d\mu(y)d\mu(x)$ $\displaystyle\lesssim$
$\displaystyle 2^{2k}\ell(Q)\rho(Q)\mu(Q)b(Q_{0})$ (8.37)
$\displaystyle\lesssim$ $\displaystyle 2^{2k}m(U^{*}_{Q})b(Q_{0})\approx
m(U^{*}_{Q})\ell(Q_{0})^{-2}b(Q_{0})$

by (5.17). We sum over the cubes $Q\subset Q_{0}$ to obtain

$\begin{split}\sum_{Q\subset Q_{0}}V_{Q}^{k}\lesssim 2^{2k}\sum_{Q\subset
Q_{0}}m(U^{*}_{Q})b(Q_{0})\approx
2^{2k}m(T_{Q_{0}})b(Q_{0})\approx\frac{\rho(Q_{0})\mu(Q_{0})}{\ell(Q_{0})}b(Q_{0})\end{split}$

because the $U^{*}_{Q}$ are contained in $T_{Q_{0}}$ (see (5.12) and (5.11))
and have bounded overlap. We now use the definition (8.31) of $b(Q_{0})$ and
the doubling property (H3) to get

$\sum_{Q\subset
Q_{0}}V_{Q}^{k}\lesssim\int_{2Q_{0}}\int_{4Q_{0}}\frac{\rho(Q_{0})|g(y)-g(z)|^{2}}{\mu(4Q_{0})\ell(Q_{0})}\,d\mu(y)\,d\mu(z).$

Observe that $\mu(B(y,|y-z|))\leq C\mu(4Q_{0})$. Besides, due to (H5),

$\frac{\rho(Q_{0})}{\ell(Q_{0})}\lesssim\frac{\rho(y,|y-z|)}{|y-z|}\left(\frac{\ell(Q_{0})}{|x-z|}\right)^{-\epsilon}\lesssim\frac{\rho(y,|y-z|)}{|y-z|}\qquad\text{
for $(y,z)\in 2Q_{0}\times 4Q_{0}$}.$

It follows that

$\begin{split}\sum_{Q\subset
Q_{0}}V_{Q}^{k}&\lesssim\int_{2Q_{0}}\int_{4Q_{0}}\frac{\rho(y,|y-z|)|g(y)-g(z)|^{2}}{\mu(B(y,|y-z|))|y-z|}\,d\mu(y)\,d\mu(z)\\\
&\quad=\int_{2Q_{0}}\int_{4Q_{0}}\frac{\rho(y,|y-z|)^{2}|g(y)-g(z)|^{2}}{m(B(y,|y-z|)\cap\Omega)}\,d\mu(y)\,d\mu(z)\end{split}$

by (2.6). Notice that $|y-z|\leq 8\ell(Q_{0})=2^{3-k}$ when $y\in 2Q_{0}$ and
$z\in 4Q_{0}$. Also, for a given pair $(y,z)\in\Gamma\times\Gamma$, with
$y\neq z$, the set of cubes $Q_{0}$ of generation $k$ for which $y\in Q_{0}$
and $z\in 2Q_{0}$ has less than $C$ elements; because of this,

(8.38)
$\begin{split}\sum_{j>k}\sum_{Q\in\mathbb{D}_{j}}V_{Q}^{k}&=\sum_{Q_{0}\in\mathbb{D}_{k}}\sum_{Q\subset
Q_{0}}V_{Q}^{k}\\\ &\lesssim\iint_{x,y\in\Gamma\,;\,|x-y|\leq
2^{3-k}}\frac{\rho(y,|y-z|)^{2}|g(y)-g(z)|^{2}}{m(B(y,|y-z|)\cap\Omega)}\,d\mu(y)\,d\mu(z)=J(k)\end{split}$

The combination of (8.27), (8.30), and (8.38) gives that $\sum_{j\geq
0}T_{j}^{k}\lesssim J(k)$, and hence by (8.17)
$\|g-g_{k}\|^{2}_{H}=\sum_{j}T_{j}^{k}\leq CJ(k)$, as needed for (8.15).

This completes our proof of the density of smooth functions in $H$. The fact
that $g_{k}$ converges to $g$ in $L^{2}_{loc}(\Gamma,\mu)$ has been shown in
(8.25), and up to a subsequence, we can also assume that $g_{k}$ also
converges to $g$ $\mu$-a.e. on $\Gamma$. Lemma 8.12 follows. $\Box$

##  9\. Completeness of $W$ and density of smooth functions

In all of this section, $(\Omega,m,\mu)$ satisfies (H1)–(H6).

First we talk about completeness. In fact $W$ cannot really be a Banach space,
because $\|.\|_{W}$ is not a norm on $W$, only a semi-norm. Thus we need to
quotient $W$ by the functions $u$ such that $\|u\|_{W}=0$, that is, thanks to
Lemma 4.10, by the constant functions. So we work with the homogeneous space
$\dot{W}$ defined as the quotient space $W/\mathbb{R}$ \- i.e. element of
$\dot{W}$ are classes $\dot{u}:=\\{u+c\\}_{c\in\mathbb{R}}$ \- and outfitted
with the quotient norm that we still call $\|.\|_{W}$ by notation abuse.

######  Lemma 9.1.

Then the quotient space $\dot{W}=W/\mathbb{R}$, equipped with the quotient
norm $\|\cdot\|_{W}$, is complete.

Also, , if a sequence $\\{u_{k}\\}_{k=1}^{\infty}$ in $W$ and a function
$u_{\infty}\in W$ are such that
$\lim_{k\to+\infty}\|u_{k}-u_{\infty}\|_{W}=0$, then there exists constants
$c_{k}\in\mathbb{R}$ such that $u_{k}-c_{k}\to u$ in $L^{1}_{loc}(\Omega,m)$.

######  Remark 9.2.

The measure $\mu$ does not play any role in this lemma, and we might be able
to remove the assumption (H3). However, it will be convenient to use the
dyadic decomposition (and Theorem 5.24) given in Section 5.

Proof. We follow the arguments of [DFM2, Lemma 5.1]. Let
$\\{\dot{u}_{k}\\}_{k\in\mathbb{N}}$ be a Cauchy sequence in $\dot{W}$. We
need to show that

  1. (i)

for every sequence $\\{u_{k}\\}_{k\in\mathbb{N}}$, with $u_{k}\in\dot{u}_{k}$,
there exists $u\in W$ and $\\{c_{k}\\}_{k\in\mathbb{N}}$ such that
$u_{k}-c_{k}\to u$ in $L^{1}_{loc}(\Omega,m)$ and $\nabla
u=\lim_{k\to+\infty}\nabla u_{k}$ in $L^{2}(\Omega;m)$;

  2. (ii)

if $u_{\infty},u^{\prime}_{\infty}\in W$ are such that there exist
$\\{u_{k}\\}_{k\in\mathbb{N}}$ and $\\{u^{\prime}_{k}\\}_{k\in\mathbb{N}}$
such that $u_{k},u^{\prime}_{k}\in\dot{u}_{k}$ for all $k\in\mathbb{N}$ and

$\lim_{k\to\infty}\|u_{k}-u_{\infty}\|_{W}=\lim_{k\to\infty}\|u^{\prime}_{k}-u^{\prime}_{\infty}\|_{W}=0,$

then $\dot{u}_{\infty}=\dot{u}^{\prime}_{\infty}$.

Proof. First we prove (ii). Let $u_{\infty}$, $u^{\prime}_{\infty}$,
$\\{u_{k}\\}_{k\in\mathbb{N}}$, and $\\{u^{\prime}_{k}\\}_{k\in\mathbb{N}}$ be
as in (ii). Notice that $\nabla(u_{k}-u^{\prime}_{k})=0$ (in the sense of $W$)
because $u_{k}$ and $u^{\prime}_{k}$ represent the same class; by Lemma 4.10
this also means that $u_{k}-u^{\prime}_{k}$ is a constant. By assumption,
$u_{\infty}\in W$ and $\nabla u_{\infty}$ (in the sense of $H$) is the limit
of $\nabla u_{k}$ in $L^{2}(\Omega;m)$. Similarly, $\nabla
u^{\prime}_{\infty}$ is the limit of $\nabla u^{\prime}_{k}$. Hence $\nabla
u_{\infty}=\nabla u^{\prime}_{\infty}$ in $L^{2}$, and again this means that
$u_{\infty}-u^{\prime}_{\infty}$ is constant; (ii) follows.

Let us turn to the proof of (i). Pick a central point $x_{0}\in\Gamma$ and,
for $j\in\mathbb{Z}$, denote by $Q^{j}$ the only cube in $\mathbb{D}_{j}$ that
contains $x_{0}$. Observe that when $j$ tends to $-\infty$, the sets
$T_{2Q^{j}}$ defined by (5.12) grow to $\Omega$, i.e., eventually contain any
compact subset of $\Omega$. Thus the convergence in each $L^{1}(T_{2Q^{j}},m)$
implies the convergence in $L^{1}_{loc}(\Omega,m)$.

We shall restrict our attention to $j\leq 0$. Observe that because of Theorem
5.24 (Poincaré’s inequality), applied with $p=2$ and $D=T_{2Q^{j}}$, there
exists constants $C_{j}:=C(C_{1},C_{2},C_{4},C_{6},j)$ such that for any $f\in
W$,

(9.3) $\fint_{T_{2Q^{j}}}|f-f^{0}|\,dm\lesssim C_{j}\int_{\Omega}|\nabla
f|^{2}\,dm,$

where we can take $f^{0}=\fint_{U^{*}_{Q^{0}}}f\,dm$, i.e., take the fixed set
$E=U^{*}_{Q^{0}}$, which is contained in $T_{2Q^{j}}$ because $j\geq 0$.

Now let $u_{k}\in\dot{u}_{k}$ be as in the statement, and set

(9.4) $c_{k}=\fint_{U^{*}_{Q^{0}}}u_{k}\,dm.$

By (9.3), $(u_{k}-c_{k})_{k}$ is a Cauchy sequence in $L^{1}(T_{Q^{j}})$ for
each integer $j\leq 0$. Hence, there exists $u^{j}\in L^{1}(T_{Q^{j}})$ such
that $u_{k}-c_{k}$ converges to $u^{j}$ in $L^{1}(T_{Q^{j}})$. By uniqueness
of the limit, $u^{j}=u^{i}$ almost everywhere on $T_{Q^{j}}\cap T_{Q^{i}}$, so
we can define $u\in L^{1}_{loc}(\Omega,m)$ such that $u=u^{j}$ a.e. on
$T_{Q^{j}}$.

It remains to check that $u\in W$ and $u_{k}\to u$ in $W$. Since
$u_{k}-c_{k}\in W$, Definition 4.1 gives us a smooth function $\varphi_{k}\in
C^{\infty}(\overline{\Omega})\cap W$ such that

(9.5) $\fint_{U^{*}_{Q^{0}}}|u_{k}-c_{k}-\varphi_{k}|\,dm\leq\frac{1}{k}$

and

(9.6) $\int_{\Omega}|\nabla u_{k}-\nabla\varphi_{k}|^{2}\,dm\leq\frac{1}{k}.$

Set $d_{k}=\fint_{U^{*}_{Q_{0}}}\varphi_{k}\,dm$. By (9.5) and (9.4),

(9.7)
$|d_{k}|\leq\frac{1}{k}+\Big{|}\fint_{U^{*}_{Q^{0}}}(u_{k}-c_{k})\,dm\Big{|}=\frac{1}{k}.$

We are now ready to prove that $\varphi_{k}\to u$ in $L^{1}_{loc}(\Omega,m)$.
Write

$\begin{split}\fint_{T_{2Q^{j}}}|u-\varphi_{k}|\,dm&\leq\fint_{T_{2Q^{j}}}|u-(u_{k}-c_{k})|\,dm+\fint_{T_{2Q^{j}}}|(u_{k}-\varphi_{k})-(c_{k}-d_{k})|\,dm+|d_{k}|\\\
&:=T_{1}+T_{2}+T_{3}.\end{split}$

The term $T_{1}$ tends to 0 as $k\to\infty$ because by construction
$u_{k}-c_{k}\to u=u^{j}$ in $L^{1}(T_{Q^{j}})$. The term $T_{2}$ tends also to
$0$, thanks to (9.3) and (9.6). The term $T_{3}$ converges to 0 because of
(9.7). We conclude, since $T_{2Q^{j}}\uparrow\Omega$, that $\varphi_{k}\to u$
in $L^{1}_{loc}(\Omega,m)$; in particular

(9.8) $\lim_{k\to\infty}\int_{B}|\varphi_{k}-u|\,dm=0\quad\text{ for any ball
$B$ satisfying $2B\subset\Omega$.}$

In addition, $\\{\dot{u}_{k}\\}_{k\in\mathbb{N}}$ is a Cauchy sequence in $W$,
and if we combine this fact with (9.6), we get that
$\\{\nabla\varphi_{k}\\}_{k\in\mathbb{N}}$ is a Cauchy sequence in
$L^{2}(\Omega,m)$. We conclude that there exists $v$ such that

(9.9) $\lim_{k\to\infty}\int_{\Omega}|\nabla\varphi_{k}-v|^{2}\,dm=0.$

We may now use our smooth functions $\varphi_{k}\in
C^{\infty}(\overline{\Omega})\cap W$ to check that $u\in W$, as in Definition
4.1, and with the gradient $v$; indeed (4.3) comes from (9.8), and (4.4) comes
from (9.9). Also, $u_{k}\to u$ in $W$ because $\nabla u_{k}\to v$ in
$L^{2}(m)$, by by (9.6) and (9.9). Lemma 9.1 follows. $\Box$

######  Lemma 9.10.

Let $\\{u_{i}\\}_{i\in\mathbb{N}}$ be a Cauchy sequence in $W$, i.e.,
$\|u_{i}-u_{j}\|_{W}\to 0$ as $i,j\to\infty$. If $u_{i}$ converges to $u$ in
$L^{1}_{loc}(\Omega,m)$, then $u\in W$ and $\|u_{i}-u\|_{W}\to 0$ as
$i\to\infty$.

In connection with this result, we’ll say that $\\{u_{i}\\}_{i\in\mathbb{N}}$
converges to $u$ in $W$ and $L^{1}_{loc}(\Omega,m)$ if
$\\{u_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence in $W$ as above and
$u_{i}\to u$ in $L^{1}_{loc}(\Omega,m)$.

Proof. Keep the same sets $Q^{j}$ as in the proof of Lemma 9.1 and let
$\\{u_{i}\\}$ be as in the statement. As before, by Definition of $u_{i}\in
W$, we can find $\varphi_{i}\in C^{\infty}(\overline{\Omega})\cap W$ such that

(9.11) $\fint_{U^{*}_{Q^{0}}}|u_{i}-\varphi_{i}|\,dm\leq\frac{1}{i}$

and

(9.12) $\int_{\Omega}|\nabla u_{i}-\nabla\varphi_{i}|^{2}\,dm\leq\frac{1}{i}.$

Set $c_{i}=\fint_{U^{*}_{Q_{0}}}(u_{i}-\varphi_{i})\,dm$; by (9.11),
$|c_{i}|\leq\frac{1}{i}$. For each fixed $j\geq 0$, and as in the proof of
Lemma 9.1,

$\begin{split}\fint_{T_{2Q^{j}}}|u-\varphi_{i}|\,dm&\leq\fint_{T_{2Q^{j}}}|u-u_{i}|\,dm+\fint_{T_{2Q^{j}}}|(u_{i}-\varphi_{i})-c_{i}|\,dm+|c_{i}|,\end{split}$

which tends to $0$ as $i\to\infty$. This proves that $\varphi_{i}\to u$ in
$L^{1}_{loc}(\Omega,m)$. Moreover, by (9.12) and the fact that $\\{\nabla
u_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence in $L^{2}(\Omega,m)$,
$\\{\nabla\varphi_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence in
$L^{2}(\Omega,m)$, hence there exists $v\in L^{2}(\Omega,m)$ such that
$\nabla\varphi_{i}\to v$ in $L^{2}(\Omega,m)$. These two convergences - the
convergence in $L^{1}_{loc}(\Omega,m)$ and the convergence of the gradients in
$L^{2}(\Omega,m)$ \- entail by definition of $W$ that $u\in W$, and by
uniqueness of the gradient that $v=\nabla u$. The lemma follows. $\Box$

######  Lemma 9.13.

Let $\\{u_{i}\\}_{i\in\mathbb{N}}$ be a sequence of functions in $W$, and let
$u\in W$. If $u_{i}$ converges to $u$ in both $W$ and $L^{1}_{loc}(\Omega,m)$,
then $\operatorname{Tr}u_{i}$ converges to $\operatorname{Tr}u$ in $H$ and
$L^{2}_{loc}(\Gamma,\mu)$.

Proof. The convergence of the traces in $H$ is a direct consequence of Theorem
6.6 and the convergence of the initial sequence in $W$, and actually does not
need the convergence in $L^{1}_{loc}(\Omega,m)$.

The convergence of the traces in $L^{2}_{loc}(\Omega,m)$ is the analogue of
(5.16) in [DFM2]. Let us write $g$ for $\operatorname{Tr}u$ and $g_{i}$ for
$\operatorname{Tr}u_{i}$. Since the operator $\operatorname{Tr}$ is linear,
without loss of generality, we can assume that $u\equiv 0$ and thus $g\equiv
0$. So we want to prove that $\\{g_{i}\\}$ converges to 0 in
$L^{2}_{loc}(\Gamma,\mu)$. That is, if $x_{0}$ is a fixed point in $\Gamma$
and $Q_{j}$ is the only set in $\mathbb{D}_{j}$ containing $x_{0}$, we want to
show that for $j\in\mathbb{N}$ and $\epsilon>0$, there exists
$i_{0}\in\mathbb{N}$ such that

(9.14) $\int_{Q_{j}}|g_{i}|^{2}d\mu\leq\epsilon\qquad\text{ for }i\geq i_{0}.$

We introduce $g_{i}^{k}:=\operatorname{Tr}_{k}u_{i}$, where
$\operatorname{Tr}_{k}$ is defined in the proof of Theorem 6.6. Then

$\begin{split}\int_{Q_{j}}|g_{i}|^{2}d\mu&\leq
2\int_{Q_{j}}|g_{i}-g_{i}^{k}|^{2}d\mu+2\int_{Q_{j}}|g_{i}^{k}|^{2}d\mu\\\
&\lesssim\frac{2^{-2j}\mu(Q_{j})}{m(U^{*}_{Q_{j}})}2^{-(k-j)\epsilon}\int_{\Omega}|\nabla
u_{i}|^{2}dm+\int_{x\in Q_{j}}\left|\fint_{B_{x}^{k}}u_{i}\right|^{2}d\mu(x)\
:=T_{1}+T_{2}.\end{split}$

where we invoke (6.13) for the second line and $B_{x}^{k}$ is the ball used to
define $\operatorname{Tr}_{k}u(x)$. The values of
$\|u_{i}\|_{W}^{2}=\int_{\Omega}|\nabla u_{i}|^{2}\,dm$ are uniformly bounded,
since $\\{u_{i}\\}$ converges in $W$. So we can fix $k$, so large that that
$T_{1}\leq\epsilon/2$ uniformly in $i\in\mathbb{N}$. As for $T_{2}$, observe
that

$T_{2}\leq C_{j,k}\int_{E_{j,k}}|u_{i}|$

where $E_{j,k}=\bigcup_{Q\in\mathbb{D}_{k}:\,Q\subset Q_{j}}U^{*}_{Q}$ is
relatively compact in $\Omega$. Since the values of $j,k$ are fixed and
$\\{u_{i}\\}$ converges to $u\equiv 0$ in $L^{1}(E_{j,k})$, we can choose
$i_{0}$ such that $T_{2}\leq\epsilon/2$ for $i\geq i_{0}$. Lemma 9.13 follows.
$\Box$

######  Lemma 9.15.

The space

(9.16) $W_{0}:=\\{u\in W,\,\operatorname{Tr}u=0\\}$

equipped with the scalar product $\left<u,v\right>_{W}:=\int_{\Omega}\nabla
u\cdot\nabla v\,dm$ (and the norm $\|.\|_{W}$) is a Hilbert space.

Proof. Notice that $\|.\|_{W}$ is indeed a norm for $W_{0}$, because the only
constant that is allowed in $W_{0}$ is $0$. The proof will be similar to
[DFM2, Lemma 5.2], and use Lemmas 9.1 and 9.13.

Let $\\{u_{i}\\}_{i\in\mathbb{N}}$ be a Cauchy sequence in $W_{0}$. By the
proof of Lemma 9.1, there exists $\bar{u}\in W$ and a sequence of constants
$c_{i}=\fint_{U^{*}_{Q^{0}}}u_{i}\,dm$ (see (9.4)) such that

(9.17) $u_{i}-c_{i}\to\bar{u}\quad\text{ in }L^{1}_{loc}(\Omega,m).$

Let us prove that $\\{c_{i}\\}$ is a Cauchy sequence in $\mathbb{R}$. For
$i,j\geq 0$,

$|c_{i}-c_{j}|\leq\fint_{U^{*}_{Q^{0}}}|u_{i}-u_{j}|\,dm\leq
C\Big{\\{}\fint_{U^{*}_{Q^{0}}}|\nabla u_{i}-\nabla
u_{j}|^{2}\,dm\Big{\\}}^{1/2}\leq C\|u_{i}-u_{j}\|_{W},$

where the second inequality is due to the Poincaré inequality (Theorem 5.24).
So $\\{c_{i}\\}_{i\in\mathbb{N}}$ is indeed a Cauchy sequence, and thus
converges to a constant $c\in\mathbb{R}$. Define $u\in W$ as $\bar{u}-c$; then
(9.17) says that $u_{i}\to u$ in $L^{1}_{loc}(\Omega,m)$, but the convergence
also holds in $W$ by definition of $\bar{u}$. Lemma 9.13 implies now that
$\operatorname{Tr}u$ is the limit in $L^{2}_{loc}(\Gamma,\mu)$ of
$\operatorname{Tr}u_{i}\equiv 0$, that is $\operatorname{Tr}u\equiv 0$ and
hence $u\in W_{0}$. The lemma follows. $\Box$

Recall from Lemma 6.20 that $u\varphi\in W$ when $u\in W$ and $\varphi\in
C^{\infty}_{0}(\mathbb{R}^{n})$, and that we have the product rule
$\nabla(u\varphi)=\varphi\nabla u+u\nabla\varphi$ for its derivative. Also,
the trace of $u\varphi$ is $\varphi Tru$. We can use this to prove that
$C^{\infty}_{0}(\Omega)$ is dense in $W_{0}$, as in the following lemma.

######  Lemma 9.18.

The completion of $C^{\infty}_{0}(\Omega)$ for the norm $\|.\|_{W}$ is
$W_{0}$.

Moreover, if $E\subset\mathbb{R}^{n}$ is an open set and $u\in W$ is compactly
supported in $E\cap\overline{\Omega}$, then $u$ can be approximated in the
norm $\|.\|_{W}$ by functions in $C^{\infty}_{0}(E\cap\Omega)$.

Proof. This result is entirely similar to [DFM2, Lemma 5.5] and we refer to it
for a complete proof. The main steps are:

  1. (i)

we use cut-off functions $\varphi_{r}$ to approach $u\in W_{0}$ by functions
that are equal to 0 on $\Gamma_{r}:=\\{X\in\Omega,\,\delta(X)\leq r\\}$,

  2. (ii)

we use cut-off functions $\phi_{R}$ to approach the functions $u\varphi_{r}$
obtained in (i) by functions compactly supported in $\Omega$,

  3. (iii)

we use a mollifier to smooth the functions $u\varphi_{r}\phi_{R}$.

And obviously, in order to deal with the functions $u\varphi_{r}$ or
$u\varphi_{r}\phi_{R}$, we use in a crucial manner the aforementioned Lemma
6.20. $\Box$

######  Lemma 9.19.

The set $C^{\infty}(\Omega)\cap C^{0}(\overline{\Omega})\cap W$ is dense in
$W$. That is, for any $u\in W$, there exists a sequence
$\\{u_{i}\\}_{i\in\mathbb{N}}$ in $C^{\infty}(\Omega)\cap
C^{0}(\overline{\Omega})\cap W$ such that $\\{u_{i}\\}$ converges to $u$
pointwise a.e. and in $L^{1}_{loc}(\Omega,m)$, and

$\|u_{i}-u\|_{W}\longrightarrow 0\quad\text{ as }i\to+\infty.$

Proof. In [DFM2], the analogue of this result is given by [DFM2, Lemma 5.3],
but we cannot follow the same approach here (in [DFM2], the functions we
considered were in $L^{1}_{loc}(\mathbb{R}^{n})$, and thus allowed us to
simply use a mollifier).

However, most of the job is already done by Lemmas 8.12 and 9.18. We take
$u\in W$, and we want to find a smooth approximating sequence $\\{u_{i}\\}$.
First write $u=v+w$ where

$w=\operatorname{Ext}\circ\operatorname{Tr}u\quad\text{ and }\quad v=u-w.$

Theorems 6.6 and 8.5 imply that $w$ \- and thus $v$ \- lies in $W$. Moreover,
thanks to (8.6), $\operatorname{Tr}w=\operatorname{Tr}u$ and hence
$\operatorname{Tr}v=0$; that is, $v\in W_{0}$.

Thanks to Lemma 9.18, we can find a sequence $\\{v_{i}\\}_{i\geq 0}$ in
$C^{\infty}_{0}(\Omega)\subset C^{\infty}(\Omega)\cap
C^{0}(\overline{\Omega})\cap W$ such that $\|v_{i}-v\|_{W}$ tends to $0$. And
since $W_{0}$ continuously injects in $L^{1}_{loc}(\Omega,m)$ ( by Theorem
7.1), the sequence $\\{v_{i}\\}$ converges also in $L^{1}_{loc}(\Omega,m)$
and, up to a subsequence, pointwise a.e. This takes care of $v$.

We use Lemma 8.12 to approximate $\operatorname{Tr}u$ by some functions
$(g_{i})_{i\geq 0}$ in $C^{\infty}(\mathbb{R}^{n})\cap H$. Then we construct
$w_{i}\in C^{\infty}(\Omega)$ as $\operatorname{Ext}g_{i}$. Thanks to Theorem
8.5, $\|w_{i}-w\|_{W}$ tends to $0$ as $i$ goes to $+\infty$. Besides it is
easy to check from the definition of the extension operator
$\operatorname{Ext}$ that the convergence of $g_{i}$ to $\operatorname{Tr}u$
in $L^{1}_{loc}(\Gamma,\mu)$ (also given by Lemma 8.12) implies that $w_{i}$
converges to $w$ uniformly on an compact subsets of $\Omega$, and thus also
pointwise a.e. and in $L^{1}_{loc}(\Omega,m)$.

If we set $u_{i}=v_{i}+w_{i}$, we showed above the right convergences (in $W$,
$L^{1}_{loc}(\Omega)$, and a.e. pointwise) of $u_{i}$ to $u$. The only
unproved fact is that $w_{i}$ is continuous up to the boundary, that is
$w_{i}\in C^{0}(\overline{\Omega})$. We skip this part because it is very
classical (see for instance in Section VI.2.2 of [Ste]). $\Box$

The next result states some basic properties of the derivative of $f\circ u$
when $u\in W$ (chain rule), and the fact that $uv$ lies in $W\cap
L^{\infty}(\Omega)$ as soon as $u$ and $v$ both lie in $W\cap
L^{\infty}(\Omega)$.

######  Lemma 9.20.

The following properties hold:

  1. (i)

Let $f\in C^{1}(\mathbb{R})$ be such that $f^{\prime}$ is bounded, and let
$u\in W$. Then $f\circ u\in W$,

$\nabla(f\circ u)=f^{\prime}(u)\nabla u,\quad\text{ and
}\quad\operatorname{Tr}(f\circ u)=f\circ(Tu),$

where the last two equalities hold in the $m$-a.e. and $\mu$-a.e. sense,
respectively.

  2. (ii)

Let $u,v\in W$. Then $\max\\{u,v\\}$ and $\min\\{u,v\\}$ lie in $W$,

$\nabla\max\\{u,v\\}(x)=\left\\{\begin{array}[]{ll}\nabla u(x)&\text{ if
}u(x)\geq v(x)\\\ \nabla v(x)&\text{ if }v(x)\geq u(x),\\\ \end{array}\right.$
$\nabla\min\\{u,v\\}(x)=\left\\{\begin{array}[]{ll}\nabla u(x)&\text{ if
}u(x)\leq v(x)\\\ \nabla v(x)&\text{ if }v(x)\leq u(x),\\\ \end{array}\right.$
$\operatorname{Tr}(\max\\{u,v\\})=\max\\{\operatorname{Tr}u,\operatorname{Tr}v\\},\quad\text{
and
}\quad\operatorname{Tr}(\min\\{u,v\\})=\min\\{\operatorname{Tr}u,\operatorname{Tr}v\\},$

where the first two equalities hold $m$-a.e., and the last two $\mu$-a.e.

  3. (iii)

If $\\{u_{k}\\}_{k\in\mathbb{N}}$, $\\{v_{k}\\}_{k\in\mathbb{N}}$ are two
sequences of functions in $W$ that converge to $u$, $v\in W$ both in
$L^{1}_{loc}(\Omega,m)$, pointwise a.e., and in $W$ (that is,
$\|u_{k}-u\|_{W}+\|v_{k}-v\|_{W}\to 0$ as $k\to\infty$), then
$\max\\{u_{k},v_{k}\\}$ and $\min\\{u_{k},v_{k}\\}$ lie in $W$ and converge to
$\max\\{u,v\\}$ and $\min\\{u_{k},v_{k}\\}$ in $L^{1}_{loc}(\Omega,m)$,
pointwise a.e., and in $W$. In addition,
$\operatorname{Tr}\max\\{u_{k},v_{k}\\}$ tends to
$\max\\{\operatorname{Tr}u,\operatorname{Tr}v\\}$ and
$\operatorname{Tr}\min\\{u_{k},v_{k}\\}$ tends to
$\min\\{\operatorname{Tr}u,\operatorname{Tr}v\\}$, in both case in
$L^{2}_{loc}(\Gamma,\mu)$.

Proof. Point (i) and (ii) are the analogues of Lemmas 6.1 in [DFM2]. The proof
is the same as in [DFM2] (which is itself based on the proof of results 1.18
to 1.23 in [HKM]), and strongly relies on Lemma 9.19 (the approximation of
elements in $W$ by smooth functions) and Lemma 9.13 (the convergence in $W$
implies the convergence of traces). The conclusion (iii) is an intermediate
result for (ii), proved as (6.16) and (6.17) in [DFM2]. $\Box$

######  Lemma 9.21.

Let $u,v\in W\cap L^{\infty}(\Omega)$. Then $uv\in W\cap L^{\infty}(\Omega)$,
with $\nabla[uv]=v\nabla u+u\nabla v$, and
$\operatorname{Tr}(uv)=\operatorname{Tr}u\cdot\operatorname{Tr}v$.

Proof. If $u$ or $v$ is the zero constant, there is nothing to prove.
Otherwise, we can divide $u$ and $v$ by their respective $L^{\infty}$ norm,
and thus, without loss of generality, we can assume that
$\|u\|_{\infty}=\|v\|_{\infty}=1$.

By Lemma 9.19, we can find two sequences
$\\{\tilde{u}_{k}\\}_{k\in\mathbb{N}}$ and $\\{\tilde{v}_{k}\in\mathbb{N}\\}$
in $C^{\infty}(\Omega)\cap C^{0}(\overline{\Omega})\cap W$ such that
$\tilde{u}_{k}\to u$ and $\tilde{v}_{k}\to v$ in $W$, $L^{1}_{loc}(\Omega,m)$,
and pointwise a.e. By (iii) of Lemma 9.20, the truncated functions

$u_{k}:=\max\\{-1,\min\\{1,\tilde{u}_{k}\\}\\}\quad\text{ and
}v_{k}:=\max\\{-1,\min\\{1,\tilde{v}_{k}\\}\\}$

lie in $C^{0}(\overline{\Omega})\cap W$, are locally Lipschitz, and converge
to respectively $u$ and $v$ in $W$, $L^{1}_{loc}(\Omega,m)$, and pointwise
a.e.

Since the derivative is a local object, we can use (4.5) and the classical
product rule to say that

$\nabla[u_{k}v_{k}]=u_{k}\nabla v_{k}+v_{k}\nabla u_{k}.$

We conclude by showing, as in the proof of [DFM2, Lemma 6.3] that
$u_{k}v_{k}\to uv$ in $L^{1}_{loc}(\Omega,m)$, $u_{k}\nabla v_{k}+v_{k}\nabla
u_{k}\to u\nabla v+v\nabla u$ in $L^{2}(\Omega,m)$, and
$u_{k}v_{k}=\operatorname{Tr}(u_{k}v_{k})\to\operatorname{Tr}u\cdot\operatorname{Tr}v$
in $L^{1}_{loc}(\Gamma,\mu)$. The lemma follows then from Lemma 9.10 and
Theorem 6.6. $\Box$

##  10\. The localized versions $W_{r}(E)$ of our energy space $W$

The aim of this short section is to define local versions of $W$, which will
be useful to study local solutions to our degenerate elliptic equations. As in
the previous section, we assume throughout that $(\Omega,m,\mu)$ satisfies
(H1)–(H6).

In general, we want to localize $W$ with an open set $E^{\prime}$ of
$\mathbb{R}^{n}$, we set

(10.1) $E=E^{\prime}\cap\overline{\Omega},$

and define the space of functions $W_{r}(E)$ by

(10.2) $W_{r}(E):=\\{u\in L^{1}_{loc}(E\cap\Omega,m):\,\varphi u\in W\text{
for all }\varphi\in C^{\infty}_{0}(E^{\prime})\\}.$

It is natural to call this space $W_{r}(E)$, as opposed to
$W_{r}(E^{\prime})$, because it does not depend on the part of $E^{\prime}$
that leaves away from $\overline{\Omega}$. But there is an important special
case, when $E^{\prime}\subset\Omega$ and so $E=E^{\prime}$ is an open subset
of $\Omega$. In this case, the information that $f\in W_{r}(E)$ does not give
any control on $f$ at the boundary $\partial E$ (which may intersect
$\Gamma$), and $W_{r}(E)$ will be mainly used to give interior estimates for
weak solutions (that will be defined soon). In the general case, $E$ may
contain pieces of the boundary $\Gamma$, and then the fact that $f\in
W_{r}(E)$ gives some information on the behavior of $f$ near $E\cap\Gamma$, in
the same way as the fact that $f\in W$ gives a global information on $f$ near
$\Gamma$. For instance, we can can take for $E$ (the interior in
$\overline{\Omega}$ of) the set $T_{Q}\cup Q$, for some dyadic cube
$Q\in\mathbb{D}$. Obviously $W_{r}(E)\subset W_{r}(F)$ when $F\subset E$, and
in particular $W_{r}(T_{Q}\cup Q)\subset W_{r}(T_{Q})$. In addition, if
$F\subsetneq E$, it is not very hard to find a function $u\in
W_{r}(F)\setminus W_{r}(E)$ \- just make $|\nabla u(X)|$ blows up when $X$
gets close to $E\setminus F$ \- and thus the local spaces $W_{r}(E)$ are all
different. Thus for instance

$W\subsetneq W_{r}(\overline{\Omega})\subsetneq W_{r}(\Omega),$

smooth functions on $\Omega$ that possibly explode along $\Gamma$ lie in the
last space, while they only lie in $W_{r}(\overline{\Omega})$ when they are
locally controlled near $\Gamma$, and they only lie in $W$ when in addition
their gradient lies in $L^{2}(\Omega,m)$.

Functions in $W_{r}(E)$ are not necessarily in $L^{1}_{loc}(E)$ (see Section 4
where we defined $W$). They still have the a notion of gradient - that may be
different from the distributional gradient - inherited from $W$. Indeed, if
$E^{\prime}$ is an open subset of $\mathbb{R}^{n}$ such that
$E=E^{\prime}\cap\overline{\Omega}$, consider $K$ any compact subset of
$E^{\prime}$ and take $\varphi_{K}\in C^{\infty}(E^{\prime})$ such that
$\varphi_{K}=1$ on $K$, then we construct the $W$-gradient of $u\in W_{r}(E)$
on $K$ as the $W$-gradient of $\varphi_{K}u$. As an easy consequence, for
$u\in W_{r}(E)$, we have $\nabla u\in L^{2}_{loc}(E,m)$ (where in fact we just
integrate on $E\cap\Omega$, but local means in terms of the open set
$E^{\prime}$, or $E=E^{\prime}\cap\overline{\Omega}$) and then $u\in
L^{2}_{loc}(E,m)$ by Theorem 5.24. These observations are summarized in the
next lemma.

######  Lemma 10.3.

Let $E=E^{\prime}\cap\overline{\Omega}$, for some open set
$E^{\prime}\subset\mathbb{R}^{n}$. Then every function $u\in W_{r}(E)$ lies in
$L^{1}_{loc}(E,m)$, and its gradient lies in $L^{2}_{loc}(E,m)$.

######  Remark 10.4.

We don’t have many doubts that the reverse inclusion

(10.5) $W_{r}(E)\supset\\{u\in L^{1}_{loc}(E,m):\,\nabla u\in
L^{2}_{loc}(E,m)\\}.$

also holds. The idea of the proof of (10.5) would be to take $u\in
L^{1}_{loc}(E)$ that satisfies $\nabla u\in L^{2}_{loc}(E,m)$, and $\varphi\in
C^{\infty}_{0}(E^{\prime})$. We would set then
$K^{\prime}:=\operatorname{supp}\,\varphi$ which has a smooth boundary, and we
would say that $u|_{K^{\prime}}$ can be extended to a function $\bar{u}\in W$
such that $\bar{u}=u$ a.e. on $K^{\prime}$. Then we would use Lemma 6.20 in
order to show that $\varphi u=\varphi\bar{u}\in W$. The problem with this
proof is that we don’t know any reference for the extension theorem needed to
built $\bar{u}$ in weighted Sobolev spaces (an analogue of [Maz, Section
1.1.17] in the unweighted case), and we do not want to spend time on something
that we will not need.

The next lemma allows us to speak about traces for functions in the local
Sobolev spaces $W_{r}(E)$.

######  Lemma 10.6.

Let $E^{\prime}\subset\mathbb{R}^{n}$ be open, and set
$E=E^{\prime}\cap\overline{\Omega}$ as in (10.1). For every function $u\in
W_{r}(E)$, we can define the trace of $u$ on $\Gamma\cap E$ by

(10.7) $\operatorname{Tr}u(x)=\lim_{X\in\gamma(x)\atop\delta(X)\to
0}\fint_{B(X,\delta(X)/2)}u\qquad\text{ for $\mu$-almost every }x\in\Gamma\cap
E,$

and $\operatorname{Tr}u\in L^{2}_{loc}(\Gamma\cap E^{\prime},\mu)$. Moreover,
for every choice of $f\in W_{r}(E)$ and

$\varphi\in C^{\infty}(E^{\prime})$, $\varphi u\in W_{r}(E)$ and

(10.8) $\operatorname{Tr}(\varphi u)(x)=\varphi(x)\operatorname{Tr}u(x)\
\text{ for $\mu$-almost every }x\in\Gamma\cap E.$

Proof. None of this is too surprising; the trace is a local notion, and
$W_{r}(E)$ is designed to ba a local space. Let $E^{\prime}$, $E$, and $f$ be
as in the statement, and let $B$ a compact ball in $E^{\prime}$, and choose
$\psi\in C^{\infty}_{0}(E)$ such that $\psi\equiv 1$ near $B$. Then $\psi u\in
W$ by (10.2), and the analogue of (10.7) for $\psi u$ comes with the
construction of the trace. This implies the existence of the same limit for
$u$, almost everywhere in $\Gamma\cap B$.

In addition, since $\psi u\in W$, Theorem 6.6 says that $Tr(\psi u)\in H$, and
then $Tr(u)=Tr(\psi u)\in L^{2}(B,d\mu)$ (see the definition (6.5)). Therefore
$Tr(u)\in L^{2}_{loc}(E^{\prime},d\mu)$, as announced.

The fact that $\varphi u\in W_{r}(E)$ when $u\in W_{r}(E)$ and $\varphi\in
C^{\infty}(E^{\prime})$ comes right from the definition (10.2) and Lemma 6.20,
and (10.8) is immediate because when $B$ and $\psi$ are as above and
$\mu$-almost everywhere on $B$,

$Tr(\varphi u)=Tr(\psi^{2}\varphi u)=\psi\varphi Tr(\psi u)=\varphi Tr(u)$

by (10.7), the formula for the trace of a product of $\psi u\in W$ and a
$\psi\varphi\in C^{\infty}_{0}(\mathbb{R}^{n})$, and the fact that in $B$, the
formula (10.7) does not see the cut-off $\psi$. $\Box$

##  11\. Definitions of solutions and their properties

We now have all the functional analysis needed to deal with the main goal of
this section, which is to define weak solutions to appropriate degenerate
elliptic operators, and give their first properties. We will follow Section 8
in [DFM2] (which itself copies the frame of the first sections of [Ken]), and
we will refer to [DFM2] for most of the proofs. As in the previous section, we
systematically assume that $(\Omega,m,\mu)$ satisfies (H1)–(H6).

Recall that we intend to work with the degenerate elliptic operators
$L=-\mathop{\operatorname{div}}A\nabla$, where the matrix
$A:\Omega\to\mathbb{M}_{n}(\mathbb{R})$ satisfies the following elliptic and
boundedness conditions:

(11.1) $A(X)\xi\cdot\xi\geq C_{A}^{-1}w(x)|\xi|^{2}\quad\text{ for
}X\in\Omega\text{ and }\xi\in\mathbb{R}^{n}$

and

(11.2) $A(X)\xi\cdot\zeta\leq C_{A}w(x)|\xi||\zeta|\quad\text{ for
}X\in\Omega\text{ and }\xi,\zeta\in\mathbb{R}^{n},$

where $w$ is the weight associated to the measure $m$ given in as part of
(H4). We shall also use the “normalized” matrix $\mathcal{A}:=w^{-1}A$ which
satisfies the unweighted ellipticity and boundedness conditions

(11.3) $\mathcal{A}(X)\xi\cdot\xi\geq C_{A}^{-1}|\xi|^{2}\quad\text{ for
}X\in\Omega\text{ and }\xi\in\mathbb{R}^{n}$

and

(11.4) $\mathcal{A}(X)\xi\cdot\zeta\leq C_{A}|\xi||\zeta|\quad\text{ for
}X\in\Omega\text{ and }\xi,\zeta\in\mathbb{R}^{n}.$

We introduce the bilinear form $a$ defined by

(11.5) $\begin{split}a(u,v):=\int_{\Omega}A\nabla u\cdot\nabla
v&=\int_{\Omega}\mathcal{A}\nabla u\cdot\nabla v\,dm\\\ &\qquad\text{ for any
$u,v$ that satisfies }\int_{\Omega}|\nabla u||\nabla
v|\,dm<+\infty.\end{split}$

The conditions (11.1)–(11.2) entail that $a$ is bounded on
$\dot{W}\times\dot{W}$ (the homogeneous quotient space) and coercive on
$\dot{W}$, i.e.,

(11.6) $a(u,u)\geq C_{A}^{-1}\|u\|_{W}^{2}\quad\text{ and }\quad a(u,v)\leq
C_{A}\|u\|_{W}\|v\|_{W}\qquad\text{ for }u,v\in W.$

It is also coercive on $W_{0}$ (no need to take a quotient, because $W_{0}$
does not contain nontrivial constant functions).

######  Definition 11.7.

Let $E\subset\Omega$ be a open set. We say that $u\in W_{r}(E)$ is a (weak)
solution to $Lu=0$ in $E$ when

(11.8) $a(u,\varphi)=\int_{\Omega}A\nabla
u\cdot\nabla\varphi=\int_{\Omega}\mathcal{A}\nabla
u\cdot\nabla\varphi\,dm=0\qquad\text{ for any }\varphi\in
C^{\infty}_{0}(\Omega).$

Similarly $u\in W_{r}(E)$ is a subsolution (respectively supersolution) to
$Lu=0$ in $E$ when

(11.9) $a(u,\varphi)\leq 0\ \text{(resp. $\geq 0$)}\qquad\text{ for any
}\varphi\in C^{\infty}_{0}(\Omega)\text{ that satisfies }\varphi\geq 0.$

In the rest of the section, we present the analogues of the results in [DFM2,
Section 8], and we discuss the differences in the proofs when needed.

The first result enlarges the class of possible test functions.

######  Lemma 11.10.

Let $E\subset\Omega$ be an open set and let $u\in W_{r}(E)$ be a solution to
$Lu=0$ in $E$. We write $E^{\Gamma}$ for $E\cup(\Gamma\cap\partial E)$, that
is, $E^{\Gamma}$ is the union of $E$ with the part of its boundary that
intersects $\Gamma$. The identity (11.8) holds:

  * •

when $\varphi\in W_{0}$ is compactly supported in $E$;

  * •

when $\varphi\in W_{0}$ is compactly supported in $E^{\Gamma}$ and $u\in
W_{r}(E^{\Gamma})$;

  * •

when $E=\Omega$, $\varphi\in W_{0}$, and $u\in W$.

In addition, (11.9) holds when $u$ is a subsolution (resp. supersolution) in
$E$, $\varphi$ is a non-negative test function, and the couple $(u,\varphi)$
satisfies one of the above conditions.

Proof. See the proof of [DFM2, Lemma 8.3]. This lemma is a consequence of
Lemma 9.18, that gives that the functions in $W_{0}$ can be approximated by
smooth functions. $\Box$

The next result proves the stability of subsolutions/supersolutions under
$\max$/$\min$.

######  Lemma 11.11.

Let $E\subset\Omega$ be an open set.

  * •

If $u,v\in W_{r}(E)$ are subsolutions (to $Lu=0$) in $E$, then
$t=\max\\{u,v\\}$ is also a subsolution in $E$.

  * •

If $u,v\in W_{r}(E)$ are supersolutions in $E$, then $t=\min\\{u,v\\}$ is also
a supersolution in $E$.

In particular if $k\in\mathbb{R}$, then $(u-k)_{+}:=\max\\{u-k,0\\}$ is a
subsolution in $E$ whenever $u\in W_{r}(E)$ is a subsolution in $E$ and
$\min\\{u,k\\}$ is a supersolution in $E$ whenever $u\in W_{r}(E)$ is a
supersolution in $E$.

Proof. The proof is the same as the one of [DFM2, Lemma 8.23] and [Sta,
Theorem 3.5]. Lemma 8.5 in [DFM2] shows that the result can be localized into
a relatively compact open subset $F$ of $E$. Theorem 3.5 in [Sta] relies on
the fact the bilinear form $a$ is coercive and continuous (on appropriate
local spaces) and on convex analysis. $\Box$

In the sequel, the notation $\sup$ and $\inf$ are used for the essential
supremum and essential infimum, since they are the definitions that makes
sense for the functions in $W$ or in $W_{r}(E)$, for
$E=E^{\prime}\cap\overline{\Omega}$ and $E^{\prime}\subset\mathbb{R}^{n}$
open.

In addition, the expression “$\operatorname{Tr}u=0$ a.e. on $B$”, for a
function $u\in W_{r}(B\cap\overline{\Omega})$, means that
$\operatorname{Tr}u$, which is defined on $\Gamma\cap B$ and lies in
$L^{1}_{loc}(B\cap\Gamma,\mu)$ thanks to Lemma 10.6, is equal to $0$
$\mu$-almost everywhere on $\Gamma\cap B$. The expression
“$\operatorname{Tr}u\geq 0$ a.e. on $B$” is defined similarly.

We now state some classical regularity results inside the domain and at the
boundary.

######  Lemma 11.12 (interior Caccioppoli inequality).

Let $E\subset\Omega$ be an open set, and let $u\in W_{r}(E)$ be a non-negative
subsolution in $E$. Then for any $\alpha\in C^{\infty}_{0}(E)$,

(11.13) $\int_{\Omega}\alpha^{2}|\nabla u|^{2}dm\leq
C\int_{\Omega}|\nabla\alpha|^{2}u^{2}dm,$

where $C$ depends only upon the constant $C_{A}$.

In particular, if $B$ is a ball of radius $r$ such that $2B\subset\Omega$ and
$u\in W_{r}(2B)$ is a non-negative subsolution in $2B$, then

(11.14) $\int_{B}|\nabla u|^{2}dm\leq Cr^{-2}\int_{2B}u^{2}dm.$

######  Lemma 11.15 (Caccioppoli inequality on the boundary).

Let $B\subset\mathbb{R}^{n}$ be a ball of radius $r$ centered on $\Gamma$, and
let $u\in W_{r}(2B\cap\overline{\Omega})$ be a non-negative subsolution in
$2B\cap\Omega$ such that $\operatorname{Tr}u=0$ a.e. on $2B$. Then for any
$\alpha\in C^{\infty}_{0}(2B)$,

(11.16) $\int_{2B\cap\Omega}\alpha^{2}|\nabla u|^{2}dm\leq
C\int_{2B\cap\Omega}|\nabla\alpha|^{2}u^{2}dm,$

where $C$ depends only on the constant $C_{A}$. In particular, we can take
$\alpha\equiv 1$ on $B$ and $|\nabla\alpha|\leq\frac{2}{r}$, which gives

(11.17) $\int_{B\cap\Omega}|\nabla u|^{2}dm\leq
Cr^{-2}\int_{2B\cap\Omega}u^{2}dm.$

Proof. The proofs of the two lemmas are similar to Lemma 8.6 and Lemma 8.11 in
[DFM2]. There is not any difficulty here, maybe it is worth saying that we use
$\varphi=\alpha^{2}u$, where $\alpha$ is an appropriate cut-off function; and
$\varphi$ is a valid test function due to Lemma 11.10 and, for the boundary
version, Lemma 10.6. $\Box$

Let us turn to the statement of the Moser estimates.

######  Lemma 11.18 (interior Moser estimate).

Let $p>0$ and $B$ be a ball such that $2B\subset\Omega$. If $u\in W_{r}(2B)$
is a non-negative subsolution in $2B$, then

(11.19) $\sup_{B}u\leq
C\left(\frac{1}{m(2B)}\int_{2B}u^{p}\,dm\right)^{\frac{1}{p}},$

where $C$ depends on $n$, $C_{4}$, $C_{6}$, $C_{A}$, and $p$.

######  Lemma 11.20 (Moser estimates on the boundary).

Let $p>0$, $B$ be a ball centered on $\Gamma$, and $u\in
W_{r}(2B\cap\overline{\Omega})$ be a non-negative subsolution in
$2B\cap\Omega$ such that $\operatorname{Tr}u=0$ a.e. on $2B$. Then

(11.21) $\sup_{B\cap\Omega}u\leq
C_{p}\left(m(2B)^{-1}\int_{2B\cap\Omega}|u|^{p}dm\right)^{\frac{1}{p}},$

where $C_{p}$ depends only on $n$, $C_{1}$ to $C_{6}$, $C_{A}$, and $p$.

Proof. The proofs for these two results are analogous to the ones of [DFM2,
Lemmas 8.7 and 8.12], and relies on the so-called Moser iterations.

What we need are Lemma 11.11, a Cacciopoli inequality (Lemma 11.12 or Lemma
11.15, according to the version we want to prove), a Sobolev-Poincaré
inequality (Theorem 5.24 or Corollary 7.9, the balls in the right-hand side of
(7.10) are slightly bigger than the ones in the left-hand side, but the
argument can easily be adapted), and the doubling property (H4). $\Box$

The next step is the Hölder continuity of solutions. We shall give a few
intermediate results, starting by the density lemmas.

######  Lemma 11.22 (Density lemma inside the domain).

Let $B$ be a ball such that $4B\subset\Omega$ and $u\in W_{r}(4B)$ be a non-
negative supersolution in $4B$ such that

$m(\\{X\in 2B,\,u(X)\geq 1\\})\geq\epsilon m(2B).$

Then

(11.23) $\inf_{B}u\geq C^{-1},$

where $C>0$ depends only on $n$, $C_{4}$, $C_{6}$, $C_{A}$, and $\epsilon$.

######  Lemma 11.24 (Density lemma on the boundary).

Let $B$ be a ball centered on $\Gamma$ and $u\in
W_{r}(4B\cap\overline{\Omega})$ be a non-negative supersolution in
$4B\cap\Omega$ such that $\operatorname{Tr}u=1$ a.e. on $4B$. Then

(11.25) $\inf_{B\cap\Omega}u\geq C^{-1},$

where $C>0$ depends only on $n$, $C_{1}$ to $C_{6}$ and $C_{A}$.

Proof. The proof of Lemma 11.22 can be copied from the one of Density Theorem
(Section 4.3, Theorem 4.9) in [HL]. The proof of Lemma 11.24 is similar to the
one Lemma 8.14 in [DFM2] (which is itself inspired from the Density Theorem in
[HL]).

Formally, the ideas of the proof are to say that $v=-\ln u$ is a subsolution
that satisfies $\operatorname{Tr}u=0$ a.e. on $4B$ (if needed), and then to
use Moser estimates (Lemma 11.18 or Lemma 11.20) and a Poincaré inequality
(Theorem 5.24 or Corollary 7.9) in an appropriate way.

Of course, we need to be very careful: for instance when constructing $v$, we
want to use Lemma 9.20 in order to verify that $v$ is indeed in $W_{r}(2B)$,
yet the function $-\ln$ is not Lipschitz… But the pitfalls are the same as in
the proof of [DFM2, Lemma 8.14]. $\Box$

Next comes oscillation estimates.

######  Lemma 11.26 (interior Oscillation estimates).

Let $B$ be a ball such that $4B\subset\Omega$ and $u\in W_{r}(4B)$ be a
solution in $4B$. Then, there exists $\eta\in(0,1)$ such that

(11.27) $\operatorname*{osc}_{B}u\leq\eta\operatorname*{osc}_{4B}u,$

where the constant $\eta$ depends only on on $n$, $C_{4}$, $C_{6}$, and
$C_{A}$.

######  Lemma 11.28 (Oscillation estimates on the boundary).

Let $B$ be a ball centered on $\Gamma$ and $u\in
W_{r}(4B\cap\overline{\Omega})$ be a solution in $4B\cap\Omega$ such that
$\operatorname{Tr}u$ is uniformly bounded on $4B\cap\Gamma$. Then, there
exists $\eta\in(0,1)$ such that

(11.29)
$\operatorname*{osc}_{B\cap\Omega}u\leq\eta\operatorname*{osc}_{4B\cap\Omega}u+(1-\eta)\operatorname*{osc}_{\Gamma\cap
4B}\operatorname{Tr}u.$

The constant $\eta$ depends only on on $n$, $C_{1}$ to $C_{6}$, and $C_{A}$.

Proof. Lemma 11.26 and Lemma 11.28 can be proved respectively as Theorem 2.4
in [HL, Section 4.3] and as Lemma 8.15 in [DFM2]. The proofs work as long as
Lemma 11.22 or Lemma 11.24 is true. $\Box$

We shall now present the Hölder regularity of solutions.

######  Lemma 11.30 (interior Hölder continuity).

Let $x\in\Omega$ and $R>0$ be such that $B(x,2R)\subset\Omega$, and let $u\in
W_{r}(B(x,2R))$ be a solution to $Lu=0$ in $B(x,2R)$. Set

$\operatorname*{osc}_{B}u:=\sup_{B}u-\inf_{B}u.$

Then there exists $\alpha\in(0,1]$ and $C>0$ such that for any $0<r<R$,

(11.31) $\operatorname*{osc}_{B(x,r)}u\leq
C\left(\frac{r}{R}\right)^{\alpha}\left(\frac{1}{m(B(x,R))}\int_{B(x,R)}u^{2}\,dm\right)^{\frac{1}{2}},$

where $\alpha$ and $C$ depend only on $n$, $C_{4}$, $C_{6}$, and $C_{A}$.
Hence $u$ is (possibly after modifying it on a set of measure $0$) locally
Hölder continuous with exponent $\alpha$.

######  Lemma 11.32.

Let $B=B(x,r)$ be a ball centered on $\Gamma$ and $u\in
W_{r}(B\cap\overline{\Omega})$ be a solution in $B\cap\Omega$ such that
$\operatorname{Tr}u$ is continuous and bounded on $B\cap\Gamma$. There exists
$\alpha>0$ such that for $0<s<r$,

(11.33) $\operatorname*{osc}_{B(x,s)\cap\Omega}u\leq
C\left(\frac{s}{r}\right)^{\alpha}\operatorname*{osc}_{B(x,r)\cap\Omega}u+C\operatorname*{osc}_{B(x,\sqrt{sr})\cap\Gamma}\operatorname{Tr}u$

where the constants $\alpha,C$ depend only on $n$, $C_{1}$ to $C_{6}$, and
$C_{A}$. In particular, $u$ (possibly after modification on a set of measure
0) is continuous on $B\cap\Omega$, can be extended by continuity on
$B\cap\Gamma$, and the values of this extension on $B\cap\Gamma$ are
$\operatorname{Tr}u$.

If, in addition, $\operatorname{Tr}u\equiv 0$ on $B$, then for any $0<s<r/2$

(11.34) $\operatorname*{osc}_{B(x,s)\cap\Omega}u\leq
C\left(\frac{s}{r}\right)^{\alpha}\left(\frac{1}{m(B)}\int_{B\cap\Omega}|u|^{2}dm\right)^{\frac{1}{2}}.$

Proof. The proof of the two last lemmas are the same as the ones of Theorem
2.5 in [HL, Section 4.3] and Lemma 8.16 in [DFM2]. $\Box$

It remains to treat the Harnack inequality.

######  Lemma 11.35 (Harnack Inequality ).

Let $B$ be a ball such that $2B\subset\Omega$, and let $u\in W_{r}(2B)$ be a
non-negative solution to $Lu=0$ in $2B$. Then

(11.36) $\sup_{B}u\leq C\inf_{B}u,$

where $C$ depends only on $n$, $C_{4}$, $C_{6}$, and $C_{A}$.

Proof. The proof in [DFM2] uses, roughtly speaking, the condition (H6’) to say
that the Harnack inequality can be proved using the classical theory of
uniformly elliptic operators in divergence form.

If we were to have (H6’) instead of (H6), we could proceed in a similar
manner. Fortunately for us, the proof in the classical theory can easily
adapted to our setting. This observation was already made in [FKS], but since
our conditions are slightly weaker than [FKS], we sketch the proof to check
that we don’t have any extra difficulties.

Step 1: The John-Nirenberg lemma.

Let $O$ be an open subset of $\Omega$. Suppose that $w\in L^{1}(O,m)$ lies in
$BMO(O,m)$, in the sense that that for every ball $B\subset O$

(11.37) $\fint_{B}|w-w_{B}|\,dm\leq C_{JN}$

for a constant $C_{JN}$ independent of $B$, and where $w_{B}$ denotes $\fint
w\,dm$. Then we claim that for any $B\subset O$,

(11.38) $\fint_{B}\exp\left(\frac{p_{0}}{C_{JN}}|w-w_{B}|\right)\,dm\leq C,$

where $p_{0}$ and $C$ depend only on $C_{4}$ (and $n$).

The claim is the John-Nirenberg lemma, whose proof uses only the Calderón-
Zygmund decomposition (see for instance [HL, Chapter 3, Theorem 1.5]).

Step 2: The weak Harnack inequality.

Suppose $2B\subset\Omega$ and let $u\in W(2B)$ be a non-negative supersolution
to $Lu=0$. Then we claim that there exists $p_{1}>0$ such that

(11.39) $\inf_{B}u\geq
C^{-1}\left(\fint_{2B}u^{p_{1}}\,dm\right)^{\frac{1}{p_{1}}},$

where $C^{-1}$ depends only on $C_{4}$, $C_{6}$ $C_{A}$, and $n$.

For any $\epsilon>0$, we consider the supersolution $\bar{u}=u+\epsilon>0$ and
then $v=\bar{u}^{-1}$. For any $\varphi\in C^{\infty}_{0}(2B)$, the function
$v^{2}\varphi$ belongs to $W_{0}$ thanks to Lemmas 6.20 and 9.20, and is
compactly supported in $\Omega$. So $v^{2}\varphi$ can be used as a test
function, by Lemma 11.10, hence

$\int_{2B}\mathcal{A}\nabla\bar{u}\cdot\nabla[v^{2}\varphi]\,dm\geq 0,$

that is,

$\int_{2B}\bar{u}^{-2}(\mathcal{A}\nabla\bar{u}\cdot\nabla\varphi)\,dm\geq
2\int_{2B}\bar{u}^{-3}\varphi(\mathcal{A}\nabla\bar{u}\cdot\nabla u)\,dm$

hence, by the positivity of $\mathcal{A}$

(11.40) $\int_{2B}(\mathcal{A}\nabla
v\cdot\nabla\varphi)\,dm\leq-2\int_{2B}\bar{u}^{-3}\varphi(\mathcal{A}\nabla\bar{u}\cdot\nabla
u)\,dm\leq 0.$

We deduce that $v$ is a non-negative subsolution in $2B$, so using Moser’s
estimate (Lemma 11.18), we get that for any $p>0$

$\sup_{B}v\leq C_{p}\left(\fint_{\frac{3}{2}B}v^{p}\,dm\right)^{\frac{1}{p}}$

where $C_{p}$ depends on $C_{4}$, $C_{6}$, $C_{A}$, $n$, and $p$. Using the
fact that $v=\bar{u}^{-1}$, we deduce that

(11.41) $\inf_{B}\bar{u}\geq
C_{p}\left(\fint_{\frac{3}{2}B}\bar{u}^{-p}\,dm\right)^{-\frac{1}{p}}.$

The claim (11.39) will be established as soon as we prove that for some
$p_{1}>0$, one has

(11.42)
$\left(\fint_{\frac{3}{2}B}\bar{u}^{-p_{1}}\,dm\right)\left(\fint_{\frac{3}{2}B}\bar{u}^{p_{1}}\,dm\right)\leq
C,$

with a bound $C$ independent of $u$ and

the $\epsilon$ used to define $\bar{u}$, and we shall now prove (11.42) using
the John-Nirenberg inequality.

Take $w=\log\bar{u}$; we want to check that $w\in BMO(\frac{3}{2}B)$. We test
$\bar{u}$ against the test function $\bar{u}^{-1}\varphi^{2}$, where
$\varphi\in C^{\infty}_{0}(2B)$ to obtain

$2\int_{2B}\varphi\bar{u}^{-1}(\mathcal{A}\nabla\bar{u}\cdot\nabla\varphi)\,dm-\int_{2B}\varphi^{2}\bar{u}^{-2}(\mathcal{A}\nabla\bar{u}\cdot\nabla\bar{u})\,dm\geq
0.$

We use the fact that $\nabla w=\bar{u}^{-1}\nabla u$ and the ellipticity
conditions (11.3)–(11.4) to obtain

$\int_{2B}\varphi^{2}|\nabla w|^{2}\,dm\leq C\int_{2B}\varphi|\nabla
w||\nabla\varphi|\,dm,$

which implies, by the Cauchy-Schwarz inequality, that

(11.43) $\int_{2B}\varphi^{2}|\nabla w|^{2}\,dm\leq
C\int_{2B}|\nabla\varphi|^{2}\,dm.$

For any ball $B^{\prime}\subset\frac{3}{2}B$ of radius $r^{\prime}$, we can
built a smooth function $\varphi$ such that $\varphi\equiv 1$ on $B^{\prime}$,
$\varphi\equiv 0$ on $\frac{9}{8}B^{\prime}$, and $|\nabla\varphi|\leq
10/r^{\prime}$. Using those test functions in (11.43) gives that for any
$B^{\prime}\subset\frac{3}{2}B$,

(11.44) $\fint_{B^{\prime}}|\nabla w|^{2}\,dm\leq C(r^{\prime})^{-2},$

where $C$ depends only on $C_{A}$. The assumption (H6), i.e. the Poincaré
inequality, infers now that

$\fint_{B^{\prime}}|w-w_{B^{\prime}}|\,dm\leq C,$

as in (11.37). From step 1, the inequality (11.38) thus holds, that is we can
find a $p_{1}>0$ such that

(11.45)
$\fint_{\frac{3}{2}B}\exp\left(p_{1}|w-w_{\frac{3}{2}B}|\right)\,dm\leq C.$

We are now ready for the proof of (11.42). Indeed, just observe that

$\begin{split}\left(\fint_{\frac{3}{2}B}u^{-p_{1}}\,dm\right)&\left(\fint_{\frac{3}{2}B}u^{p_{1}}\,dm\right)=\left(\fint_{\frac{3}{2}B}\exp(-p_{1}w)\,dm\right)\left(\fint_{\frac{3}{2}B}\exp(p_{1}w)\,dm\right)\\\
&=\left(\fint_{\frac{3}{2}B}\exp(-p_{1}[w-w_{\frac{3}{2}B}])\,dm\right)\left(\fint_{\frac{3}{2}B}\exp(p_{1}[w-w_{\frac{3}{2}B}])\,dm\right)\\\
&\leq\left(\fint_{\frac{3}{2}B}\exp(p_{1}|w-w_{\frac{3}{2}B}|)\,dm\right)^{2}\lesssim
1,\end{split}$

by (11.45).

Step 3: Conclusion.

We combine (11.39) with Lemma 11.18 \- the Moser inequality inside the domain,
to get the desired Harnack inequality

$\sup_{B}u\leq C\inf_{B}u$

We should require $B$ to satisfy $4B\subset\Omega$, but we can easily solve
this issue by covering $B$ by balls $B^{\prime}$ of smaller radius that
satisfy $4B^{\prime}\subset 2B\subset\Omega$. $\Box$

We shall also need the following version of the Harnack inequality, which will
be useful to define the harmonic measure.

######  Lemma 11.46.

Let $K$ be a compact subset of $\Omega$ and let $u\in W_{r}(\Omega)$ be a non-
negative solution in $\Omega$. Then

(11.47) $\sup_{K}u\leq C_{K}\inf_{K}u,$

where $C_{K}$ depends only on $n$, $C_{1}$, $C_{2}$, $C_{4}$, $C_{6}$,
$C_{A}$, and $\operatorname{diam}\,K/\,\mathrm{dist}(K,\Gamma)$.

Proof. The proof is the same as the one of [DFM2, Lemma 8.10]. The topological
conditions (H1)–(H2) allow us to connect any couple of points in $K$ by a
chain of balls that says away from the boundary (see Proposition 2.18). The
length of the chain can be bounded by a constant depending only on
$\operatorname{diam}K/\,\mathrm{dist}(K,\partial\Omega)$. We then use the
Harnack inequality above on those balls. $\Box$

##  12\. Construction of the harmonic measure

We follow Section 9 in [DFM2] and, as in the previous section, we will refer
to [DFM2] when the proofs do not require any new argument.

The objective of the section is, as the title suggests, to construct a
harmonic measure associated to our degenerate operator
$L=-\mathop{\operatorname{div}}A\nabla$ that still satisfies (11.1)–(11.2). By
harmonic measure, we mean a family of measures $\omega^{X}_{L}$, where
$X\in\Omega$ is called pole of the harmonic measure, such that for any Borel
set $E\subset\Gamma$, the function $u_{E}$ defined as
$u_{E}(X)=\omega^{X}_{L}$ solves the Dirichlet problem

(12.1) $\left\\{\begin{array}[]{ll}Lu_{E}=0&\text{ in }\Omega\\\
u_{E}={\mathds{1}}_{E}&\text{ on }\Gamma.\end{array}\right.$

But (12.1) does not make a lot of sense for the moment. The part “$Lu_{E}=0$
in $\Omega$” is easy to interpret: we want $u_{E}$ to lie in $W_{r}(\Omega)$
and to be a solution to $Lu=0$ in $\Omega$. The part “$u_{E}={\mathds{1}}_{E}$
on $\Gamma$” is harder to understand: we could hope that the meaning is
$\operatorname{Tr}u_{E}={\mathds{1}}_{E}$ $\mu$-a.e. on $\Gamma$, but it is
unclear that it is possible at this point.

Another issue is the uniqueness: just take $\Omega=\mathbb{R}^{n}_{+}$ \- i.e.
$\Gamma=\mathbb{R}^{n-1}$ \- and $E=\emptyset$ \- i.e. ${\mathds{1}}_{E}\equiv
0$ \- and we can find at least two solutions ($u_{1}\equiv 0$ and
$u_{2}=\delta$) to $Lu=0$ that satisfy both $\operatorname{Tr}u=0$. Imposing
that $u$ lies in $W$ is not immediately possible, since characteristic
functions of non-trivial sets do not always lie in $H$.

Our salvation will come from the maximum principle. And instead of (12.1), we
shall say that the harmonic measure $\omega_{L}^{X}$ is built such that for
any $g\in C^{\infty}_{0}(\mathbb{R}^{n})$, the function defined by

$u(X)=\int_{\Gamma}g(y)d\omega^{X}_{L}(y)$

lies in $W$, is a solution to $Lu=0$, and satisfies $\operatorname{Tr}u=g$.
Let us now give a full presentation.

We say that $f\in W^{-1}$ if $f$ is a linear form on $W_{0}$ that satisfies

$|\left<f,v\right>_{W^{-1},W_{0}}|\leq C_{f}\|v\|_{W},$

where we anticipate slightly and denote by $\left<f,v\right>_{W^{-1},W_{0}}$
the effect of $f$ on $v$. The best constant $C_{f}$ in the inequality above is
denoted $\|f\|_{W^{-1}}$.

Let us give first the existence and uniqueness of solutions $u\in W$ to $Lu=f$
and $\operatorname{Tr}u=g$, where $f\in W^{-1}$ and $g\in H$ are given.

######  Lemma 12.2.

For any $f\in W^{-1}$ and any $g\in H$, there exists a unique $u\in W$ such
that

(12.3) $\int_{\Omega}\mathcal{A}\nabla u\cdot\nabla
v\,dm=\left<f,v\right>_{W^{-1},W_{0}}\qquad\text{ for all }v\in W_{0},$

and

(12.4) $\operatorname{Tr}u=g\text{ a.e. on }\Gamma.$

Moreover, there exists $C>0$, independent of $f$ and $g$, such that

(12.5) $\|u\|_{W}\leq C(\|g\|_{H}+\|f\|_{W^{-1}}).$

Proof. The lemma follows from the extension theorem (Theorem 8.5) and the Lax-
Milgram theorem. Details are given in the proof of [DFM2, Lemma 9.1]. $\Box$

The next result needed is a maximum principle. In its weak form, the maximum
principle is as follows.

######  Lemma 12.6.

Let $u\in W$ be a supersolution in $\Omega$ satisfying $\operatorname{Tr}u\geq
0$ $\mu$-a.e. on $\Gamma$. Then $u\geq 0$ a.e. in $\Omega$.

Proof. Take $v:=\min\\{u,0\\}\leq 0$; we want to prove that $v\equiv 0$. Lemma
9.20 allows us to say that $v\in W$, $\nabla v=\nabla
u{\mathds{1}}_{\\{u<0\\}}$, and $\operatorname{Tr}v=0$ a.e. in $\Gamma$. In
particular $v\in W_{0}$, which makes $v$ a valid test function to be tested
against the supersolution $u$ (see Lemma 11.10). This gives

(12.7) $0\geq\int_{\Omega}\mathcal{A}\nabla u\cdot\nabla
v\,dm=\int_{\\{u<0\\}}\mathcal{A}\nabla u\cdot\nabla
u\,dm=\int_{\Omega}\mathcal{A}\nabla v\cdot\nabla v\,dm\geq C_{A}^{-1}\|\nabla
v\|_{W}^{2}\geq 0.$

that is, $\|\nabla v\|_{W}=0$. Yet, $\|.\|_{W}$ is a norm on $W_{0}\ni v$,
hence $v=0$ a.e. in $\Omega$. $\Box$

Here is a stronger form of the maximum principle.

######  Lemma 12.8 (Maximum principle).

Let $u\in W$ be a solution to $Lu=0$ in $\Omega$. Then

(12.9) $\sup_{\Omega}u\leq\sup_{\Gamma}\operatorname{Tr}u$

and

(12.10) $\inf_{\Omega}u\geq\inf_{\Gamma}\operatorname{Tr}u,$

where we recall that $\sup$ and $\inf$ actually essential supremum and
infimum. In particular, if $\operatorname{Tr}u$ is essentially bounded (for
the measure $\mu$), then

(12.11) $\sup_{\Omega}|u|\leq\sup_{\Gamma}|\operatorname{Tr}u|.$

Proof. Let us prove (12.9). Write $M$ for the essential supremum of
$\operatorname{Tr}u$ on $\Gamma$; we may assume that $M<+\infty$, because
otherwise (12.9) is trivial. Then $M-u\in W$ and $\operatorname{Tr}(M-u)\geq
0$ a.e. on $\Gamma$. Lemma 12.6 yields $M-u\geq 0$ a.e. in $\Omega$, that is

(12.12) $\sup_{\Omega}u\leq\sup_{\Gamma}Tu.$

The lower bound (12.10) is similar and (12.11) follows. $\Box$

The harmonic measure will be defined with the help of the Riesz representation
theorem (for measures), so we need a linear form on $C^{0}_{0}(\Gamma)$, the
space of compactly supported continuous functions on $\Gamma$. We also write
$C^{0}_{b}(\overline{\Omega})$ for the space of continuous bounded functions
on $\overline{\Omega}$.

######  Lemma 12.13.

There exists a unique linear operator

(12.14) $U:\,C^{0}_{0}(\Gamma)\to C^{0}_{b}(\overline{\Omega})$

such that, for every every $g\in C^{0}_{0}(\Gamma)$,

  1. (i)

if $g\in C^{0}_{0}(\Gamma)\cap H$, then $Ug\in W$, and it is the solution of
(12.3)–(12.4), with $f=0$, provided by Lemma 12.2;

  2. (ii)

$\displaystyle\sup_{\Omega}Ug=\sup_{\Gamma}g$ and
$\displaystyle\inf_{\Omega}Ug=\inf_{\Gamma}g$;

In addition, $U$ enjoys the following properties:

  1. (iii)

the restriction of $Ug$ to $\Gamma$ is $g$;

  2. (iv)

$Ug\in W_{r}(\Omega)$ and is a solution to $Lu=0$ in $\Omega$;

  3. (v)

if $B$ is a ball centered on $\Gamma$ and $g\equiv 0$ on $B$, then $Ug$ lies
in $W_{r}(B\cap\overline{\Omega})$;

Proof. The proof of the existence of $U$ and its properties is similar to the
one of Lemma 9.4 in [DFM2]. Still, let us give a sketch of the proof of
existence for $U$.

First, we use $(i)$ to define $U$ on $C^{0}_{0}(\Gamma)\cap H$. Lemma 12.8
proves that $(ii)$ is satisfied for any $g\in C^{0}_{0}(\Gamma)\cap H$, and in
particular

$U:\,C^{0}_{0}(\Gamma)\cap H\to C^{0}_{b}(\overline{\Omega})$

is a continuous operator if we equip both $C^{0}_{0}(\Gamma)\cap H$ and
$C^{0}_{b}(\overline{\Omega})$ with the norm $\|.\|_{\infty}$. Then, observe
that the space $\mathcal{C}_{0}^{0}(\Gamma)\cap H$ contains all the
restrictions to $\Gamma$ of functions in $C^{\infty}_{0}(\mathbb{R}^{n})$, and
hence $\mathcal{C}_{0}^{0}(\Gamma)\cap H$ is dense in $C_{0}^{0}(\Gamma)$
(equipped with the norm $\|.\|_{\infty}$). We define $U:\,C^{0}_{0}(\Gamma)\to
C^{0}_{b}(\overline{\Omega})$ as the only bounded extension of
$U:\,C^{0}_{0}(\Gamma)\cap H\to C^{0}_{b}(\overline{\Omega})$, and in
particular $(ii)$ is preserved.

The property $(iii)$ is true when $g\in C^{0}_{0}(\Gamma)\cap H$ thanks to
Lemma 11.32, and the property is kept by the extension. The property $(iv)$ is
true when $g\in C^{0}_{0}(\Gamma)\cap H$ by Lemma 12.2, and can be extended
for all $g\in C^{0}_{0}(\Gamma)$ with the help of Cacciopoli’s inequality
(Lemma 11.12). As for the property $(v)$ \- which is immediate by construction
for any $g\in C^{0}_{0}(\Gamma)\cap H$ \- we prove it by approaching $g\in
C^{0}_{0}(\Gamma)$ by functions in $C^{0}_{0}(\Gamma)\cap H$ and we use Lemma
11.15 to control the estimate on the gradient when we take the limit.

The uniqueness of $U$ is also immediate, since $(ii)$ forces
$U:\,C^{0}_{0}(\Gamma)\to C^{0}_{b}(\overline{\Omega})$ to be the continuous
extension of $U:\,C^{0}_{0}(\Gamma)\cap H\to C^{0}_{b}(\overline{\Omega})$
given by $(i)$. $\Box$

######  Lemma 12.15.

For any $X\in\Omega$, there exists a unique positive regular Borel measure
$\omega^{X}:=\omega_{L}^{X}$ on $\Gamma$ such that

(12.16) $Ug(X)=\int_{\Gamma}g(y)d\omega^{X}(y)$

for any $g\in C^{0}_{0}(\Gamma)$. Besides, for any Borel set $E\subset\Gamma$,

(12.17) $\omega^{X}(E)=\sup\\{\omega^{X}(K):\,E\supset K,\,K\text{
compact}\\}=\inf\\{\omega^{X}(V):\,E\subset V,\,V\text{ open}\\}.$

In addition, the harmonic measure is a probability measure, that is

(12.18) $\omega^{X}(\Gamma)=1.$

Proof. The first part, that is the existence of a positive regular Borel
measure satisfying (12.16), and the property (12.17), is immediate by applying
the Riesz representation theorem (see for instance [Rud, Theorem 2.14]) to
$U$. The positivity of the harmonic measure comes from
$\displaystyle\inf_{\Omega}Ug=\inf_{\Gamma}g$ given by Lemma 12.13 $(ii)$.

The fact that $\omega^{X}(\Gamma)\leq 1$ comes from the fact that
$\displaystyle\sup_{\Omega}Ug=\sup_{\Gamma}g$. We can prove that
$\omega^{X}(\Gamma)\geq 1$ by using the Hölder regularity at the boundary
(Lemma 11.32). See the proof of Lemma 9.6 in [DFM2] for details. $\Box$

######  Lemma 12.19.

Let $E\subset\Gamma$ be a Borel set and define the function $u_{E}$ on
$\Omega$ by $u_{E}(X)=\omega^{X}(E)$. Then

  1. (i)

if there exists $X\in\Omega$ such that $u_{E}(X)=0$, then $u_{E}\equiv 0$;

  2. (ii)

the function $u_{E}$ lies in $W_{r}(\Omega)$ and is a solution in $\Omega$;

  3. (iii)

if $B\subset\mathbb{R}^{n}$ is a ball such that $E\cap B=\emptyset$, then
$u_{E}\in W_{r}(B\cap\overline{\Omega})$ and $\operatorname{Tr}u_{E}=0$ a.e.
on $B$.

Proof. The proof of this result is analogous to the one of Lemma 9.7 in
[DFM2]. Here are some of the main ideas. The proof of $(i)$ is quite easy. We
approach ${\mathds{1}}_{E}$ by $g\in C^{0}_{0}(\Gamma)$, and compare $u_{E}$
with $u_{g}=Ug$. We get that $\left|u_{g}(X)-u_{E}(X)\right|=u_{g}(X)$ is as
small as we want. Then we use Lemma 11.46 to say that $0\leq u_{E}(Y)\lesssim
u_{g}(Y)\lesssim u_{g}(X)\leq\epsilon$, and we let $\epsilon$ tend to $0$.

The proofs of $(ii)$ and $(iii)$ are longer. They consist in approaching
$u_{E}$ by functions $u_{g}=Ug$ that have all the desired properties by Lemma
12.13, then controlling $\nabla u_{g}$ uniformly with the help of Lemma 11.12
and Lemma 11.15. We eventually use Lemma 11.30, Lemma 11.32, and Lemma 12.8 to
ensure that the $u_{g}$’s are nice functions that satisfy $g\leq h\Rightarrow
u_{g}\leq u_{h}$. $\Box$

##  13\. Bounded boundaries

So far, in Section 5 and hence in all the sections following it, we have been
working with a boundary set $\Gamma$ which is unbounded, and hence an
unbounded domain $\Omega$ too. But it is some times interesting, and not too
difficult, to deal with bounded sets $\Gamma$. In this section we describe how
to modify our assumptions, and some times the proofs, to extend the results of
this paper to the case of bounded $\Gamma$. So let us assume now that
$\Gamma=\partial\Omega$ is bounded, and (to normalize things) that

(13.1) $0\in\Gamma\ \text{ and }\ \operatorname{diam}(\Gamma)=R_{0}>0.$

There will be two slightly different cases to consider, Case 1 when $\Omega$
also is bounded (and connected - due to (H1)), and Case 2 when $\Omega$ is the
unbounded component of $\mathbb{R}^{n}\setminus\Gamma$. When the dimension of
$\Gamma$ is smaller than $n-1$, we are in Case 2, but Case 1 is interesting
too, especially in the context of mixed co-dimensions, where we may do it on
purpose to add pieces of boundary that isolate some parts of a domain. That
is, even if we start with the unbounded component of
$\mathbb{R}^{n}\setminus\Gamma$, we could for instance add to $\Gamma$ a large
sphere like $S=\partial B(0,2R_{0})$ to $\Gamma$, and restrict our attention
to the bounded component of $\mathbb{R}^{n}\setminus(\Gamma\cup S)$ that
touches $S$ because this is simpler.

Most of the results above are local, in the sense that they rely on
computations that do not go too far. The only difference that it makes on our
assumptions is that - if $\Omega$ is bounded - we need to take the $r$ in (H1)
not too large, for instance not bigger than $\operatorname{diam}\Omega$, while
the unbounded case will require that $r$ to be taken in the full range
$(0,+\infty)$.

Observe also that in the case where $\Gamma$ is bounded, we just need the
analogue of (H1)-(H6), where we keep the same statement as before but only ask
(H1) and (H3) to hold when $B(x,r)\subset B_{0}=B(0,2R_{0})$, and (H2) to hold
for points $X,Y\in B_{0}$ (the other case would follow anyway). When $\Omega$
is also bounded, we can restrict to $B(0,2R_{0})$ in the definition of (H4)
(the absolute continuity and doubling property for $m$) and (H6) (the density
and weak Poicaré inequality for $m$). One could see such apparent weakening as
an improvement, but it is easy to check that the cases that we dropped are
automatically true for bounded $\Gamma$ and/or $\Omega$. However the truth is
that we are not really interested in studying wild weights $w$ far from
$\Gamma$, and a simple monomial equivalent at infinity would probably be
enough.

With these assumptions, most of our local estimates still hold, with very
little changes in the proofs. Let us be a little more specific.

We keep the definition of $W$ as it was. Notice that constant functions still
lie in $W$ (with a vanishing norm); depending on the behavior of $m$ far from
$\Gamma$, the functions $u\in W$ may have a more or less rich behavior near
infinity, but let us not bother yet. Section 4 goes through without
modification (we kept the same assumptions on $m$ alone).

The definition of dyadic pseudocubes has to be changed a little bit: we only
use the partition $\Gamma=\bigcup_{j\in{\mathcal{J}}_{k}}Q_{j}^{k}$ for $k\geq
k_{0}$, where $k_{0}$ is such that $2^{-k_{0}}\sim 4R_{0}$, and also it is
customary to take for $k=k_{0}$ the trivial decomposition into a unique cube
$Q_{0}=\Gamma$. Of course, all the subsequent sums in $k$ will be restricted
to $k\geq k_{0}$.

Then, even in the definition of the access regions $\gamma(x)$ (as in (5.8))
for unbounded domains $\Omega$, we will only consider cubes of size at most
$CR_{0}$, and so our access regions will be bounded. We are not shocked
because for many of our results we already considered the truncated regions of
(5.11). The results of Section 5, and in particular the improved Poincaré
inequality in Theorem 5.25, are local and stay the same, but we only consider
sets that are contained in $CB_{0}$. Recall however that the case of balls $B$
such that $2B\subset\Omega$, even when $B$ is large, is taken care of in Lemma
4.7, so we will never be in real trouble anywhere.

Our definition (6.5) of the Hilbert space $H$ on $\Gamma$ stays the same; as
before constants lie in $H$, with a vanishing norm. Theorem 6.6 on the
existence of a trace operator is still valid with the same proof. The proof
does not use the values of $u\in W$ at distance more than $CR_{0}$ from
$\Gamma$, so we may even forget the corresponding part of $||f||_{W}$ in the
estimate for $\|\operatorname{Tr}(f)\|_{H}$. That is,

(13.2) $\|\operatorname{Tr}u\|_{H}^{2}\lesssim\int_{\\{Z\in\Omega\cap
B(0,CR_{0})\\}}|\nabla u(Z)|^{2}\,dm(Z).$

Another way (softer but just a bit more complicated technically) to check this
is to notice that, when $\Omega$ is unbounded, we may always truncate any
$u\in W$ in the following way. We select a smooth cut-off function $\varphi$
such that $\varphi=1$ in $B_{0}$ and $\varphi=0$ outside of $2B_{0}$, pick a
ball $B_{1}$ of radius $R_{0}$ such that $B_{1}\subset 2B_{0}\setminus B_{0}$
and $B_{1}$ touches $\partial B_{0}$, let $m_{1}$ denote the average of $u$ on
$B_{1}$, and consider the “truncated” function $\widetilde{u}=\varphi
u+(1-\varphi)m_{1}$. Obviously $\widetilde{u}$ has the same trace, and is
would be easy to see, using the extension of Lemma 4.7 to a $(2,2)$-Poincaré
inequality, as in Theorem 5.24 with $p=2$, that $\widetilde{f}\in W$, with
$||\widetilde{f}||_{W}^{2}\leq C\int_{2B_{0}}|\nabla f|^{2}dm$. Of course all
this is much easier if $w$ is reasonably smooth on $2B_{0}\setminus B_{0}$.

The product rule for the trace and gradient (Lemma 6.20), as well as all the
local algebraic formulas, go through. Similarly, the Poincaré inequalities on
the boundary (Theorem 7.1 and Corollary 7.9) stay the same, except that we
restrict to balls of radius at most $10R_{0}$, say.

Our extension theorem (Theorem 8.5) is still true; the construction also
easily gives that $\operatorname{Ext}(f)$ is constant on
$\mathbb{R}^{n}\setminus CB_{0}$ (when $\Omega$ is unbounded), and we can take
the constant equal to the average $m_{0}=\fint_{\Gamma}fd\mu$ . The simplest
way to see this is to consider $f-m_{0}$ and use the formula (8.3), but
restrict the sum to Whitney cubes of size at most $CR_{0}$. Or said
differently, for the function $f-m_{0}$ we can use $y_{I}=0$ for all the large
Whitney cubes.

There is no difficulty with the density or algebraic results of Section 9, and
the local spaces of Section 10 are (just a bit) simpler. The definition of
solutions is local, and all the regularity theorems for solutions found in
Section 11 stay the same. This statement may look obvious, we are saying since
the beginning of the section that all the results are exactly the same for
bounded $\Gamma$ and unbounded $\Gamma$, but let us observe the following
interesting fact. The boundary regularity results, such as Lemmas 11.20 and
11.32, hold for all balls $B$ centered at the boundary even when the radius of
$B$ is way bigger than the diameter of $\Gamma$, and so can be applied for
instance to the Green functions - that we shall introduce in a next paper.

Let us now review the basic features of the harmonic measure. Its construction
given in Section 12 still goes through; that is, the existence of solutions as
in Lemma 12.2 is still valid, by Lax Milgram, and so is the maximum principle
that allows one to solve the Dirichlet problem for $f\in
C^{0}_{0}(\Gamma)=C^{0}(\Gamma)$. Thus $\omega^{X}$ is defined (by the Riesz
representation theorem), and is again a probability measure because the best
extension of the function $1$ is $1$.

If $\Gamma$ is bounded and $\Omega$ is the unbounded component of
$\mathbb{R}^{n}\setminus\Gamma$, then Brownian paths leaving from $X\in\Omega$
have a nonzero probability of never touching $\Gamma$ before going to
infinity. It means that the classical harmonic measure - defined from the
Laplacian - is not a probability measure. This simple case is however not
included in our theory; indeed the assumption (H5) fails for large $r$ when we
take $\mu$ as the surface measure on the bounded set $\Gamma$ and $m$ as the
Lebesgue measure on $\Omega$. On the contrary, our theory roughly says that a
modified Brownian motion, that imposes a drift in the direction of $\Gamma$
when we are far from it, is sufficient to guarantee to touch $\Gamma$ with
probability one.

For the two last sections, where we study the Green functions and the harmonic
measure, leading to a comparison principle, we do not assume that $\Gamma$ is
unbounded anymore. So both $\Gamma$ and $\Omega$ can be bounded or unbounded,
and we believe that the case where $\Gamma$ is bounded and $\Omega$ is the
unbounded component of $\mathbb{R}^{n}\setminus\Gamma$ is the most tricky one.

##  14\. Green functions

We associate Green functions to the degenerate elliptic operator $L$. A Green
function is, formally, a function $g$ defined on $\Omega\times\Omega$ and such
that for any $y\in\Omega$, the function $g(.,y)$ satisfies (12.3) and (12.4)
with $f=\delta_{y}$ \- the Dirac distribution at the point $y$ \- and $g\equiv
0$.

The harmonic measure can be seen as a fundamental tool to solve the problem
$Lu=0$ in $\Omega$ with $\operatorname{Tr}u=g$ on $\Gamma$, while the Green
function is a key ingredient to be able to solve $Lu=f$ in $\Omega$ with
$\operatorname{Tr}u=0$ on $\Gamma$. Their properties are actually related, as
we shall see in Section 15.

Let us recall that, as in the previous sections, we assume (H1)–(H6), and
(11.1)–(11.2).

In order to define the Green function in our context, we will follow closely
the proof of Grüter and Widman [GW] (as in [DFM2]). In the article [GW], the
authors proved the existence of the Green functions $g(.,y)$ by taking a weak
limit of some $g^{\rho}(.,y)$ that solves $Lu=f^{\rho}$ and
$\operatorname{Tr}u\equiv 0$ for some $f^{\rho}$ that ‘approximates’ the delta
distribution $\delta_{y}$.

Some difficulties appears when we try to get ‘local’ estimates, i.e. when the
distance between $x$ and $y$ is small compared to the distance of both points
to the boundary. Those estimates are needed to show that our $g^{\rho}(.,y)$
are uniformly bounded in some good space. We solve those difficulties by using
methods inspired from [FJK], where the authors deal with degenerate elliptic
operators but they define Green function via another method.

For short, we claim here that Grüter and Widman’s method can be applied - up
to few changes - to obtain Green functions in a large varieties of situations,
and for instance doesn’t require to have a global Sobolev inequality.

Instead of giving a big theorem for the start, as in [DFM2], we choose here to
divide the work, and prove plenty of small lemmas, whose proofs are sometimes
omitted because they are the same as in [DFM2]. The important results are
gathered at the end of the section, in Theorem 14.60.

######  Definition 14.1.

Let $y\in\Omega$ and $\rho>0$ satisfy $100\rho<\delta(y)$. The function
$g^{\rho}(.,y)$ is the function in $W_{0}$ that satisfies

(14.2) $\int_{\Omega}\mathcal{A}\nabla g^{\rho}(.,y)\cdot\nabla
v\,dm=\fint_{B(y,\rho)}v\,dm\qquad\text{ for all }v\in W_{0},$

as given by Lemma 12.2.

Notice that the definition makes sense, because $v\to\fint_{B(y,\rho)}v\,dm$
is a bounded linear form on $W_{0}$ (and hence an element $f^{\rho}\in W^{-1}$
to which we apply the lemma), by the doubling condition (H4) and the Poincaré
inequality Corollary 7.9. The norm of $f^{\rho}$ in $W^{-1}$ depends on $y$
and $\rho$, but it doesn’t matter.

Since $y$ will be fixed for a long part of our section, we write in the sequel
$\mathfrak{g}^{\rho}$ for $g^{\rho}(.,y)$ and $B_{\rho}$ for $B(y,\rho)$. Then
the condition (14.2) in the definition becomes

(14.3) $\int_{\Omega}\mathcal{A}\nabla\mathfrak{g}^{\rho}\cdot\nabla
v\,dm=\fint_{B_{\rho}}v\,dm\qquad\text{ for all }v\in W_{0},$

We deduce at once from the definition that

(14.4) $\mathfrak{g}^{\rho}\in W_{0}$ is a solution to $Lu=0$ in
$\Omega\setminus\overline{B_{\rho}}$.

In particular, by Lemmas 11.30 and 11.32, the function $\mathfrak{g}^{\rho}$
is continuous on $\overline{\Omega}\setminus\overline{B_{\rho}}$.

######  Lemma 14.5.

For all $y\in\Omega$, the function $\mathfrak{g}^{\rho}=g^{\rho}(.,y)$ is
nonnegative.

Proof. The proof of this fact is identical to the one given for [DFM2, Lemma
10.1], and relies only on the stability of $W_{0}$ provided by Lemma 9.20.
$\Box$

We now prove pointwise estimates on $\mathfrak{g}^{\rho}$ and start with the
case when $x$ is far from $y$.

######  Lemma 14.6.

If $x,y\in\Omega$ are such that $10|x-y|\geq\delta(y)$, then

$\mathfrak{g}^{\rho}(x)\leq C\frac{|x-y|^{2}}{m(B(y,|x-y|)\cap\Omega)},$

where $C$ depends on $C_{1}$ to $C_{6}$, $C_{A}$, and $n$.

Proof. Let $R\geq\delta(y)>100\rho>0$, and write $B_{R}$ for $B(y,R)$. Let $p$
be in the range given by Corollary 7.9; that is, any $p\in[1,2k)$, where $k$
is a constant that depends on the geometry, will do. We want to prove that for
all $t>0$,

(14.7) $\frac{m(\\{x\in
B_{R},\,\mathfrak{g}^{\rho}(x)>t\\})}{m(B_{R}\cap\Omega)}\leq
C\left(\frac{t\,m(B_{R}\cap\Omega)}{R^{2}}\right)^{-\frac{p}{2}}$

with a constant $C$ independent of $\rho$, $t$ and $R$. The proof of the claim
is analogous to the one in [DFM2], but we repeat it because we will use
similar computations later on. We use (14.3) with the test function

(14.8)
$\varphi(z):=\left(\frac{2}{t}-\frac{1}{\mathfrak{g}^{\rho}(z)}\right)^{+}=\max\left\\{0,\frac{2}{t}-\frac{1}{\mathfrak{g}^{\rho}(z)}\right\\}$

(and $\varphi(z)=0$ if $\mathfrak{g}^{\rho}(z)=0$), which lies in $W_{0}$ by
Lemma 9.20. Set
$\Omega_{s}:=\big{\\{}z\in\Omega,\,\mathfrak{g}^{\rho}(z)>s\big{\\}}$ and
observe that $\varphi$ is supported in $\Omega_{t/2}$. Hence

(14.9)
$a(\mathfrak{g}^{\rho},\varphi)=\int_{\Omega_{t/2}}\frac{\mathcal{A}\nabla\mathfrak{g}^{\rho}\cdot\nabla\mathfrak{g}^{\rho}}{(\mathfrak{g}^{\rho})^{2}}\,dm=\fint_{B_{\rho}}\varphi\,dm\leq\frac{2}{t}$

and then, thanks to the ellipticity condition (11.3),

(14.10)
$\int_{\Omega_{t/2}}\frac{|\nabla\mathfrak{g}^{\rho}|^{2}}{(\mathfrak{g}^{\rho})^{2}}\,dm\leq\frac{C}{t}.$

Pick a point $y_{0}\in\Gamma$ such that $|y-y_{0}|=\delta(y)$. Set
$\widetilde{B}_{R}$ for $B(y_{0},2R)\supset B_{R}$. Define $v$ by
$v(z):=(\ln(\mathfrak{g}^{\rho}(z))-\ln t+\ln 2)^{+}$; then $v\in W_{0}$ too,
thanks to Lemma 9.20, and $|\nabla
v|^{2}=|\nabla\mathfrak{g}^{\rho}|^{2}/(g^{\rho})^{2}$. Corollary 7.9 (the
Sobolev-Poincaré inequality at the boundary) implies that

(14.11)
$\begin{split}\left(\int_{\Omega_{t/2}\cap\widetilde{B}_{R}}|v|^{p}\,dm\right)^{\frac{1}{p}}&\leq
CR\,m(\widetilde{B}_{R}\cap\Omega)^{\frac{1}{p}-\frac{1}{2}}\left(\int_{\Omega_{t/2}\cap
2\widetilde{B}_{R}}|\nabla v|^{2}\,dm\right)^{\frac{1}{2}}\\\ &\leq
CR\,m(B_{R}\cap\Omega)^{\frac{1}{p}-\frac{1}{2}}t^{-\frac{1}{2}}\end{split}$

by (14.10) and (H4). Yet, $v\geq\ln(2)$ on $\Omega_{t}$, and thus the above
inequality gives that

(14.12) $m(\Omega_{t}\cap B_{R})\leq
m(\Omega_{t}\cap\widetilde{B}_{R})\lesssim
R^{p}[m(B_{R}\cap\Omega)]^{1-\frac{p}{2}}t^{-\frac{p}{2}}\leq
m(B_{R}\cap\Omega)\left(\frac{m(B_{R}\cap\Omega)t}{R^{2}}\right)^{-\frac{p}{2}}.$

The claim (14.7) follows. We are now ready to establish pointwise estimates on
$\mathfrak{g}^{\rho}$ when $x$ is far from $y$. We now aim to prove (14.7)
with a constant independent of $\rho$. Set $R=10|x-y|\geq\delta(y)$. By
(14.4), $\mathfrak{g}^{\rho}\in W_{0}$ is a solution to $Lu=0$ in
$\Omega\setminus\overline{B_{\rho}}$, so we can use Moser’s estimates; we
claim that we get that

(14.13)
$\mathfrak{g}^{\rho}(x)\leq\frac{C}{m(B(x,R/2)\cap\Omega)}\int_{B(x,R/2)\cap\Omega}\mathfrak{g}^{\rho}\,dm.$

When $\delta(x)\geq R/50$ we apply Lemma 11.18 in the ball $B(x,R/100)$), and
when $\delta(x)\leq R/50$ we apply Lemma 11.20 in the ball $B(x_{0},R/25)$,
where $x_{0}$ is such that $|x-x_{0}|=\delta(x)$. We can use (H4) to replace
the measure of the ball by $m(B(x,R/2)\cap\Omega)$.

We can use now the fact that $B(x,R/2)\subset B_{R}$ and Cavalieri’s formula
(see for instance [Duo, p. 28, Proposition 2.3]) to get that

(14.14)
$\mathfrak{g}^{\rho}(x)\lesssim\int_{0}^{+\infty}\frac{m(\Omega_{t}\cap
B_{R})}{m(\Omega\cap B_{R})}dt.$

Take $s>0$, to be chosen later. We bound the interior of the integral above by
$1$ when $t<s$, and for $t\geq s$ we use (14.7), which we apply with some
$p>2$ (this is possible); we get that

(14.15)
$\begin{split}\mathfrak{g}^{\rho}(x)&\lesssim\int_{0}^{s}\frac{m(\Omega_{t}\cap
B_{R})}{m(\Omega\cap B_{R})}dt+\int_{s}^{+\infty}\frac{m(\Omega_{t}\cap
B_{R})}{m(\Omega\cap B_{R})}dt\\\
&\lesssim\int_{0}^{s}1\,dt+\left(\frac{m(B_{R}\cap\Omega))}{R^{2}}\right)^{-\frac{p}{2}}\int_{s}^{+\infty}t^{-\frac{p}{2}}dt\\\
&\lesssim
s\left[1+\left(\frac{s\,m(B_{R}\cap\Omega)}{R^{2}}\right)^{-\frac{p}{2}}\right].\end{split}$

Now we minimize the right-hand side in $s$. We find $s\approx
R^{2}/m(B_{R}\cap\Omega)$ and then $\mathfrak{g}^{\rho}(x)\lesssim
R^{2}/m(B_{R}\cap\Omega)$. Since $R=10|x-y|$, the lemma follows from (H4).
$\Box$

The next result deals with the case when $x$ and $y$ are close to each other.

######  Lemma 14.16.

If $x,y\in\Omega$ are such that $40\rho\leq 2|x-y|\leq\delta(y)$, then

$\mathfrak{g}^{\rho}(x)\leq
C\int_{|x-y|}^{\delta(y)}\frac{r^{2}}{m(B(y,r))}\,\frac{dr}{r},$

where $C$ depends on $C_{1}$ to $C_{6}$, $C_{A}$, and $n$.

Proof. The proof of this result in the classical case, at least the one in
[GW], uses a global Sobolev equality. In our setting, we don’t have Sobolev
embeddings, only a Sobolev-Poincaé inequality (Theorem 4.7), and the $L^{q}$
norm given in the right-hand side of our Sobolev-Poincaré inequality may just
be $L^{2+\epsilon}$. In particular, we have no reason to get close to the
desired $L^{2n/(n-2)}$.

Fortunately, the slight improvement in the exponent of the $L^{p}$ space given
by Theorem 4.7 will be - as for Lemma 14.6 \- sufficient. Even better, the
proof will follow the same ideas as Lemma 14.6.

Let $j_{0}\geq 0$ be the biggest integer such that
$2^{j_{0}+1}|x-y|\leq\delta(y)$. To lighten the notation, we write $B^{j}$ for
$B(y,2^{j}|x-y|)$. We shall prove that for any $j$ between $0$ and $j_{0}-1$,

(14.17) $\sup_{B^{j+1}\setminus
B^{j}}\mathfrak{g}^{\rho}-\sup_{B^{j+2}\setminus
B^{j+1}}\mathfrak{g}^{\rho}\leq C\,\frac{(2^{j}|x-y|)^{2}}{m(B_{j})}.$

We write $\mathfrak{g}_{j}^{\rho}$ for