# Green function estimates on complements of low-dimensional uniformly
rectifiable sets

Guy David  Guy David. Université Paris-Saclay, CNRS, Laboratoire de
mathématiques d’Orsay, 91405, Orsay, France  guy.david@universite-paris-
saclay.fr ,  Joseph Feneuil  Joseph Feneuil. Université Paris-Saclay, CNRS,
Laboratoire de mathématiques d’Orsay, 91405, Orsay, France
<EMAIL_ADDRESS>and  Svitlana Mayboroda  Svitlana Mayboroda. School
of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
<EMAIL_ADDRESS>
###### Abstract.

It has been recently established in [DM2] that on uniformly rectifiable sets
the Green function is almost affine in the weak sense, and moreover, in some
scenarios such Green function estimates are equivalent to the uniform
rectifiability of a set. The present paper tackles a strong analogue of these
results, starting with the “flagship” degenerate operators on sets with lower
dimensional boundaries.

We consider the elliptic operators
$L_{\beta,\gamma}=-\operatorname{div}D^{d+1+\gamma-n}\nabla$ associated to a
domain $\Omega\subset\mathbb{R}^{n}$ with a uniformly rectifiable boundary
$\Gamma$ of dimension $d<n-1$, the now usual distance to the boundary
$D=D_{\beta}$ given by
$D_{\beta}(X)^{-\beta}=\int_{\Gamma}|X-y|^{-d-\beta}d\sigma(y)$ for
$X\in\Omega$, where $\beta>0$ and $\gamma\in(-1,1)$. In this paper we show
that the Green function $G$ for $L_{\beta,\gamma}$, with pole at infinity, is
well approximated by multiples of $D^{1-\gamma}$, in the sense that the
function
$\big{|}D\nabla\big{(}\ln\big{(}\frac{G}{D^{1-\gamma}}\big{)}\big{)}\big{|}^{2}$
satisfies a Carleson measure estimate on $\Omega$. We underline that the
strong and the weak results are different in nature and, of course, at the
level of the proofs: the latter extensively used compactness arguments, while
the present paper relies on some intricate integration by parts and the
properties of the “magical” distance function from [DEM].

Résumé. Dans [DM2] il est démontré que pour les domaines à bord uniformément
rectifiable, la fonction de Green vérifie des estimations faibles de bonne
approximation par des fonctions affines, avec une réciproque vraie dans
certains cas encourageants. Ici on part de la rectifiabilité uniforme et on
démontre les estimations fortes naturelles d’approximation de la fonction de
Green, et aussi des solutions, par des applications affines (ou, de manière
équivalente, des multiples de la distance au bord adoucie). L’étude inclut les
analogues naturels du Laplacien dans les domaine dont la frontière est de
grande co-dimension.

On considère les opérateurs elliptiques
$L_{\beta,\gamma}=\operatorname{div}D^{d+1+\gamma-n}\nabla$ associés à un
domaine $\Omega\subset\mathbb{R}^{n}$ dont le bord $\Gamma$ est Ahlfors
régulier et uniformément rectifiable de dimension $d<n-1$ et à la distance au
bord maintenant usuelle $D=D_{\beta}$ définie par
$D_{\beta}(X)^{-\beta}=\int_{\Gamma}|X-y|^{-d-\beta}d\sigma(y)$ pour
$X\in\Omega$, où $\beta>0$ et $\gamma\in(-1,1)$ sont des paramètres et
$\sigma$ une mesure Ahlfors régulière sur $\Gamma$. Les auteurs ont montré
précédemment que la mesure elliptique associée à $L_{\beta,\gamma}$ est bien
définie et est mutuellement absolument continue par rapport à $\sigma$, avec
un poids de $A_{\infty}$. Ici on démontre que la fonction de Green $G$ avec
pôle à l’infini associée à $L_{\beta,\gamma}$ est bien approchée par les
multiples de $D$, au sens où la fonction
$\big{|}D\nabla\big{(}\ln\big{(}\frac{G}{D^{1-\gamma}}\big{)}\big{)}\big{|}^{2}$
vérifie une condition de Carleson sur $\Omega$. Ces nouvelles estimations sont
différentes en nature. Les estimations de [DM2] reposaient sur des arguments
de compacité; ici on a besoin d’estimations plus précises, obtenues par
intégration par parties et en utilisant les propriétés algébriques de la
fonction $D_{\alpha}$ dans le cas“magique” de [DEM].

David was partially supported by the European Community H2020 grant GHAIA
777822, and the Simons Foundation grant 601941, GD. Mayboroda was supported in
part by the NSF grant DMS 1839077 and the Simons foundation grant 563916, SM

Key words/Mots clés. Uniformly rectifiable sets, degenerate elliptic
operators, Estimates on Green functions, harmonic measure in higher
codimension.

AMS classification: 42B37, 31B25, 35J25, 35J70.

###### Contents

  1. 1 Introduction
  2. 2 Definitions and anterior results
  3. 3 Proof of Theorem 2.21

##  1\. Introduction

Rectifiable sets are an important notion in geometric measure theory and the
calculus of variation, in particular because the sets that minimize an energy
often enter this category. In the past decades, many mathematicians worked on
finding characterizations of rectifiability by properties apparently unrelated
to geometric measure theory. In the early 90’s, the quantifiable version -
uniform rectifiability - was introduced in [DS1, DS2] along with many
characterizations in terms of geometry (such as big pieces of Lipschitz
images, or using Peter Jones’ $\beta$ numbers) and in terms of singular
integrals. Later, it was observed that uniformly rectifiable sets may be the
right extension of Lipschitz graphs for elliptic boundary value problems, that
is, if $\Omega\subset\mathbb{R}^{n}$ is an open set with uniformly rectifiable
boundary and $\Omega$ provides enough access its the boundary, then we can
control the oscillations of the harmonic functions in $\Omega$. It is even
more noteworthy that a criterion for rectifiability can be obtained using
harmonic functions. Indeed, Hofmann, Martell, and Uriarte-Tuero proved in
[HM1, HMU] that under some conditions regarding the access to the boundary,
$\partial\Omega$ is uniformly rectifiable if and only if the harmonic measure
on $\partial\Omega$ is absolutely continuous in a quantitative way - called
$A_{\infty}$ \- with respect to the surface measure (see also [AHMNT]). The
optimal topological conditions in this regard have been identified in [AHMMT].
These were accompanied but a rich array of beautiful and difficult alternative
characterizations, exploring Carleson estimates for the solutions, behavior of
the singular integral operators, extensions to more general elliptic
operators, to mention just a few. Here we do not aim to provide a survey of
the related literature; the reader can consult, e.g., [DFM4] for a more
detailed presentation of the literature.

A weakness of the above theory is the fact that the harmonic measure on
$\Gamma$ only makes sense when $\Gamma\subset\mathbb{R}^{n}$ is of dimension
$d>n-2$, because lower dimensional sets have probability zero to be hit by a
Brownian motion. As a consequence, rectifiability, a notion that exists for
all integer dimensions, can only be characterized by means of the harmonic
measure for sets $\Gamma\subset\mathbb{R}^{n}$ of dimension $d=n-1$. To
overcome this obstacle, the authors of the present article developed a theory
of degenerate elliptic operators. The idea was to define a ‘harmonic measure’
on a set $\Gamma$ of dimension $d<n-1$ by replacing the Laplacian by an
operator on $\Omega:=\mathbb{R}^{n}\setminus\Gamma$ in the form
$L=-\operatorname{div}w(x)\nabla$, where $w(x)$ goes to infinity at an
appropriate rate when $x$ is approaching $\Gamma$, so that the corresponding
‘Brownian motion’ is attracted by $\Gamma$ and hits the boundary with
probability one. The articles [DFM2, DFM5] set the elliptic theory for this.
In particular an appropriate elliptic measure is constructed for a large class
of sets of any dimension $d<n$ (or even mixed dimensions) and operators $L$ as
above, with the usual nice properties such as the non-degeneracy and doubling
properties for the harmonic measure, the Harnack inequality and the comparison
principle for solutions, and estimates for the change of poles.

Then we tested the relevancy of our new elliptic measure. Dahlberg proved in
[Da] that the classical harmonic measure is absolutely continuous with respect
to the surface measure, and even $A_{\infty}$, whenever the boundary is
Lipschitz. In [DFM3] we assumed that $\Gamma$ is the graph of a Lipschitz
function $\varphi:\mathbb{R}^{d}\to\mathbb{R}^{n-d}$ with small Lipschitz
constant, and proved the same $A_{\infty}$ property for correctly chosen
operators. Yet we had to be careful about the operator we picked because as
showed in [DFM4], based on counterexamples from [CFK, MM], not every operator
$L$ will work. We wanted an explicit operator, which could be constructed by a
single method for any set of dimension $d$. We set our choice on

(1.1) $L_{\beta,\gamma}:=-\operatorname{div}(D_{\beta})^{d+1+\gamma-n}\nabla,$

where $\beta>0$, $\gamma\in(-1,1)$, and $D_{\beta}$ is defined on $\Omega$ as

(1.2)
$D_{\beta}(X):=\left(\int_{\Gamma}|X-y|^{-d-\beta}d\sigma(y)\right)^{-1/\beta}.$

The quantity $D_{\beta}$ is equivalent to the distance to the boundary, i.e.,
there exists $C>0$ such that

(1.3) $C^{-1}\,\mathrm{dist}(X,\Gamma)\leq D_{\beta}\leq
C\,\mathrm{dist}(X,\Gamma)\qquad\text{ for }X\in\Omega,$

but the advantage of $D_{\beta}$ over $\,\mathrm{dist}(.,\Gamma)$ is to always
be smooth in a certain quantitative way.

When $\gamma=0$, the level of the degeneracy $D_{\beta}^{d+1-n}$ in the
coefficients of the operator
$L_{\beta,0}:=-\operatorname{div}(D_{\beta})^{d+1-n}\nabla$ makes it a perfect
analogue of the Laplacian for the sets with a $d$-dimensional boundary when
$d<n-1$. In particular, we proved that the harmonic measure associated to
$L_{\beta,\gamma}$ is $A_{\infty}$ with respect to the $d$-dimensional
Hausdorff measure. Moreover, recently, in two distinct papers [DM1, Fen1] we
extended this result to the more general case where
$\Gamma\subset\mathbb{R}^{n}$ is uniformly rectifiable. Studying these
operators and the dimension of the support of the corresponding elliptic
measure, we were naturally drawn to introducing a parameter $\gamma$ which
seemingly unbalances the situation. Besides, for $d=n-1$, the operator
$L_{\beta,\gamma}:=-\operatorname{div}(D_{\beta})^{\gamma}\nabla$ is the
celebrated Caffarelli-Silvester extension of the fractional Laplacian operator
(cf. [CS]). However, we were surprised to realize that the argument in [DFM3]
extends to all $\gamma\in(-1,1)$ rather simply, and the generalization to
uniformly rectifiable set is stated for any $\gamma\in(-1,1)$ in [Fen1]. The
proof in [DM1] relies on geometric arguments, such as corona decompositions
and the construction of sawtooth domains, and an extrapolation argument, which
allow one to reduce to the case of small Lipschitz graphs. The proof of [Fen1]
is substantially simpler and more direct, and is based on a trick unique to
the case where the dimension of $\Gamma$ is at most $n-2$.

######  Theorem 1.4 ([DM1, Fen1]).

Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly
rectifiable set with $d<n-1$, and let $\sigma$ be an Ahlfors regular measure
that satisfies (2.1). Take $\beta>0$, $\gamma\in(-1,1)$, define
$L_{\beta,\gamma}$ as in (1.1), and construct the associated harmonic measure
$\omega_{\beta,\gamma}^{X}$ as in Definition 2.20. Then
$\omega_{\beta,\gamma}^{X}$ is $A_{\infty}$-absolutely continuous with respect
to $\sigma$. This means that for every choice of $\epsilon\in(0,1)$, there
exists $\delta\in(0,1)$, that depends only on $C_{\sigma}$, $C_{0}$,
$\epsilon$, $n$, $d$, $\beta$, and $\gamma$, such that for each choice of
$x\in\Gamma$, $r>0$, a Borel set $E\subset B(x,r)\cap\Gamma$, and a corkscrew
point $X=A_{x,r}$ as in (2.3),

(1.5)
$\frac{\sigma(E)}{\sigma(B(x,r)\cap\Gamma)}<\delta\Rightarrow\frac{\omega^{X}_{\Omega,L}(E)}{\omega^{X}_{\Omega,L}(B(x,r)\cap\Gamma)}<\epsilon.$

It is known that in the present context where all our measures are doubling,
the $A_{\infty}$ condition also implies that under the assumptions of (1.5),

(1.6)
$\frac{\omega^{X}_{\Omega,L}(E)}{\omega^{X}_{\Omega,L}(B(x,r)\cap\Gamma)}<\delta\Rightarrow\frac{\sigma(E)}{\sigma(B(x,r)\cap\Gamma)}<\epsilon.$

Observe that in the previous theorem, contrary to the case of co-dimension 1,
i.e. when $d=n-1$, we don’t assume any topological condition. And that is
perfectly natural, since the domain $\Omega:=\mathbb{R}^{n}\setminus\Gamma$
has a lot of paths and ample access to the boundary (see Lemmas 2.1 and 11.6
in [DFM2]). The next big objective would be to prove the reverse implication,
meaning that if the harmonic measure on $\Gamma$ is $A_{\infty}$ with respect
to the Hausdorff measure, then $\Gamma$ is uniformly rectifiable.
Unfortunately (and surprisingly) this fails brutally when $d+\beta=n+2$ (see
[DEM]), and although we expect this to be the only exception, the methods used
to prove the converse in co-dimension 1 do not appear to be adaptable to the
higher codimension case.

The purpose of the present paper is to provide different estimates on the
harmonic functions, which we hope will ultimately furnish one side of the
desired criterion. Indeed, as established in [DM2], some weak bounds on the
Green function are equivalent to the uniform rectifiability even in lower
dimensional settings. The general idea is that instead of trying to
characterize rectifiable sets using the harmonic measure, we would do so using
the property that the Green function behaves like a distance to the boundary.
This is not a surprising idea, as the Green functions and harmonic measure are
deeply connected, and are both prominent in the analysis of the free boundary
problems, and the proof of the properties of the harmonic measure in [DFM2,
DFM5] heavily relies on a comparison between the harmonic measure and Green
functions. Yet, the results we are about to prove are not known in the
“classical” setting of domains with an $n-1$ dimensional boundary, except for
a few simple situations and certainly not in the generality of the uniformly
rectifiable sets which we attack in this paper.

We will use the Green function with a pole at infinity, which is constructed
in [DEM, Definition 6.2, Lemma 6.5] with the following properties.

######  Proposition 1.7 ([DEM]).

Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly
rectifiable set with $d<n-1$. Take $\beta>0$, $\gamma\in(-1,1)$, and construct
$L_{\beta,\gamma}$ on $\Omega:=\mathbb{R}^{n}\setminus\Gamma$ as in (1.1).
There exists a continuous function $G^{\infty}=G_{\beta,\gamma}^{\infty}$ on
$\mathbb{R}^{n}$ such that

  1. (i)

$G^{\infty}>0$ on $\Omega$,

  2. (ii)

$G^{\infty}=0$ on $\Gamma$,

  3. (iii)

there exists a positive Borel measure
$\omega_{\infty}=\omega^{\infty}_{\beta,\gamma}$ on $\Gamma$ such that

$\int_{\Omega}\nabla
G^{\infty}\cdot\nabla\varphi\,D_{\beta}^{d+1-n+\gamma}dx=\int_{\Gamma}\varphi(y)d\omega^{\infty}(y)\qquad\text{
for }\varphi\in C^{\infty}_{0}(\mathbb{R}^{n}).$

We call $G^{\infty}$ the Green function with pole at infinity, and
$G^{\infty}$ is unique up to multiplication by a scalar constant.

The Green function with pole at infinity for the Laplacian in
$\mathbb{R}^{n}_{+}=\\{(x,t)\in\mathbb{R}^{n-1}\times(0,+\infty)\\}$ is
$G^{\infty}(x,t)=t$. The one for the operator $-\operatorname{div}|t|^{d+1-n}$
in
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}=\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{n-d},\,t\neq
0\\}$ is $G^{\infty}(x,t)=|t|$. There is a third case where $G^{\infty}$ can
be computed, which is when $\Gamma$ is a $d$-Ahlfors regular set with $d<n-2$,
and we choose the specific operator $L_{n-d-2,0}$; then one can easily compute
that $G^{\infty}=D_{n-d-2}$. In [DEM] this is called the magic case. The rough
idea of what follows is that, when $\gamma=0$, a good Green function at
infinity should behave a bit like the distance to the boundary $\Gamma$, or
like $D_{\beta}$ which is its smooth substitute. When $\gamma\neq 0$, similar
homogeneity considerations lead us to expect that in the good cases,
$G^{\infty}$ will behave like $D_{\beta}^{1-\gamma}$ instead. Here is our main
result.

######  Theorem 1.8.

Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly
rectifiable set with $d<n-1$, and let $\sigma$ be an Ahlfors regular measure
on $\Gamma$ that satisfies (2.1). For $\beta>0$ and $\gamma\in(-1,1)$, let
$L_{\beta,\gamma}$ be as in (1.1) and define $G^{\infty}_{\beta,\gamma}$ as in
Proposition 1.7. Then for any $\alpha>0$, there exists $C>0$ that depends only
on $C_{\sigma}$, $C_{0}$, $\alpha$, $\beta$, $\gamma$, $d$, $n$ such that for
any ball $B:=B(x,r)$ centered on $\Gamma$, one has

(1.9)
$\int_{B}\left|\nabla\ln\left(\frac{G^{\infty}_{\beta,\gamma}}{D_{\alpha}^{1-\gamma}}\right)\right|^{2}\,D_{\alpha}^{d+2-n}\,dX\leq
C\sigma(B).$

Remarks:

  * •

Remember that $G^{\infty}_{\beta,\gamma}$ is only defined up to a constant.
Yet, the above theorem makes perfect sense, because the value of left-hand
side of (1.9) does not change when we replace $G^{\infty}_{\beta,\gamma}$ by
$KG^{\infty}_{\beta,\gamma}$, where $K$ is a positive constant.

  * •

Theorem 1.8 is just the application of Theorem 2.21 to the Green function at
infinity: we shall generalize the estimate of Theorem 1.8 to a large class of
solutions that is interesting by itself. However, we decided to highlight the
above statement, which was our true purpose in our search for
characterizations of rectifiability.

  * •

Here we decided to use the same measure $\sigma$ for the definitions of
$L_{\beta,\gamma}$ and $D_{\alpha}$, but a minor modification of the proof
would allow us to take two different Ahlfors regular measures $\sigma$ and
$\widetilde{\sigma}$ to define $L_{\beta,\gamma}$ and and $D_{\alpha}$.

  * •

As we have pointed out above, these results are not known for $d=n-1$ in the
generality of the uniformly rectifiable sets. In the half-space one can see
somewhat similar estimates in [FKP] and the bounds on the second derivatives
of the Green function for a special class of operators were, in disguise,
obtained in [HMT]. However, it is not clear how to deduce from [HMT] a co-
dimension 1 analogue of (1.9), directly or using interpolation.

In the present paper we only show that good geometric properties (the uniform
rectifiability of $\Gamma$) imply precise approximation properties of
$G^{\infty}$ by $D_{\alpha}$, and the proof will rely heavily on the
$A_{\infty}$ property of the harmonic measure and some intermediate results in
[Fen1] concerning the uniform rectifiability of $\Gamma$. We do not address
the issue of the converse in this paper. Yet there are good reasons to believe
that it may be easier to prove than for the absolute continuity of the
harmonic measure. In a parallel paper [DM2], the authors study a less precise
(weaker) approximation property of the Green function, and show that in some
cases (but where $d>n-2$) it already implies the uniform rectifiability of
$\Gamma$, while in the case of the present paper it implies another strange
property of potentials defined on $\Gamma$. But we did not manage to prove
that this strange property is impossible to obtain when $d+\alpha\neq n+2$.

The reader should be aware that even though we think of the Green function
estimate (1.9) as a possible alternative to the $A_{\infty}$ absolute
continuity of the harmonic measure for the characterization of uniform
rectifiability, we already know that in the general context of elliptic
operator in the form $L=-\operatorname{div}A\nabla$, the $A_{\infty}$ absolute
continuity (1.5) - or (1.6) - is not equivalent to (1.9). A counterexample in
$\mathbb{R}^{n}_{+}$, that can be extended to any codimensions using the
construction (4.6) in [DFM4], is given in the next lines. Denote the running
point in $\mathbb{R}^{n}_{+}$ as $(x,r)\in\mathbb{R}^{n-1}\times(0,+\infty)$,
and observe that in this case $D_{\alpha}(x,r)=c_{\alpha}r$. We set
$b(r):=1/(2+\cos(r))$ and we construct $L_{b}:=-\operatorname{div}b(r)\nabla$.
By uniqueness of the Green function with pole at infinity (Proposition 1.7),
we have

$G^{\infty}(x,r)=\int_{0}^{r}\frac{ds}{b(s)}=2r+\sin(r).$

On one hand, $r\leq G^{\infty}(x,r)\leq 3r$ for all
$(x,r)\in\mathbb{R}^{n}_{+}$, and that is enough, by using a comparison
principle with the harmonic measure, to show that the harmonic measure
associated to $L$ is equivalent to the surface measure on $\mathbb{R}^{n-1}$
(see for instance Theorem 1.17 in [Fen2] for details), hence it is
$A_{\infty}$ absolutely continuous with respect to the surface measure. On the
other hand, we have

$\begin{split}\left|\nabla\ln\Big{(}\frac{G_{\infty}}{D_{\alpha}}\Big{)}\right|=\left|\frac{\nabla
G^{\infty}}{G^{\infty}}-\frac{\nabla
D_{\alpha}}{D_{\alpha}}\right|=\left|\frac{2+\cos(r)}{2r+\sin(r)}-\frac{1}{r}\right|\geq\frac{1}{3r}\left|\cos(r)-\frac{\sin(r)}{r}\right|.\end{split}$

That is, for any ball $B(x,R)\subset\mathbb{R}^{n}$ centered on the boundary
$\mathbb{R}^{n-1}$, we have

$\begin{split}\int_{B(x,R)}\left|\nabla\ln\Big{(}\frac{G_{\infty}}{D_{\alpha}}\Big{)}\right|^{2}D_{\alpha}\,dX\geq
c_{n}R^{n-1}\int_{0}^{R/2}\left|\cos(r)-\frac{\sin(r)}{r}\right|^{2}\frac{dr}{r}\geq
c^{\prime}_{n}R^{n-1}\ln(R),\end{split}$

when $R$ is large (e.g. $R\geq 100$), and where $c_{n},c^{\prime}_{n}$ are
constants that depends only on $n$. As a consequence, (1.9) fails for the
elliptic operator $L_{b}$. This highlights the difference between Green
function and harmonic measure estimates for general elliptic operators.

Coming back to the Green function, the reader might wonder about the
comparison with the results in [DM2]. It is not easy to describe the results
in [DM2] avoiding technicalities, but roughly speaking, we only proved there
the weak statement that the set where the Green function behaves like a
distance to the boundary is a Carleson-prevalent set. This carries the
structural information saying that there are a lot of points and a lot of
scales where the desired estimate is true, but it carries no norm control.
Respectively, the methods heavily rely on compactness arguments. In the
present paper the arguments and the results are completely different, aiming
at a strong norm control in the form of (1.9) and using a new idea that the
“magical” distance function identified in [DEM], being an explicit solution to
a certain PDE, can be effectively used in the integration-by-parts arguments
(cf. [Fen1]). A good comparison is a familiar to many experts integration by
parts with the weight $t$ which disappears when the Laplacian hits $t$,
although the details in our case are necessarily considerably more involved.

In the next section, we shall give the definitions that we skipped up to now
to lighten the introduction, such as the definition of Ahlfors regular and
uniformly rectifiable sets. We also introduce the results from [Fen1] which we
will rely upon. The remainder of the article will be devoted to the proof of
our main result.

We shall use the notation $A(x)\lesssim B(x)$ when $A(x)\leq CB(x)$ and $C$ is
a constant whose dependence into the various parameters will be either
recalled or obvious from context. We also write $A(x)\approx B(x)$ when
$A(x)\lesssim B(x)$ and $A(x)\gtrsim B(x)$.

##  2\. Definitions and anterior results

For the rest of the article, we take $\Gamma\subset\mathbb{R}^{n}$ and
$\Omega:=\mathbb{R}^{n}\setminus\Gamma$. We assume that $\Gamma$ is a
$d$-Ahlfors regular set with $d<n-1$, that is $\Gamma$ is closed and there
exists a measure $\sigma$ supported on $\Gamma$ and a constant $C_{\sigma}\geq
1$ such that

(2.1) $C_{\sigma}^{-1}r^{d}\leq\sigma(B(x,r))\leq C_{\sigma}r^{d}\qquad\text{
for }x\in\Gamma,\ r>0.$

It is known that if the above property (2.1) is true for some measure
$\sigma$, then it is also true when $\sigma$ is replaced by
$\mathcal{H}^{d}|_{\Gamma}$ \- the $d$-dimensional Hausdorff measure
restricted to $\Gamma$.

The Ahlfors regularity of $\Gamma$ and the low dimension $d<n-1$ are
sufficient conditions to obtain the aforementioned equivalence (1.3). Indeed,
Lemma 5.1 in [DFM3] gives us that

(2.2) $C^{-1}\,\mathrm{dist}(X,\Gamma)\leq D_{\beta}\leq
C\,\mathrm{dist}(X,\Gamma)\qquad\text{ for }X\in\Omega,$

where the constant $C>0$ above depends only on $\beta>0$, $C_{\sigma}$ and
$n-d>1$.

Moreover, Lemma 11.6 in [DFM2] entails the existence of a constant $C$ that
depends only on $C_{\sigma}$ and $n-d>1$ such that for any $x\in\Gamma$ and
$r>0$, we can find a point $A_{x,r}$ such that

(2.3) $C^{-1}r\leq\,\mathrm{dist}(A_{x,r},\Gamma)\leq|A_{x,r}-x|\leq Cr.$

In other words, when $\Gamma$ is $d$-Ahlfors regular with $d<n-1$, its
complement automatically satisfies the interior corkscrew condition.

We shall also assume that $\Gamma$ is uniformly rectifiable. Equivalent
definitions of uniform rectifiability were given in [DS1, DS2], and the reader
may use their preferred one, but since we will only use the uniform
rectifiability of $\Gamma$ via results from [Fen1] that rely on the
summability properties of Tolsa’s $\alpha$-numbers, we will use these
properties as a definition of uniform rectifiability. We need some notation
first.

We denote by $\Xi$ the set of affine $d$-dimensional planes in
$\mathbb{R}^{n}$. Each plane $P\in\Xi$ is associated with a measure $\mu_{P}$,
which is the restriction to $P$ of the $d$-dimensional Hausdorff measure (i.e.
$\mu_{P}$ is the Lebesgue measure on the plane). A flat measure is a measure
$\mu$ that can be written $\mu=c\mu_{P}$ where $c$ is a positive constant and
$P\in\Xi$. The set of flat measure is called $\mathcal{F}$.

We need the following variant of Wasserstein distances to quantify the
difference between two measures, and then measure how far $\sigma$ is from
flat measures.

######  Definition 2.4.

For $x\in\mathbb{R}^{n}$ and $r>0$, denote by $Lip(x,r)$ the set of
$1$-Lipschitz functions $f$ supported in $\overline{B(x,r)}$, that is the set
of functions $f:\mathbb{R}^{n}\to\mathbb{R}$ such that $f(y)=0$ for
$y\in\mathbb{R}^{n}\setminus B(x,r)$ and $|f(y)-f(z)|\leq|y-z|$ for
$y,z\in\mathbb{R}^{n}$. The normalized Wasserstein distance in B(x,r) between
two measures $\sigma$ and $\mu$ is

(2.5) $\,\mathrm{dist}_{x,r}(\mu,\sigma)=r^{-d-1}\sup_{f\in
Lip(x,r)}\Big{|}\int fd\sigma-\int fd\mu\Big{|}.$

The distance to flat measures is then defined by

(2.6)
$\alpha_{\sigma}(x,r)=\inf_{\mu\in{\mathcal{F}}}\,\mathrm{dist}_{x,r}(\mu,\sigma).$

One can easily check that when (2.1) holds, the quantity $\alpha_{\sigma}$ is
uniformly bounded, i.e. there exists a constant $C$ that depends only on $d$,
$n$, and $C_{\sigma}$ such that $\alpha_{\sigma}(x,r)\leq C$ for $x\in\Gamma$
and $r>0$.

Let $\Gamma$ be a $d$-Ahlfors regular set, and $\sigma$ a measure that
satisfies (2.1). Tolsa’s characterization of uniform rectifiability, Theorem
1.2 in [Tol], is as follows111Tolsa’s characterization of rectifiability in
[Tol] is given with dyadic cubes but one can easily check that our bound (2.7)
is equivalent to Tolsa’s one.: $\Gamma$ is uniformly rectifiable if and only
if there exists a constant $C_{0}>0$ such that

(2.7) $\int_{0}^{r}\int_{\Gamma\cap
B(x,r)}|\alpha_{\sigma}(y,s)|^{2}\,d\sigma(y)\,\frac{ds}{s}\leq
C_{0}\sigma(B(x,r))\qquad\text{ for }x\in\Gamma\text{ and }r>0.$

Here we will only use the fact that (2.7) holds when $\Gamma$ is uniformly
rectifiable. That is, we will only use (2.7) and do not need to know other
properties of uniformly rectifiable sets. The property (2.7) will allow us to
obtain additional estimates on the smooth distance $D_{\beta}$. The
presentation of those bounds will be easier after the following definition.

######  Definition 2.8.

Let the function $f$ be defined on $\Omega$. We say that $f$ satisfies the
Carleson measure condition when $f\in L^{\infty}(\Omega)$ and
$|f(X)|^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}dX$ is a Carleson measure, that is,

(2.9) $\int_{B(x,r)}|f(X)|^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}dX\leq
C\sigma(B(x,r))$

for $x\in\Gamma$ and $r>0$, with a constant $C$ that does not depend on $x$ or
$r$.

Thus this is actually a quadratic Carleson condition. For short, we shall
write $f\in CM$, or $f\in CM(C)$ when we want to refer to the constant in
(2.9).

Due to (2.2), we can replace $\,\mathrm{dist}(X,\Gamma)^{d-n}$ with
$D_{\beta}^{d-n}(X)$, and even choose $\beta$ to fit our purposes; we shall
often do this without additional explanations. We shall rely strongly on Lemma
1.24 in [Fen1], which says the following.

######  Lemma 2.10.

Let $\Gamma$ be uniformly rectifiable, so that (2.1) and (2.7) hold. Let
$\beta>0$. Then there exist a scalar function $b$ and a vector function
$\mathcal{V}$, both defined on $\Omega$, such that

(2.11) $\int_{\Gamma}|X-y|^{-n}(X-y)d\sigma(y)=(b\nabla
D_{\beta}+\mathcal{V})D_{\beta}^{d+1-n}\qquad\text{ for }X\in\Omega$

and a constant $C_{1}$ that depends only on $C_{\sigma}$, $C_{0}$, $\beta$,
$n$, and $d$, such that

(2.12) $C_{1}^{-1}\leq b\leq C_{1},$ (2.13) $D_{\beta}\nabla b\in CM(C_{1}),$
(2.14) $|\mathcal{V}|\leq C_{1},$

and

(2.15) $\mathcal{V}\in CM(C_{1}).$

Observe that the left-hand side of (2.11) is divergence free. So if we use
(2.11) and we write the divergence free condition in weak terms, we obtain

(2.16) $\int_{\Omega}(b\nabla
D_{\beta}+\mathcal{V})\cdot\nabla\varphi\,D_{\beta}^{d+1-n}\,dX=0\qquad\text{
for }\varphi\in C^{\infty}_{0}(\Omega).$

We also need the following, which is Lemma 1.26 in [Fen1].

######  Lemma 2.17.

Let $\Gamma$ be uniformly rectifiable, i.e., assume that (2.1) and (2.7) hold.
Let $\alpha,\beta>0$. Then $D_{\alpha}\nabla[D_{\beta}/D_{\alpha}]$ satisfies
the Carleson measure condition with a constant that depends only on
$C_{\sigma}$, $C_{0}$, $\alpha$, $\beta$, $n$, and $d$.

We are now finished with the geometric background and turn to the elliptic
theory. Pick $\beta>0$ and $\gamma\in(-1,1)$. We whall use the operator
$L_{\beta,\gamma}$ constructed in (1.1); hence $L_{\beta,\gamma}$ enters the
scope of the theory developed in [DFM5], and in particular there is an
elliptic measure $\omega^{X}$ which we shall describe now.

We first need a Hilbert space $W_{\gamma}$, which is the same as in [DFM5],
Definition 3.1, but is more easily defined as

(2.18) $W_{\gamma}=\big{\\{}u\in
L^{1}_{loc}(\mathbb{R}^{n}),\|u\|_{\gamma}:=\int_{\Omega}|\nabla
u(X)|^{2}\,\mathrm{dist}(X,\Gamma)^{d+1+\gamma-n}dX<+\infty\big{\\}};$

the equivalence between the two definition is proved as in [DFM2, Lemma 3.3
and Lemma 5.21].

Each $f\in W_{\gamma}$ has a trace on $\Gamma$, which lies in a corresponding
Sobolev space $H_{\gamma}$ (which is equal to $H^{1/2}(\Gamma)$ when
$\gamma=0$); then we denote by $W_{\gamma,0}$ the set of functions in
$W_{\gamma}$ with zero trace; $W_{\gamma,0}$ is also the completion of
$C^{\infty}_{0}(\Omega)$ under the norm $\|.\|_{\gamma}$. And for any open set
$E\subset\mathbb{R}^{n}$, we write that $u\in W_{\gamma}(E)$ [respectively,
$u\in W_{\gamma,0}(E)$] if $u\varphi\in W_{\gamma}$ [respectively,
$u\varphi\in W_{\gamma,0}$] for any $\varphi\in C^{\infty}_{0}(E)$.

Then there is a notion of weak solution for $L_{\beta,\gamma}=0$, which the
reader may also find in [DFM5], such that in particular

(2.19) $\int_{\Omega}(\nabla
u\cdot\nabla\varphi)\,D_{\beta}^{d+1+\gamma-n}=0\qquad\text{ for }\varphi\in
C^{\infty}_{0}(\Omega).$

when $u$ is a weak solution for $L_{\beta,\gamma}=0$ (in $\Omega$). Here and
below, we remove the variable $X$ and the integration symbol $dX$ from the
notation when they are not entirely needed; unless otherwise specified, all
our integrals on $\Omega$ will be against the Lebesgue measure $dX$.

With all this notation, the main properties of our elliptic measures
$\omega^{X}$ are as follows.

######  Definition 2.20.

For each $X\in\Omega$, we can define a unique probability measure
$\omega^{X}:=\omega^{X}_{\beta,\gamma}$ on $\Gamma$ with the following
properties. For any $g\in C_{0}(\Gamma)$ (i.e., continuous function on
$\Gamma$ and compactly supported), the function $u_{g}$ defined as

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

is a weak solution to $L_{\beta,\gamma}$ and, if in addition $g$ lies in the
Sobolev space $H_{\gamma}$, then $u_{g}\in W_{\gamma}$ and the trace of
$u_{g}$ is equal to $g$.

The space $H_{\gamma}\cap C_{0}(\Gamma)$ is dense in $C_{0}(\Gamma)$ (with the
sup norm), so the last condition is our way of solving a Dirichlet problem. We
are now ready to state the general version of our main theorem.

######  Theorem 2.21.

Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly
rectifiable set with $d<n-1$, and let $\sigma$ be an Ahlfors regular measure
on $\Gamma$ that satisfies (2.1). For $\beta>0$ and $\gamma\in(-1,1)$, define
$L_{\beta,\gamma}$ as in (1.1). Then for any $\alpha>0$, there exists $C>0$
that depends only on $C_{\sigma}$, $C_{0}$, $\alpha$, $\beta$, $\gamma$, $d$,
$n$, such that for any ball $B:=B(x,r)$ centered on $\Gamma$ and any non-
negative non identically zero weak solution $u$ of $L_{\beta,\gamma}u=0$ in
$\Omega\cap 3B$ which lies in $W_{\gamma,0}(3B)$, one has

(2.22)
$\int_{B}\left|\nabla\ln\left(\frac{u}{D_{\alpha}^{1-\gamma}}\right)\right|^{2}\,D_{\alpha}^{d+2-n}\leq
C\sigma(B).$

The proof of the Theorem will use the uniform rectifiability of $\Gamma$ via
Theorem 1.4 and Lemma 2.10. The Lemma will be used to estimate the left-hand
side of (2.22) via simple integration techniques, except for one more
complicated term in the form

$\int\left|\nabla\ln\left(\frac{u}{D_{\alpha}^{1-\gamma}}\right)\right||\nabla\phi_{B,\epsilon}|\,D_{\alpha}^{d+1-n},$

where $\phi_{B,\epsilon}$ is a well chosen cut off function, which is related
to the integral of the logarithm of the Poisson kernel and that will be
estimated using Theorem 1.4, that is, the $A_{\infty}$ absolute continuity of
the harmonic measure.

##  3\. Proof of Theorem 2.21

Let us recall

######  Lemma 3.1.

Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly
rectifiable set with $d<n-2$, and let $\sigma$ be an Ahlfors regular measure
on $\Gamma$ that satisfies (2.1). Define $L_{\beta,\gamma}$ as in (1.1). Then
there exist $C>0$ and $\theta\in(0,1]$, that depend only on $C_{\sigma}$,
$C_{0}$, $\beta$, $\gamma$, $n$ and $d$, such that for each choice of
$x\in\Gamma$, $r>0$, any Borel set $E\subset B(x,r)\cap\Gamma$, and any
corkscrew point $X=A_{x,r}$ as in (2.3), one has

(3.2)
$\frac{\omega_{\beta,\gamma}^{X}(E)}{\omega_{\beta,\gamma}^{X}(B(x,r))}\leq
C\left(\frac{\sigma(E)}{\sigma(B(x,r))}\right)^{\theta}$

and

(3.3) $\frac{\sigma(E)}{\sigma(B(x,r))}\leq
C\left(\frac{\omega_{\beta,\gamma}^{X}(E)}{\omega_{\beta,\gamma}^{X}(B(x,r))}\right)^{\theta}.$

Here and below, we assume implicitly that the constant $C$ in (2.3) (the
definition of corkscrew points) is chosen to depend on $C_{\sigma}$, $n$ and
$d$ only. If we allow a larger $C$ in (2.3), the constants in (3.2) and (3.3)
depend on $C$ as well.

Proof. The conditions (3.2) and (3.3) are another characterizations of the
fact that $\omega^{X}\in A_{\infty}(\sigma)$, which is true by Theorem 1.4.
The fact that (3.2)–(3.3) is implied by (1.6) can be found in [Ken, Theorem
1.4.13] and its proof in [CF, Lemma 5]. $\Box$

For our next result, we want to establish that the logarithm of the Poisson
kernel, that is, $\ln(\frac{d\omega^{X}}{d\sigma})$, is integrable. We want a
quantitative version, and moreover, we shall state this in a form that is more
directly applicable when we need it (in the proof of Proposition 3.8).

######  Lemma 3.4.

Let $\Gamma$, $\sigma$, and $L_{\beta,\gamma}$ as in Lemma 3.1.

Take $B:=B(x,r)$, a ball centered on $\Gamma$, and $X=A_{x,r}$, a corkscrew
point as in (2.3). If $\\{Q_{i}\\}_{i\in\mathcal{I}}$ is a finitely
overlapping collection of Borel subsets of $B\cap\Gamma$, then

(3.5)
$\sum_{i\in\mathcal{I}}\left|\ln\left(\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\right)\right|\sigma(Q_{i})\leq
C\sigma(B),$

where $C$ depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$, $d$,
and the maximal number of overlaps in the collection
$\\{Q_{i}\\}_{i\in\mathcal{I}}$.

Proof. We introduce for $k\in\mathbb{Z}$,

$\mathcal{I}_{k}:=\left\\{i\in\mathcal{I},\,2^{k}\leq\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\leq
2^{k+1}\right\\}.$

Then we define

$E_{k}:=\bigcup_{i\in\mathcal{I}_{k}}Q_{i}.$

Since $\\{Q_{i}\\}$ is finitely overlapping, we have
$\sum_{i\in\mathcal{I}_{k}}\sigma(Q_{i})\leq C\sigma(E_{k})$, and we can thus
write

(3.6)
$\sum_{i\in\mathcal{I}}\ln\left(\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\right)\sigma(Q_{i})\lesssim\sum_{k\in\mathbb{Z}}(|k|+1)\sigma(E_{k}).$

Yet, since $\\{Q_{i}\\}_{i\in\mathcal{I}_{k}}$ is a finitely overlapping
covering of $E_{k}$,

$\omega_{\beta,\gamma}^{X}(E_{k})\approx\sum_{i\in\mathcal{I}_{k}}\omega_{\beta,\gamma}^{X}(Q_{i})\approx
2^{k}\frac{\omega_{\beta,\gamma}^{X}(B)}{\sigma(B)}\sum_{i\in\mathcal{I}_{k}}\sigma(Q_{i})\approx
2^{k}\frac{\omega_{\beta,\gamma}^{X}(B)}{\sigma(B)}\sigma(E_{k}),$

which means that

(3.7)
$\frac{\omega_{\beta,\gamma}^{X}(E_{k})}{\sigma(E_{k})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\approx
2^{k}.$

The use of (3.7) in (3.2) leads to

$\sigma(E_{k})\lesssim 2^{-k/(1-\theta)}\sigma(B)$

while the use of (3.7) in (3.3) gives

$\sigma(E_{k})\lesssim 2^{k\theta/(1-\theta)}\sigma(B).$

We use the first of the last two estimate to bound $\sigma(E_{k})$ when $k\geq
0$ and the second one to bound $\sigma(E_{k})$ when $k$ is negative. Combined
with (3.6), we deduce

$\begin{split}\sum_{i\in\mathcal{I}}\ln\left(\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\right)\sigma(Q_{i})&\lesssim\sum_{k\geq
0}(k+1)2^{-\frac{k}{1-\theta}}\sigma(B)+\sum_{k<0}(1-k)2^{-\frac{k\theta}{1-\theta}}\sigma(B)\\\
&\lesssim\sigma(B).\end{split}$

The lemma follows. $\Box$

######  Proposition 3.8.

Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly
rectifiable set with $d<n-2$, and let $\sigma$ be an Ahlfors regular measure
that satisfies (2.1). Take $\beta>0$ and $\gamma\in(-1,1)$, define
$L_{\beta,\gamma}$ as in (1.1). Then for any ball $B$ centered on $\Gamma$ and
any non-negative non identically zero weak solution $u$ to
$L_{\beta,\gamma}u=0$ in $3B\cap\Omega$, with $u\in W_{\gamma,0(3B)}$,

(3.9)
$\int_{B}\left|\nabla\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\right)\right|^{2}\,D_{\beta}^{d+2-n}\leq
C\sigma(B).$

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

Observe that Proposition 3.8 is the special case of Theorem 2.21 where we take
$\alpha=\beta$. We shall now check that conversely, Theorem 2.21 follows from
Proposition 3.8 and Lemma 2.17.

Proof of Theorem 2.21 from Proposition 3.8 with the help of Lemma 2.17. Let
$\alpha>0$. Notice that

$\begin{split}\left|\nabla\ln\left(\frac{u}{D_{\alpha}^{1-\gamma}}\right)\right|&=\left|\nabla\left[\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\right)+(1-\gamma)\ln\left(\frac{D_{\beta}}{D_{\alpha}}\right)\right]\right|\\\
&\leq\left|\nabla\left[\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\right)\right]\right|+(1-\gamma)\frac{D_{\alpha}}{D_{\beta}}\left|\nabla\left[\dfrac{D_{\beta}}{D_{\alpha}}\right]\right|.\end{split}$

So Proposition 3.8, Lemma 2.17, and (2.2) easily implies (3.9). $\Box$

Proof of Proposition 3.8. The proof of the lemma will be divided into 4 steps.
The core step is step 4, where we use the properties of the solutions. Step 3
treats in advance the most complicated term that we met in Step 4. In this
step, we will compare a solution $u$ to a Green function, then to the harmonic
measure, and finally we use Lemma 3.4. In Step 2, we construct the finitely
overlapping covering $\\{Q_{i}\\}_{i\in\mathcal{I}}$ that will be needed in
order to invoke Lemma 3.4. At last, Step 1 introduces the cut-off function
$\phi_{B,\epsilon}$ used to bound the left-hand side of (2.22) and shows that
$D_{\beta}\nabla\phi_{B,\epsilon}$ satisfies the Carleson measure condition.

Step 1: Introduction of the cut-off function $\phi_{B,\epsilon}$. Let
$B=B(x,r)$ be a ball in $\mathbb{R}^{n}$ centered on the boundary $\Gamma$ and
$\epsilon>0$ small. The proof of Theorem 2.21 is a local one and thus as usual
will involve cut off functions. Take $\psi\in C^{\infty}_{0}(\mathbb{R})$ be
such that $\psi\equiv 1$ on $[-1,1]$, $\psi$ is compactly supported in
$(-2,2)$, $0\leq\psi\leq 1$, and $|\psi^{\prime}|\leq 2$. We define the
function $\phi_{B,\epsilon}$ on $\Omega$ by

(3.10)
$\phi_{B,\epsilon}(X):=\psi\left(\frac{\,\mathrm{dist}(X,B)}{10\,\mathrm{dist}(X,\Gamma)}\right)\psi\left(\frac{2\,\mathrm{dist}(X,B)}{r}\right)\psi\left(\frac{\epsilon}{\,\mathrm{dist}(X,\Gamma)}\right).$

The support of $\phi_{B,\epsilon}$ is thus contained in

(3.11) $E_{0}=\big{\\{}X\in 2B,\,\,\mathrm{dist}(X,B)\leq
20\,\mathrm{dist}(X,\Gamma)\text{ and
}\,\mathrm{dist}(X,\Gamma)\geq\epsilon/2\big{\\}}.$

The gradient of $\phi_{B,\epsilon}$ comes from $3$ regions; the first one
(associated to the first cut-off) is

(3.12) $\displaystyle\big{\\{}X\in E_{0}\,,$ $\displaystyle
10\,\mathrm{dist}(X,\Gamma)\leq\,\mathrm{dist}(X,B)\leq
20\,\mathrm{dist}(X,\Gamma)\big{\\}}$ $\displaystyle\subset
E_{1}:=\big{\\{}X\in
2B,\,10\,\mathrm{dist}(X,\Gamma)\leq\,\mathrm{dist}(X,B)\leq
20\,\mathrm{dist}(X,\Gamma)\big{\\}},$

the second one is

(3.13) $\big{\\{}X\in E_{0}\,,\,r\leq 2\,\mathrm{dist}(X,B)\leq
2r\big{\\}}\subset E_{2}:=\big{\\{}X\in
2B,\,\,r/40\leq\,\mathrm{dist}(X,\Gamma)\leq 2r\big{\\}}$

because $\,\mathrm{dist}(X,\Gamma)\leq 2r$ when $X\in 2B$ and similarly
$\,\mathrm{dist}(X,\Gamma)\geq\frac{1}{20}\,\mathrm{dist}(X,B)\geq\frac{r}{20}$,
and the third region is contained in

(3.14) $E_{3}:=\big{\\{}X\in
2B,\,\epsilon/2\leq\,\mathrm{dist}(X,\Gamma)\leq\epsilon\big{\\}}.$

Then it is easy to check that the gradient of $\phi_{B,\epsilon}$ satisfies

(3.15)
$|\nabla\phi_{B,\epsilon}|\leq\frac{100}{\,\mathrm{dist}(X,\Gamma)}\left[{\mathds{1}}_{E_{1}}+{\mathds{1}}_{E_{2}}+{\mathds{1}}_{E_{3}}\right].$

We claim that ${\mathds{1}}_{E_{1}}$, ${\mathds{1}}_{E_{2}}$, and
${\mathds{1}}_{E_{3}}$ all satisfy the Carleson measure condition. That is, we
have

(3.16)
$\int_{B(y,s)}{\mathds{1}}_{E_{j}}^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}\leq
Cs^{d}$

for $1\leq j\leq 3$, $y\in\Gamma$, and $0<s<+\infty$. Of course the square can
be removed, and it will follows from (3.15), (3.16) (as soon as we prove it),
and (2.2) that for any $\beta>0$, $y\in\Gamma$ and $s\in(0,+\infty)$, we have

(3.17)
$\int_{B(y,s)}|D_{\beta}\nabla\phi_{B,\epsilon}|D_{\beta}^{d-n}+\int_{B(y,s)}|D_{\beta}\nabla\phi_{B,\epsilon}|^{2}D_{\beta}^{d-n}\leq
Cs^{d},$

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

We will prove the claim (3.16) in the end of the next Step.

Step 2: Construction of the collection $\\{Q_{i}\\}_{i\in\mathcal{I}}$. We
would like to have a collection of boundary cubes $Q_{i}\subset\Gamma$ and the
corresponding Whitney cubes $R_{i}\subset\Omega$ satisfying rather usual nice
properties: bounded overlap, control of the size and and the distance to the
boundary for $R_{i}$’s, reasonable placement of the corkscrew points. There
are plenty of papers that present various versions of this construction, but
here, in fact, we need something simpler than usual as we do not need to
control the cones or the related Harnack chains. For these reasons, let us
simply carry out the construction by hands.

We need a family of Whitney cubes $\mathcal{W}$, as constructed in [Ste]. We
record the basic properties of $\mathcal{W}$ that we shall need. The
collection $\mathcal{W}$ is the family of maximal dyadic cubes
$R\subset\Omega$ such that $20R\subset\Omega$, different cubes $R_{i}$ and
$R_{j}$, $i\neq j$, in $\mathcal{W}$ have disjoint interiors (by maximality),
their union covers $\Omega=\mathbb{R}^{n}\setminus\Gamma$, and for
$Q\in{\mathcal{W}}$

(3.18) $20Q\subset\Omega\text{ but }60R\cap\Gamma\neq\emptyset.$

Moreover, given $R\in\mathcal{W}$, the number of cubes
$\widetilde{R}\in\mathcal{W}$ such that

(3.19) $\,\mathrm{dist}(R,\Gamma)\approx\,\mathrm{dist}(\widetilde{R},\Gamma)\
\text{ and
}\,\,\mathrm{dist}(R,\widetilde{R})\lesssim\,\mathrm{dist}(R,\Gamma)$

is uniformly bounded by a constant that depends only on the dimensions and the
constants involved in (3.19). Also, the cubes $\widetilde{R}\in{\mathcal{W}}$
such that $3R\cap 3\widetilde{R}$, all satisfy (3.19).

We will use the cubes of ${\mathcal{W}}$ to cover the region $E_{1}\cup
E_{2}\cup E_{3}$. Consider the subset ${\mathcal{W}}_{0}\subset{\mathcal{W}}$
of cubes $R\in{\mathcal{W}}$ that meet $E_{1}\cup E_{2}\cup E_{3}$, and label
these cubes with a set $I$, so that ${\mathcal{W}}_{0}=\\{R_{i}\,;\,i\in
I\\}$. We also want to associate a boundary ball $Q_{i}=\Gamma\cap
B(x_{i},r_{i})$ to each $R_{i}$, $i\in I$. This is a classical thing to do,
but we shall do it by hand to get a slightly better control. We shall choose
the $Q_{i}$ so that

(3.20) $Q_{i}\subset 3B\cap\Gamma$

(we can do this and this will be helpful because we shall consider solutions
in $3B$),

(3.21) $\\{Q_{i}\\}_{i\in\mathcal{I}}$ has bounded overlap,

where the bound for the overlap depends only on $C_{\sigma}$, $n$, and $d$,
and

(3.22)
$r_{i}\approx\,\mathrm{dist}(R_{i},\Gamma)\approx\,\mathrm{dist}(R_{i},x_{i}),$

also with constants depend only on $C_{\sigma}$, $n$, and $d$. As a
consequence, we will be able to use any point $X_{i}\in R_{i}$ as a corkscrew
point for the pair $(x_{i},Cr_{i})$ (see (2.3)).

So let us construct the $Q_{i}$. First write $I=I_{1}\cup I_{2}\cup I_{3}$, a
disjoint union where $I_{2}=\big{\\{}i\in I\,;\,R_{i}\cap
E_{2}\neq\emptyset\big{\\}}$, then $I_{3}=\big{\\{}i\in I\setminus
I_{2}\,;\,R_{i}\cap E_{3}\neq\emptyset\big{\\}}$, and finally
$I_{1}=\big{\\{}i\in I\setminus(I_{2}\cup I_{3})\,;\,R_{i}\cap
E_{1}\neq\emptyset\big{\\}}$.

We start with $i\in I_{2}$. This is the simplest case because $E_{2}\subset
2B=B(x,2r)$, and $\,\mathrm{dist}(X,\Gamma)\geq r/40$ on $E_{2}$. By
definition of ${\mathcal{W}}$, $I_{2}$ has at most $C$ elements (see near
(3.19)), and for each $i\in I_{2}$ we take $x_{i}=x$ and $r_{i}=3r$ (and hence
$Q_{i}=\Gamma\cap 3B$); the constraints (3.20), (3.21), (3.22) for $I_{2}$ are
easily checked.

Next consider $i\in I_{3}$; thus $R_{i}$ meets $E_{3}$, where
$\varepsilon/2\leq\,\mathrm{dist}(X,\Gamma)\leq\varepsilon$. Pick
$x_{i}\in\Gamma$ such that $\,\mathrm{dist}(x_{i},E_{3}\cap
R_{i})=\,\mathrm{dist}(\Gamma,E_{3}\cap R_{i})$, and take $r_{i}=\varepsilon$.
Then $Q_{i}\subset 3B$ if $\varepsilon$ is small enough, because $E_{3}\subset
2B$ and
$\mathrm{diam}(R_{i})\approx\,\mathrm{dist}(R_{i},\Gamma)\approx\,\mathrm{dist}(E_{3}\cap
R_{i},\Gamma)\approx\varepsilon$, (3.22) holds for the same reasons, and the
$Q_{i}$, $i\in I_{3}$ have bounded overlap by the property near (3.19).

Finally, for $i\in I_{1}$, take $X_{i}\in R_{i}\cap E_{1}$ and
$x_{i}\in\Gamma$ such that $|X_{i}-x_{i}|=\,\mathrm{dist}(R_{i}\cap
E_{1},\Gamma)$, and set $Q_{i}=\Gamma\cap B(x_{i},r_{i})$, with
$r_{i}=|X_{i}-x_{i}|$. Since $X_{i}\in E_{1}$, the definition (3.12) yields

(3.23) $\,\mathrm{dist}(X_{i},B)\leq
20\,\mathrm{dist}(X,\Gamma)=20|X_{i}-x_{i}|\leq 2\,\mathrm{dist}(X_{i},B)\leq
2r,$

and in particular $x_{i}\in\frac{5}{2}B$. Also, $X_{i}\notin E_{2}$ (because
$i\notin I_{2}$), so $r_{i}=\,\mathrm{dist}(R_{i}\cap
E_{1},\Gamma)\leq\,\mathrm{dist}(X_{i},\Gamma)\leq r/40$, and hence
$Q_{i}\subset 3B$, as needed for (3.20). The Whitney property (3.22) holds
essentially by definition (and because $R_{i}$ is a Whitney cube), so we just
need to bound the overlap of the $Q_{i}$.

Assume that $Q_{j}\cap Q_{i}\neq\emptyset$ for two indices
$i,j\in\mathcal{I}_{1}$, and let $X_{i},X_{j},r_{i}$, and $r_{j}$ be as above.
Let us first check that $r_{i}\approx r_{j}$. By (3.23)

$\,\mathrm{dist}(x_{i},B)\leq|X_{i}-x_{i}|+\,\mathrm{dist}(X_{i},B)\leq
21|X_{i}-x_{i}|=21r_{i}$

and

$\,\mathrm{dist}(x_{i},B)\geq\,\mathrm{dist}(X_{i},B)-|X_{i}-x_{i}|\leq
9|X_{i}-x_{i}|=9r_{i},$

which can be summarized as

(3.24) $9r_{i}\leq\,\mathrm{dist}(x_{i},B)\leq 21r_{i}.$

Similarly,

(3.25) $9r_{j}\leq\,\mathrm{dist}(x_{j},B)\leq 21r_{j}.$

We can assume without loss of generality that
$\,\mathrm{dist}(x_{j},B)\leq\,\mathrm{dist}(x_{i},B)$, which, together with
(3.24)-(3.25), leads to $9r_{j}\leq 21r_{i}$. Moreover, since $Q_{i}\cap
Q_{j}\neq\emptyset$,

$r_{i}+r_{j}\geq|x_{j}-x_{i}|\geq\,\mathrm{dist}(x_{i},B)-\,\mathrm{dist}(x_{j},B)\geq
9r_{i}-21r_{j},$

hence $22r_{j}\geq 8r_{i}$. Recall that $9r_{j}\leq 21r_{i}$, so the two radii
are equivalent. The bounded overlap property follows, because
$|x_{j}-x_{i}|\leq r_{i}+r_{j}$ when $Q_{i}\cap Q_{j}\neq\emptyset$.

This completes our construction of Whitney cubes $R_{i}$ and associated
surface balls $Q_{i}$, with the properties (3.20)–(3.22) (notice that the
overlap constant for the $Q_{i}$, $i\in I$, is less than the sum of the
overlap constants for the $I_{j}$).

With this at hand, let us prove (3.16). This is now quite easy as for any
$y\in\Gamma$ and $0<s<+\infty$ we have

(3.26) $\int_{B(y,s)}{\mathds{1}}_{E_{1}\cup E_{2}\cup
E_{3}}^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}\\\ \leq\sum_{R_{i}:\,R_{i}\cap
B(y,s)\neq\emptyset}\int_{R_{i}}\,\mathrm{dist}(X,\Gamma)^{d-n}\lesssim\sum_{R_{i}:\,R_{i}\cap
B(y,s)\neq\emptyset}r_{i}^{d}\lesssim s^{d},$

using the bounded overlap property (3.21).

Step 3: Estimates for the integral $S$.  Take $u\in W_{\gamma,0}(3B)$, a weak
solution to $L_{\beta,\gamma}u=0$ in $3B$. Let $X_{0}\in 2B$ be a corkscrew
point for the boundary ball $2B\cap\Gamma$, and also choose another corkscrew
point $X_{1}\in\Omega\cap 8B\setminus 4B$, so that
$\,\mathrm{dist}(X_{0},\Gamma)\geq cr$ and $\,\mathrm{dist}(X_{1},\Gamma)\geq
cr$ for a small constant $c$ that depends only on $C_{\sigma}$, $d$, and $n$.
In this Step 3, we prove that

(3.27) $S:=\int_{E_{1}\cup E_{2}\cup
E_{3}}\left|\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\right)\right|D_{\beta}^{d-n}\leq
Cr^{d},$

where $C$ depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$ and
$d$.

We shall use the comparison principle to compare $u$ to the Green function,
and then estimate the Green function in terms of harmonic measure. We define
the Green function $g$ on $\Omega\times\Omega$ as in [DFM5, Section 11]. The
precise definition is not relevant for the present proof, and the properties
that we care about are the fact that $X\to g(X,X_{1})$ lies in
$W_{\gamma,0}(3B)$ and is a solution to $L_{\beta,\gamma}u=0$ in $3B$, and
that $g(X,Y)=g(Y,X)$ (which is true because the operator $L_{\beta,\gamma}$ is
selfadjoint). Theorem 15.64 in [DFM5] (the comparison theorem) yields that

(3.28)
$\frac{u(X)}{u(X_{0})}\approx\frac{g(X,X_{1})}{g(X_{0},X_{1})}\qquad\text{ for
}X\in 2B,$

with constants that depend only on $C_{\sigma}$, $C_{0}$, $n$, $d$, $\beta$
and $\gamma$. Actually, Theorem 15.64 in [DFM5] requires the solutions (in our
case $u$ and $g(.,X_{1})$) to be solutions in a larger ball $2KB\cap\Omega$,
and not in only $3B\cap\Omega$. This condition is only needed because [DFM5]
also allows sets $\Gamma$ of codimension $1$ or less, and then we need to
ensure that we can connect every component of $2B\cap\Omega$ by Harnack chains
that stays in $2KB$. Here $\Gamma$ is of dimension $d<n-1$, so $2B\cap\Omega$
is very well connected in the first place (see Lemma 2.1 in [DFM2]) , and
assuming that $u$ and $g(.,X_{1})$ are solutions in $3B$ is enough. Using the
fact that $g(X,Y)$ is symmetric, we deduce that

(3.29)
$\frac{u(X)}{u(X_{0})}\approx\frac{g(X_{1},X)}{g(X_{1},X_{0})}\qquad\text{ for
}X\in 2B.$

Recall that $X_{0}$ is a corkscrew point associated to $2B$ and that if $X\in
R_{i}$, then $\,\mathrm{dist}(X,\Gamma)\geq C^{-1}r_{i}$ and $|X-x_{i}|\leq
Cr_{i}$ by (3.22), so $X$ can be used as a corkscrew point for the boundary
ball $Q_{i}$. As a consequence, Lemma 15.28 in [DFM5], (where here
$m(B\cap\Omega)$ is the mass of $B$ for the measure
$\,\mathrm{dist}(X,\Gamma)^{d+1+\gamma-n}dX$ on $\Omega$, with
$\gamma\in(-1,1)$, so $m(B\cap\Omega)\approx r^{d+1+\gamma}$; see the
discussion in Section 3.2 of [DFM5]), and the doubling property of harmonic
measure (Lemma 15.43 of [DFM5]) give that

$g(X_{1},X_{0})\approx\,\mathrm{dist}(X_{0},\Gamma)^{1-d-\gamma}\omega^{X_{1}}(3B)$

and for each $i\in I$

$g(X_{1},X)\approx\,\mathrm{dist}(X,\Gamma)^{1-d-\gamma}\omega^{X_{1}}(Q_{i})\qquad\text{
for }X\in R_{i},$

where the constants depend only on $C_{\sigma}$, $C_{0}$, $n$, $d$, $\beta$
and $\gamma$. Using the equivalence (2.2) and the Ahlfors regularity of
$\sigma$, we deduce that

(3.30) $g(X_{1},X_{0})\approx
D_{\beta}(X_{0})^{1-\gamma}\,\frac{\omega^{X_{1}}(3B)}{\sigma(3B)}$

and for each $i\in I$,

(3.31) $g(X_{1},X)\approx
D_{\beta}(X)^{1-\gamma}\,\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\qquad\text{
for }X\in R_{i}.$

We gather (3.29), (3.30), and (3.31) to obtain that, for every $i\in I$ and
every $X\in R_{i}$,

(3.32)
$\frac{u(X)}{D_{\beta}^{1-\gamma}(X)}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\approx\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)},$

where the constants depend only on $C_{\sigma}$, $C_{0}$, $n$, $d$, $\beta$
and $\gamma$. This immediately implies that for $i\in I$ and $X\in R_{i}$,

$\left|\ln\left(\frac{u(X)}{D_{\beta}^{1-\gamma}(X)}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\right)\right|\leq
C+\left|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right|$

and then

(3.33) $\begin{split}S&\leq\sum_{i\in
I}\int_{R_{i}}\left|\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\right)\right|D_{\beta}^{d-n}\\\
&\leq\sum_{i\in
I}\left[C+\left|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right|\right]\int_{R_{i}}D_{\beta}^{d-n}.\end{split}$

By(2.2) and the fact that $R_{i}\in{\mathcal{W}}$ is a Whitney cube with the
property (3.22) (by construction), we have $D_{\beta}\approx r_{i}$ on $R_{i}$
and $|R_{i}|\lesssim r_{i}^{n}$. Hence

$\int_{R_{i}}D_{\beta}^{d-n}\lesssim r_{i}^{d}\approx\sigma(Q_{i})$

by (2.1). Using this observation in (3.33), we infer that

$\begin{split}S&\leq\sum_{i\in
I}\left[C+\left|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right|\right]\sigma(Q_{i})\\\
&=C\sum_{i\in I}\sigma(Q_{i})+\sum_{i\in
I}\left|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right|\sigma(Q_{i}).\end{split}$

The first term of the right-hand side is bounded by $C\sigma(3B)$ since the
$Q_{i}$, $i\in I$, are contained in $\Gamma\cap 3B$ (by (3.20)) and have
bounded overlap (by (3.21)). The second term is also less than $C\sigma(3B)$,
by Lemma 3.4. We conclude that

$S\lesssim\sigma(3B)\lesssim r^{d}$

by (2.1). The claim 3.27 follows.

Step 4: Core of the proof. Let us turn to the main and last step of the proof.
Set

(3.34)
$T:=\int_{\Omega}\bigg{|}\nabla\ln\Big{(}\frac{u}{D_{\beta}^{1-\gamma}}\Big{)}\bigg{|}^{2}\phi_{B,\epsilon}^{2}D_{\beta}^{d+2-n}.$

We aim to prove that

(3.35) $T\leq Cr^{d}+Cr^{d/2}T^{1/2},$

where $C$ depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$ and
$d$. The solution $u$ and the smooth distance $D_{\beta}$ are uniformly
bounded from above and from below by a positive constant on the support of the
cut-off function $\phi_{B,\epsilon}$. Thanks to this fact, the quantities of
both sides of (3.35) are finite, and (3.35) self-improves into

(3.36)
$\int_{\Omega}\Big{|}\nabla\ln\Big{(}\frac{u}{D_{\beta}^{1-\gamma}}\Big{)}\Big{|}^{2}\phi_{B,\epsilon}^{2}D_{\beta}^{d+2-n}=T\leq
Cr^{d},$

where $C>0$ depends on the same parameters as in (3.35). Once we are there,
the left-hand side is uniformly bounded in $\epsilon$, so taking $\epsilon\to
0$ leads to the desired result (3.9).

Keep in mind that

(3.37) $\nabla\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}=\frac{\nabla
u}{u}-\frac{\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}=\dfrac{D_{\beta}^{1-\gamma}\nabla
u-u\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}u}.$

We shall use Lemma 2.10 to obtain the existence of a scalar function $b$ and a
vector function $\mathcal{V}$ such that

(3.38) $H_{n-d-1}:=\int_{\Gamma}|X-y|^{-n}(X-y)d\sigma(y)=(b\nabla
D_{\beta}+\mathcal{V})D_{\beta}^{d+1-n}$

as in (2.11), with the bounds (2.12)-(2.15).

Before we start to bound $T$, let us comment on $H_{n-d-1}$ and $\nabla
D_{\beta}$. First, the vector function $H_{n-d-1}$ is smooth and divergence
free in $\Omega$, and the formulation of this fact in the weak sense is that
for any compactly supported $\varphi\in W^{1,1}(\Omega)$,

(3.39) $\int_{\Omega}H_{n-d-1}\cdot\nabla\varphi=0.$

Set $H_{\alpha}(X):=\int_{\Gamma}|X-y|^{-d-1-\alpha}(X-y)d\sigma(y)$ for
$\alpha>0$. Since
$|H_{\alpha}(X)|\leq\int_{\Gamma}|X-y|^{-d-\alpha}d\sigma(y)=D_{\alpha}(X)^{-\alpha}$,
(2.2) implies that

(3.40) $|H_{\alpha}|\lesssim D_{\alpha}^{-\alpha}\lesssim
D_{\beta}^{-\alpha}.$

Moreover, observe that $\nabla(D_{\beta}^{-\beta})=-(d+\beta)H_{\beta+1}$,
directly by (1.2); since also $\nabla(D_{\beta}^{-\beta})=-\beta
D_{\beta}^{-\beta-1}\nabla D_{\beta}$, we deduce from (3.40) that

(3.41) $|\nabla D_{\beta}|\lesssim 1.$

We turn to the proof of (3.35). Write $\phi$ instead of $\phi_{B,\epsilon}$ to
lighten the notation. By (2.12) and (3.37),

$\begin{split}T\lesssim\int_{\Omega}\bigg{|}\nabla\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{|}^{2}\phi^{2}b\,D_{\beta}^{d+2-n}&=\int_{\Omega}\frac{b\nabla
u}{u}\cdot\bigg{(}\dfrac{D_{\beta}^{1-\gamma}\nabla u-u\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}u}\bigg{)}\
\phi^{2}D_{\beta}^{d+2-n}\\\ &\hskip 56.9055pt-\int_{\Omega}\frac{b\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}D_{\beta}^{d+2-n}\\\
&:=T_{1}-T_{2}.\end{split}$

Let us start with $T_{2}$. We simplify the factors $D_{\beta}$ and we invoke
the relation (3.38) to get

$\begin{split}T_{2}&=(\gamma-1)\int_{\Omega}\frac{b\nabla
D_{\beta}}{D_{\beta}^{n-d-1}}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}\\\
&=(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}-(\gamma-1)\int_{\Omega}\mathcal{V}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}D_{\beta}^{d+1-n}\\\
&:=T_{21}+T_{22}.\end{split}$

We use the Cauchy-Schwarz inequality to bound $T_{22}$, as follows:

$\begin{split}|T_{22}|&\lesssim\Big{(}\int_{\Omega}|\mathcal{V}|^{2}\phi^{2}D_{\beta}^{d-n}\Big{)}^{\frac{1}{2}}\bigg{(}\int_{\Omega}\bigg{|}\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\bigg{|}^{2}\phi^{2}D_{\beta}^{d+2-n}\bigg{)}^{\frac{1}{2}}\\\
&\lesssim
T^{1/2}\left(\int_{2B}|\mathcal{V}|^{2}D_{\beta}^{d-n}\right)^{\frac{1}{2}}\lesssim
r^{d/2}T^{1/2}\end{split}$

by (2.15). This fits with (3.35). As for $T_{21}$, notice that its value will
not be changed if we replace $u$ by $Ku$, where $K$ is a constant. Hence

$T_{21}=(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2},$

where $K$ is a constant to be chosen later. We force $\phi^{2}$ into the
gradient, then use the fact that $H_{n-d-1}$ is divergence free (see (3.39)),
and obtain

$\displaystyle T_{21}$
$\displaystyle=(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\Big{[}\phi^{2}\ln\Big{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\Big{)}\Big{]}-2(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\phi\
\phi\ln\Big{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\Big{)}$
$\displaystyle=0-2(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\phi\
\phi\ln\Big{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\Big{)}.$

Recall that $|H_{n-d-1}|\lesssim D_{\beta}^{d+1-n}$ by (3.40), and that
$|\nabla\phi|\lesssim{\mathds{1}}_{E_{1}\cup E_{2}\cup E_{3}}/D_{\beta}$ by
(3.15) and (2.2), so

$|T_{21}|\lesssim\int_{E_{1}\cup E_{2}\cup
E_{3}}\bigg{|}\ln\bigg{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{|}D_{\beta}^{d-n}.$

We choose $K=D_{\beta}^{1-\gamma}(X_{0})/u(X_{0})$, so that the right-hand
side above is what we called $S$ in (3.27). Hence $T_{21}\lesssim r^{d}$ by
(3.27), as needed for (3.35).

We switch to the estimation of $T_{1}$. We want to use the fact that $u$ is a
solution, and for this we write

(3.42) $\begin{split}T_{1}&=\int_{\Omega}\nabla
u\cdot\frac{D_{\beta}^{1-\gamma}}{u}\bigg{(}\dfrac{D_{\beta}^{1-\gamma}\nabla
u-u\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}u}\bigg{)}\
b\phi^{2}D_{\beta}^{d+1+\gamma-n}\\\ &=-\int_{\Omega}\nabla
u\cdot\nabla\bigg{[}\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{]}\
b\phi^{2}D_{\beta}^{d+1+\gamma-n}\\\ &=-\int_{\Omega}\nabla
u\cdot\nabla\bigg{[}b\phi^{2}\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{]}\
D_{\beta}^{d+1+\gamma-n}+\int_{\Omega}\nabla u\cdot\nabla
b\bigg{(}\phi^{2}\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{)}\
D_{\beta}^{d+1+\gamma-n}\\\ &\qquad+2\int_{\Omega}\nabla
u\cdot\nabla\phi\bigg{(}\phi b\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{)}\
D_{\beta}^{d+1+\gamma-n}\\\ &:=T_{11}+T_{12}+T_{13}.\end{split}$

Notice that $T_{11}=0$ because $u$ is a weak solution to $L_{\beta,\gamma}u=0$
on $\Omega\cap 2B$ and $\phi^{2}\dfrac{D_{\beta}^{1-\gamma}}{u}$ lies in
$W^{1,2}_{loc}(\Omega)$ and compactly supported in $\Omega$. The terms
$T_{12}$ and $T_{13}$ are morally similar. In both case, we don’t like the
terms with $u$ because we don’t know so much about it, so we use (3.37) to
replace $u$ by the nice function $D_{\beta}^{1-\gamma}$, and the difference
will be controlled with the help of $T^{1/2}$, the square root of our initial
integral. We start with

(3.43) $\begin{split}T_{13}&=2\int_{\Omega}\frac{\nabla
u}{u}\cdot\nabla\phi\,\phi\,b\,D_{\beta}^{d+2-n}\\\
&=2\int_{\Omega}\left(\frac{\nabla u}{u}-\frac{\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\right)\cdot(\nabla\phi)\,\phi\,b\,D_{\beta}^{d+2-n}-2(\gamma-1)\int_{\Omega}\frac{b\nabla
D_{\beta}}{D_{\beta}^{n-d-1}}\cdot(\nabla\phi)\phi\\\
&:=T_{131}+T_{132}.\end{split}$

We use (2.12) and the fact that $D_{\beta}\nabla\phi$ satisfies the Carleson
measure property to get

$\begin{split}|T_{131}|&\lesssim\Big{(}\int_{\Omega}|\nabla\phi|^{2}D_{\alpha}^{d+2-n}\Big{)}^{1/2}\bigg{(}\int_{\Omega}\bigg{|}\frac{\nabla
u}{u}-\frac{\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\bigg{|}^{2}\phi^{2}D_{\alpha}^{d+2-n}\bigg{)}^{1/2}\\\
&\lesssim
r^{d/2}\bigg{(}\int_{\Omega}\bigg{|}\nabla\ln\bigg{(}\frac{u}{D_{\alpha}^{1-\gamma}}\bigg{)}\bigg{|}^{2}\phi^{2}D_{\alpha}^{d+2-n}\bigg{)}^{1/2}\end{split}$

by (3.17), and then (3.37). The bound of $T_{131}$ that we just obtained
appears in the right-hand side of (3.35), as desired. As for $T_{132}$, we
invoke (3.41), (2.12), and then (3.17) to write

$\begin{split}|T_{132}|&\lesssim\int_{2B}|D_{\beta}\nabla\phi|D_{\beta}^{n-d}\lesssim
r^{d}\end{split}.$

Similarly to $T_{13}$, we treat $T_{12}$ as follows:

(3.44) $\begin{split}T_{12}&=\int_{\Omega}\frac{\nabla u}{u}\cdot\nabla
b\,\phi^{2}D_{\beta}^{d+2-n}\\\ &=\int_{\Omega}\bigg{(}\frac{\nabla
u}{u}-\frac{\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\bigg{)}\cdot\nabla
b\,\phi^{2}D_{\beta}^{d+2-n}-(\gamma-1)\int_{\Omega}\nabla
D_{\beta}\cdot\nabla b\,\phi^{2}D_{\beta}^{d+1-n}\\\
&:=T_{121}+T_{122}.\end{split}$

Thanks to the definition (3.34) and then (2.13),

(3.45)
$\begin{split}|T_{121}|&\lesssim\bigg{(}\int_{\Omega}\bigg{|}\frac{\nabla
u}{u}-\frac{\nabla
D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\bigg{|}^{2}\phi^{2}D_{\beta}^{d+2-n}\bigg{)}^{1/2}\Big{(}\int_{\Omega}|\nabla
b|^{2}\phi^{2}D_{\beta}^{d+2-n}\Big{)}^{1/2}\\\ &\lesssim
T^{1/2}\left(\int_{2B}|D_{\beta}\nabla
b|^{2}D_{\beta}^{d-n}\right)^{\frac{1}{2}}\lesssim r^{d/2}T^{1/2}.\end{split}$

Now, we want to use (3.38) again, so we force the function $b$ to appear and
we place all the remaining terms in the second gradient. Then $T_{122}$
becomes

$\begin{split}T_{122}&=-(\gamma-1)\int_{\Omega}b\nabla
D_{\beta}\cdot\frac{\nabla b}{b}\,\phi^{2}D_{\beta}^{d+1-n}\\\
&=-(\gamma-1)\int_{\Omega}b\nabla
D_{\beta}\cdot\nabla[\phi^{2}\ln(b)]\,D_{\beta}^{d+1-n}-2(1-\gamma)\int_{\Omega}b\nabla
D_{\beta}\cdot\nabla\phi\,\phi\ln(b)\,D_{\beta}^{d+1-n}\\\
&:=T_{1221}+T_{1222}.\end{split}$

We start with $T_{1222}$. Since $b\approx 1$, we have that $|b\ln(b)|\lesssim
1$. Besides, $|\nabla D_{\beta}|\lesssim 1$ by (3.41). Therefore,

$|T_{1222}|\lesssim\int_{2B}|D_{\beta}\nabla\phi|D_{\beta}^{d-n},$

and then $|T_{1222}|\lesssim r^{d}$ by (3.17). We use (3.38) to write
$T_{1221}$ as

$\begin{split}T_{1221}&=(1-\gamma)\int_{\Omega}H_{n-d-1}\cdot\nabla[\phi^{2}\ln(b)]+(\gamma-1)\int_{\Omega}\mathcal{V}\cdot\nabla[\phi^{2}\ln(b)]\,D_{\beta}^{d+1-n}\\\
&=0+(\gamma-1)\int_{\Omega}\mathcal{V}\cdot\Big{[}2\phi\nabla\phi\ln(b)+\phi^{2}\frac{\nabla
b}{b}\Big{]}D_{\beta}^{d+1-n}\end{split}$

by (3.39). Hence by (2.12) and then the Cauchy-Schwarz inequality,

$\begin{split}|T_{1221}|&\lesssim\int_{2B}|\mathcal{V}|(|\nabla\phi|+|\nabla
b|)D_{\beta}^{d+1-n}\\\
&\lesssim\left(\int_{2B}|\mathcal{V}|^{2}D_{\beta}^{n-d}\right)^{\frac{1}{2}}\left(\int_{2B}(|D_{\beta}\nabla\phi|^{2}+|D_{\beta}\nabla
b|^{2})D_{\beta}^{d-n}\right)^{\frac{1}{2}}.\end{split}$

But since $\mathcal{V}$, $D_{\beta}\nabla b$, and $D_{\beta}\nabla\phi$ all
satisfies the Carleson measure condition, we conclude that $|T_{1221}|\lesssim
r^{d}$ as desired.

We bounded each term derived from $T$ by either $r^{d}$ or $r^{d/2}T^{1/2}$,
and consequently proved the claim (3.35). As was observed before, (3.9),
Proposition 3.8, and then Theorems 2.21 and 1.8 follow. $\Box$

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