# Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and
Carleson’s ${\varepsilon}^{2}$ conjecture in higher dimensions

Ian Fleschler  Department of Mathematics, Fine Hall, Princeton University,
Washington Road, Princeton, NJ 08540, USA<EMAIL_ADDRESS>,  Xavier Tolsa
ICREA (Barcelona, Catalonia)
Universitat Autònoma de Barcelona
and Centre de Recerca Matemàtica (Barcelona, Catalonia<EMAIL_ADDRESS>and  Michele Villa  University of Oulu (Oulu, Finland) and Universitat
Autònoma de Barcelona (Barcelona, Catalonia<EMAIL_ADDRESS>
###### Abstract.

The main aim of this article is to prove quantitative spectral inequalities
for the Laplacian with Dirichlet boundary conditions. More specifically, we
prove sharp quantitative stability for the Faber-Krahn inequality in terms of
Newtonian capacities and Hausdorff contents of positive codimension, thus
providing an answer to a question posed by De Philippis and Brasco.

One of our results asserts that for any bounded domain
$\Omega\subset{\mathbb{R}}^{n}$, $n\geq 3$, with Lebesgue measure equal to
that of the unit ball and whose first eigenvalue is $\lambda_{\Omega}$,
denoting by $\lambda_{B}$ the first eigenvalue for the unit ball, for any
$a\in(0,1)$ it holds

$\lambda_{\Omega}-\lambda_{B}\geq
C(a)\,\inf_{B}\bigg{(}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x)\bigg{)}^{2},$

where the infimum is taken over all balls $B$ with the same Lebesgue measure
as $\Omega$ and $\operatorname{Cap}_{n-2}$ is the Newtonian capacity of
homogeneity $n-2$. In fact, this holds for bounded subdomains of the sphere
and the hyperbolic space, as well.

In a second result, we also apply the new Faber-Krahn type inequalities to
quantify the Hayman-Friedland inequality about the characteristics of disjoint
domains in the unit sphere. Thirdly, we propose a natural extension of
Carleson’s ${\varepsilon}^{2}$-conjecture to higher dimensions in terms of a
square function involving the characteristics of certain spherical domains,
and we prove the necessity of the finiteness of such square function in the
tangent points via the Alt-Caffarelli-Friedman monotonicity formula. Finally,
we answer in the negative a question posed by Allen, Kriventsov and Neumayer
in connection to rectifiability and the positivity set of the ACF monotonicity
formula.

I.F. acknowledges the support of the National Science Foundation through the
grant FRG-1854147. X.T and M.V. are supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement 101018680). X.T. is also partially supported by
MICINN (Spain) under the grant PID2020-114167GB-I00, the María de Maeztu
Program for units of excellence (Spain) (CEX2020-001084-M), and 2021-SGR-00071
(Catalonia). M.V. is supported by the Academy of Finland via the project
“Higher dimensional Analyst’s Traveling Salesman theorems and Dorronsoro
estimates on non-smooth sets”, grant No. 347828/24304228. This work is based
upon research funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 –
390685813, while the authors were in residence at the Hausdorff Institute of
Mathematics in Spring 2022 during the program “Interactions between geometric
measure theory, singular integrals, and PDEs”.

###### Contents

  1. 1 Introduction
    1. 1.1 Quantitative Faber-Krahn inequalities
    2. 1.2 The Alt-Caffarelli-Friedman monotonicity formula and the Friedland-Hayman inequality
    3. 1.3 Carleson’s conjecture
    4. 1.4 A counterexample
  2. 2 Preliminaries
    1. 2.1 Miscellaneous notation
    2. 2.2 Capacities
    3. 2.3 Harmonic measure
  3. 3 Proof of Theorem A and beginning of the proof of Theorem B in the Wiener regular case for ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$
  4. 4 End of the proof of Theorem B in the Wiener regular case for ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$
    1. 4.1 Case 1.
    2. 4.2 Case 2.
  5. 5 The proof of Theorems A and B in the non-Wiener regular case
  6. 6 The proof of Theorem A and Theorem B for ${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ and ${\mathbb{M}}^{n}={\mathbb{H}}^{n}$.
    1. 6.1 Domains in Euclidean space
    2. 6.2 Domains in hyperbolic space
  7. 7 Proof of Theorem C
  8. 8 Existence of tangents implies finiteness of the Carleson square function
  9. 9 A counterexample to a question of Allen, Kriventsov and Neumayer

##  1\. Introduction

The main aim of this article is to prove quantitative spectral inequalities
for the Laplacian with Dirichlet boundary conditions. More specifically, we
show that the Faber-Krahn inequality is stable, where stability is quantified
in terms of Newtonian capacities and Hausdorff contents of positive
codimension. Our results hold for subdomains of the Euclidean space, the unit
sphere and the hyperbolic space, and they are sharp up to a constant factor.
We remark that Theorem A, our main result (see below), can be seen as a
geometric solution of an issue raised by Brasco and De Philippis ([BD, Open
Problem 7.23]), and it foundamentally rests on the analytic solution of the
same problem by Allen, Kriventsov and Neumayer [AKN1]. Our interest in these
versions of the Faber-Krahn inequality arise also from applications in
connection with the Alt-Caffarelli-Friedman monotonicity formula and with the
Carleson ${\varepsilon}^{2}$-conjecture about tangent points for Jordan
domains in the plane. Indeed, in this paper we also propose a natural
extension of Carleson ${\varepsilon}^{2}$-conjecture to higher dimensions in
terms of a square function involving the characteristics of certain spherical
domains, and we prove the necessity of the finiteness of such square function
at almost every tangent point. Sufficiency is shown in the companion paper
[FTV].

###  1.1. Quantitative Faber-Krahn inequalities

Given a bounded open set $\Omega\subset{\mathbb{R}}^{n}$, we say that $u\in
W^{1,2}_{0}(\Omega)$ is a Dirichlet eigenfunction of $\Omega$ (for the
Laplacian) if $u\not\equiv 0$ and

$-\Delta u=\lambda\,u,$

for some $\lambda\in{\mathbb{R}}\setminus\\{0\\}$. The number $\lambda$ is the
eigenvalue associated with $u$. It is well known that all the eigenvalues of
the Laplacian are positive and the smallest one, i.e., the first eigenvalue
$\lambda_{1}$, satisfies

$\lambda_{1}=\inf_{u\in W^{1,2}_{0}(\Omega)}\frac{\int_{\Omega}|\nabla
u|^{2}\,dx}{\int_{\Omega}|u|^{2}\,dx}.$

Further the infimum is attained by an eigenfunction $u_{0}$ which does not
change sign, and so which can be assumed to be non-negative. We will denote by
$\lambda_{\Omega}$ (or $\lambda(\Omega)$) the first eigenvalue of $\Omega$ and
by $u_{\Omega}$ the associated non-negative eigenfunction, normalized so that
$\|u_{\Omega}\|_{L^{2}(\Omega)}=1$.

The classical Faber-Krahn inequality asserts that among all bounded open sets
with a fixed volume, a ball minimizes the first eigenvalue. Following many
previous works (see, for example, [HN] and [Me]), Brasco, De Philippis, and
Velichkov proved in [BDV] the following sharp quantitative version of the
Faber-Krahn inequality.

######  Theorem 1.1.

For $n\geq 2$, let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded open set with
${\mathcal{H}}^{n}(\Omega)=1$. Then

(1.1) $\lambda_{\Omega}-\lambda_{B}\geq
c\,\inf_{B}{\mathcal{H}}^{n}(\Omega\triangle B)^{2},$

where $c$ is a positive absolute constant and the infimum is taken over all
balls $B$ with ${\mathcal{H}}^{n}(B)=1$.

The inequality above is sharp in the sense that the power $2$ on the right
hand side cannot be lowered.

The classical Faber-Krahn inequality also holds for subdomains of the sphere
${\mathbb{S}}^{n}$ or the hyperbolic space ${\mathbb{H}}^{n}$. In this context
one should take the Laplacian with respect to the Riemannian metric of the
space (i.e., the Laplace-Beltrami operator) and one should consider geodesic
balls. That is, for subdomains $\Omega$ of ${\mathbb{S}}^{n}$ or
${\mathbb{H}}^{n}$ with a given volume, the minimal value of
$\lambda_{\Omega}$ is attained again by a geodesic ball among all open bounded
domains of the same volume. Further, recently Allen, Kriventsov, and Neumayer
[AKN1] have obtained the following quantitative form.

######  Theorem 1.2.

For $n\geq 2$, let ${\mathbb{M}}^{n}$ be either ${\mathbb{R}}^{n}$,
${\mathbb{S}}^{n}$, or ${\mathbb{H}}^{n}$, and let $\beta>0$. Let $\Omega$ be
a relatively open subset of ${\mathbb{M}}^{n}$ and let $B$ be a geodesic ball
in ${\mathbb{M}}^{n}$ such that
${\mathcal{H}}^{n}(B)={\mathcal{H}}^{n}(\Omega)$. In the case
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, suppose also that
$\beta\leq{\mathcal{H}}^{n}(\Omega)\leq{\mathcal{H}}^{n}({\mathbb{S}}^{n})-\beta$,
and in the case ${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ or
${\mathbb{M}}^{n}={\mathbb{H}}^{n}$ just that
${\mathcal{H}}^{n}(\Omega)\leq\beta$. Denote by $\lambda_{\Omega}$ and
$\lambda_{B}$ the respective first Dirichlet eigenvalues of
$-\Delta_{{\mathbb{M}}^{n}}$ in $\Omega$ and $B$, and let $u_{\Omega}$ and
$u_{B}$ be the corresponding eigenfunctions normalized so that they are
positive and
$\|u_{\Omega}\|_{L^{2}({\mathbb{M}}^{n})}=\|u_{B}\|_{L^{2}({\mathbb{M}}^{n})}=1$.
Then

(1.2) $\lambda_{\Omega}-\lambda_{B}\geq
c(\beta)\,\inf_{x\in{\mathbb{M}}^{n}}\Big{(}{\mathcal{H}}^{n}(\Omega\triangle
B_{x})^{2}+\int_{{\mathbb{M}}^{n}}|u_{\Omega}-u_{B_{x}}|^{2}\,d{\mathcal{H}}^{n}\Big{)},$

where $c(\beta)>0$ and $B_{x}$ denotes a geodesic ball centered at $x$ with
the same ${\mathcal{H}}^{n}$ measure as $B$. In the case
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, (1.2) also holds with the infimum over
${\mathbb{S}}^{n}$ replaced by the evaluation at $x$ equal to
${\mathbb{S}}^{n}$-barycenter of $\Omega$ (possibly with a different constant
$c(\beta)$).

In the theorem we have denoted by $\Delta_{{\mathbb{M}}^{n}}$ the Laplace-
Beltrami operator on ${\mathbb{M}}^{n}$. For the definition of
${\mathbb{S}}^{n}$-barycenter of a ball in ${\mathbb{S}}^{n}$, see Section
2.1. The assumption involving the parameter $\beta$ is necessary to prevent
the domain from being too big and, in the case
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, also too small. Remark that in the case
${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ one can get an appropriate scaling
invariant statement by renormalising. This is not case for ${\mathbb{S}}^{n}$
or ${\mathbb{H}}^{n}$. Notice the presence of the additional term
$\int_{{\mathbb{M}}^{n}}|u_{\Omega}-u_{B}|^{2}\,d{\mathcal{H}}^{n}$ in the
inequality (1.2) when compared to (1.1). In this term we assume that the
functions $u_{\Omega}$ and $u_{B}$ vanish outside of $\Omega$ and $B$
respectively.

To state our results we need to introduce some additional notation and
terminology. For ${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ or
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$. We denote by ${\rm
dist}_{{\mathbb{M}}^{n}}$ the geodesic distance in ${\mathbb{M}}^{n}$ and by
$\partial_{{\mathbb{M}}^{n}}A$ the boundary of any set
$A\subset{\mathbb{M}}^{n}$. We also write $B_{{\mathbb{M}}^{n}}(x,r)$ to
denote an open ball in ${\mathbb{M}}^{n}$ centered in $x$, with radius $r$.
Given $B=B_{{\mathbb{M}}^{n}}(x,r)$ and $\rho>0$, we set $\rho
B=B_{{\mathbb{M}}^{n}}(x,\rho r)$ and we denote $\delta_{B}(x)={\rm
dist}_{{\mathbb{M}}^{n}}(x,\partial_{{\mathbb{M}}^{n}}B)$,

We denote by $\operatorname{Cap}_{n-2}$ the Newtonian capacity of homogeneity
$n-2$, and by $\operatorname{Cap}_{L}$ the logarithmic capacity (see Section
2.2 for the precise definitions).

Our first main result in this paper is the following.

######  Theorem A.

Given $n\geq 2$, let ${\mathbb{M}}^{n}$ be either ${\mathbb{R}}^{n}$,
${\mathbb{S}}^{n}$ or ${\mathbb{H}}^{n}$, and let $\beta>0$, $a\in(0,1)$. Let
$\Omega$ be a relatively open subset of ${\mathbb{M}}^{n}$ and let $B$ be a
geodesic ball in ${\mathbb{M}}^{n}$ such that
${\mathcal{H}}^{n}(B)={\mathcal{H}}^{n}(\Omega)$ with radius $r_{B}$. In the
case ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, suppose also that
$\beta\leq{\mathcal{H}}^{n}(\Omega)\leq{\mathcal{H}}^{n}({\mathbb{S}}^{n})-\beta$,
while in the case ${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ and
${\mathbb{M}}^{n}={\mathbb{H}}^{n}$ only assume that
${\mathcal{H}}^{n}(\Omega)\leq\beta$. Denote by $\lambda_{\Omega}$ and
$\lambda_{B}$ the first Dirichlet eigenvalues of $-\Delta_{{\mathbb{M}}^{n}}$
in $\Omega$ and $B$, respectively. Then, there is some constant $C(a,\beta)>0$
such that in the case $n\geq 3$ we have

(1.3) $\lambda_{\Omega}-\lambda_{B}\geq
C(a,\beta)\,\inf_{x\in{\mathbb{M}}^{n}}\bigg{(}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B_{x})}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x)\bigg{)}^{2},$

where $B_{x}$ denotes a ball centered in $x$ with radius $r_{B}$. In the case
$n=2$ we have

(1.4) $\lambda_{\Omega}-\lambda_{B}\geq
C(a,\beta)\,\inf_{x\in{\mathbb{M}}^{n}}\bigg{(}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{2}}((1-t)B_{x})}\frac{t\,r_{B}}{\log\frac{2tar_{B}}{\operatorname{Cap}_{L}(B_{{\mathbb{M}}^{2}}(x,atr_{B})\setminus\Omega)}}\,d{\mathcal{H}}^{1}(x)\bigg{)}^{2}.$

In the case ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, (1.3) and (1.4) also hold
with the infimum over ${\mathbb{M}}^{n}$ replaced by the evaluation at $x$
equal to ${\mathbb{S}}^{n}$-barycenter of $\Omega$ (possibly with a different
constant $C(a,\beta)$).

In the theorem we used the standard notation
$\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{A}f\,d\mu=\frac{1}{\mu(A)}\int_{A}f\,d\mu$. Notice
that in (1.3) we have

$\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\lesssim
t\,r_{B},$

because of the $(n-2)$-homogeneity of $\operatorname{Cap}_{n-2}$. Remark also
that, for all $a,a^{\prime}\in(0,1)$, we have

$\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x)\\\
\approx_{a,a^{\prime}}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,a^{\prime}tr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x).$

The same happens in the case $n=2$ concerning the integral on the right hand
side of (1.4). See Lemmas 2.4 and 2.5.

The estimates in Theorem are sharp up to a constant factor. Indeed, as in
[BDV] and [AKN2], we can consider ellipsoidal perturbations of the unit ball,
such as

$\Omega_{\varepsilon}=\\{x\in{\mathbb{R}}^{n}:(1+{\varepsilon})x_{1}^{2}+(1-{\varepsilon})x_{2}^{2}+x_{3}^{2}+\ldots+x_{n}^{2}\leq
1\\},$

with ${\varepsilon}\to 0$. As remarked in [BDV] and [BD], letting
$B_{\varepsilon}$ be a ball such that
${\mathcal{H}}^{n}(B_{\varepsilon})={\mathcal{H}}^{n}(\Omega_{\varepsilon})$,
it holds
$\lambda_{\Omega_{\varepsilon}}-\lambda_{B_{\varepsilon}}\approx{\varepsilon}^{2}$,
uniformly as ${\varepsilon}\to 0$. On the other hand, it is easy to check that
the right hand side terms of (1.3) and (1.4) are also comparable to
${\varepsilon}^{2}$, also uniformly as ${\varepsilon}\to 0$.

As far as we know, Theorem A provides the first sharp quantitative Faber-Krahn
inequality stated in terms of the capacity of (subsets of) $B\setminus\Omega$.
As remarked above, Theorem A is related to the open problem 7.23 from [BD],
which asks to prove a sharp quantitative Faber-Krahn inequality with a
suitable a capacitary asymmetry of $B$ and $\Omega$. Notice however that
Theorem A only involves the capacity of subsets of $B\setminus\Omega$, and not
of $\Omega\setminus B$. This is a natural fact because there are domains such
that $\operatorname{Cap}_{n-2}(\Omega\setminus B)$ is large while
$\lambda_{\Omega}-\lambda_{B}$ is as small as wished. Indeed, for
$0<{\varepsilon}\leq 1/10$, consider an ${\varepsilon}$-neighborhood
$U_{\varepsilon}$ of the planar set $B(0,1)\cup[1,2]$ and contract it by a
suitable factor so that the resulting domain $\Omega_{\varepsilon}$ has the
same area as $B=B(0,1)$. It is easy to check that
$\lambda_{\Omega_{\varepsilon}}\to\lambda_{B}$ as ${\varepsilon}\to 0$, but
$\operatorname{Cap}_{n-2}(\Omega_{\varepsilon}\setminus B^{\prime})$ is large
for any given ball $B^{\prime}$ with the same area as $\Omega_{\varepsilon}$,
due to the fact that $[-1,2]\subset\Omega_{\varepsilon}$, for example.

Using the connection between Hausdorff contents and capacities (see Lemma 2.1
below), we deduce the following corollary from the preceding theorem.

######  Corollary 1.3.

Under the assumptions and notation of Theorem A, for any $s>n-2$ there is some
constant $C(s,a,\beta)>0$ such that, in the case $n\geq 3$, we have

(1.5) $\lambda_{\Omega}-\lambda_{B}\geq
C(s,a,\beta)\,\inf_{x\in{\mathbb{M}}^{n}}\bigg{(}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B_{x})}\left(\frac{{\mathcal{H}}^{s}_{\infty}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{s}}\right)^{\frac{n-2}{s}}\,t\,r_{B}\,d{\mathcal{H}}^{n-1}(x)\bigg{)}^{2}.$

In the case $n=2$, for any $s>0$ we have

(1.6) $\lambda_{\Omega}-\lambda_{B}\geq
C(s,a,\beta)\,\inf_{x\in{\mathbb{M}}^{n}}\bigg{(}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{2}}((1-t)B_{x})}\frac{t\,r_{B}}{\log\left(\frac{(2atr_{B})^{s}}{{\mathcal{H}}^{s}_{\infty}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}\right)}\,d{\mathcal{H}}^{1}(x)\bigg{)}^{2}.$

In the case ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, (1.5) and (1.6) also hold
with the infimum over ${\mathbb{M}}^{n}$ replaced by the evaluation at $x$
equal to ${\mathbb{S}}^{n}$-barycenter of $\Omega$ (possibly with a different
constant $C(s,a,\beta)$).

In the case when we just quantify $\lambda_{\Omega}-\lambda_{B}$ in terms of
some integral over a suitable family of “thick points” we get a somewhat
sharper result. To state it, we need some additional notation. For given
$c_{0}>0$ and $a\in(0,1)$, in the case $n\geq 3$, we denote

(1.7) $T_{c_{0}}(\Omega,B,a)=\big{\\{}x\in
B\setminus\Omega:\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,a\,\delta_{B}(x))\setminus\Omega)\geq
c_{0}\,\delta_{B}(x)^{n-2}\big{\\}},$

and, in the case $n=2$,

(1.8) $T_{c_{0}}(\Omega,B,a)=\big{\\{}x\in
B\setminus\Omega:\operatorname{Cap}_{L}(B_{{\mathbb{M}}^{n}}(x,a\,\delta_{B}(x))\setminus\Omega)\geq
c_{0}\,\delta_{B}(x)\big{\\}}.$

We should understand the condition in the definition of
$T_{c_{0}}(\Omega,B,a)$ as a thickness type condition. In particular, if
$\Omega$ satisfies the capacity density condition or CDC (see Section 2.2), or
$\partial\Omega$ is lower $s$-content regular for some $s>n-2$, then
$T_{c_{0}}(\Omega,B,a)=B\setminus\Omega$, for $c_{0}$ small enough.

######  Theorem B.

Given $n\geq 2$ and $0<s\leq n$, let ${\mathbb{M}}^{n}$ be either
${\mathbb{R}}^{n}$, ${\mathbb{S}}^{n}$ or ${\mathbb{H}}^{n}$, and let
$c_{0},\beta>0$, $a\in(0,1)$. Let $\Omega$ be a relatively open bounded subset
of ${\mathbb{M}}^{n}$ and let $B$ be a geodesic ball in ${\mathbb{M}}^{n}$
such that ${\mathcal{H}}^{n}(B)={\mathcal{H}}^{n}(\Omega)$. For
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, suppose that
$\beta\leq{\mathcal{H}}^{n}(\Omega)\leq{\mathcal{H}}^{n}({\mathbb{S}}^{n})-\beta$,
while for ${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ and
${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ assume only that
${\mathcal{H}}^{n}(\Omega)\leq\beta$. Denote by $\lambda_{\Omega}$ and
$\lambda_{B}$ the first Dirichlet eigenvalues of $-\Delta_{{\mathbb{M}}^{n}}$
in $\Omega$ and $B$, respectively. Then,

(1.9) $\lambda_{\Omega}-\lambda_{B}\geq
C(a,s,\beta,c_{0})\,\inf_{x\in{\mathbb{M}}^{n}}\bigg{(}\int_{T_{c_{0}}(\Omega,B_{x},a)}\delta_{B}(x)^{n-s}\,d{\mathcal{H}}^{s}_{\infty}(x)\bigg{)}^{2},$

where $B_{x}$ denotes a ball centered in $x$ with radius $r_{B}$. In the case
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, (1.9) also holds with the infimum over
${\mathbb{M}}^{n}$ replaced by the evaluation at $x$ equal to
${\mathbb{S}}^{n}$-barycenter of $\Omega$ (possibly with a different constant
$C(a,s,\beta,c_{0})$).

Remark that the integral of a function $f:{\mathbb{M}}^{n}\to[0,\infty)$ with
respect to the Hausdorff content ${\mathcal{H}}^{s}_{\infty}$ is given by

$\int_{{\mathbb{M}}^{n}}f\,d{\mathcal{H}}^{s}_{\infty}=\int_{0}^{\infty}{\mathcal{H}}^{s}_{\infty}(\\{x\in{\mathbb{M}}^{n}:f(t)>t\\})\,dt.$

Observe also that in the case when $\Omega$ satisfies the CDC, from (1.9) we
deduce that, for $0<s\leq n$,

$\lambda_{\Omega}-\lambda_{B}\gtrsim\bigg{(}\int_{B\setminus\Omega}\delta_{B}(x)^{n-s}\,d{\mathcal{H}}^{s}_{\infty}(x)\bigg{)}^{2}\geq\bigg{(}\int_{B\cap\partial\Omega}\delta_{B}(x)^{n-s}\,d{\mathcal{H}}^{s}_{\infty}(x)\bigg{)}^{2}.$

The example given by the ellipsoidal perturbations of the unit ball mentioned
above shows that this estimate is also sharp. For $0<s<n$, the domain of
integration in the middle term cannot be augmented to $B\triangle\Omega$, and
the one in the last term to the full $\partial\Omega$ because of the same
discussion after the statement of Theorem A.

A basic ingredient for the proof of Theorems A and B is Theorem 1.2 from
[AKN1]. The very rough strategy of the proof of both results is the following.
We consider ${\mathbb{M}}^{n}$ embedded in ${\mathbb{R}}^{n+1}$ and we
consider suitable extensions ${\widetilde{u}}_{B}$ and
${\widetilde{u}}_{\Omega}$ of the eigenfunctions $u_{B}$ and $u_{\Omega}$
appearing in (1.2), respectively, to some open subsets
${\widetilde{B}},{\widetilde{\Omega}}\subset{\mathbb{R}}^{n+1}$, so that
${\widetilde{u}}_{B}$ is harmonic in ${\widetilde{B}}$ and
${\widetilde{u}}_{\Omega}$ is “almost harmonic” in ${\widetilde{\Omega}}$
(i.e., $\Delta{\widetilde{u}}_{\Omega}$ is very small). Then we apply apply
(1.2) by estimating ${\widetilde{u}}_{B}-{\widetilde{u}}_{\Omega}$ from below
in terms of the harmonic measure for ${\widetilde{\Omega}}$, using the fact
that ${\widetilde{u}}_{B}(x)\approx{\rm dist}(x,\partial{\widetilde{B}})$
while ${\widetilde{u}}_{\Omega}(x)=0$ in a large part of
${\widetilde{B}}\cap\partial{\widetilde{\Omega}}$. To relate this estimate to
the terms on the right hand side of (1.3), (1.4), and (1.9), we consider an
auxiliary Lipschitz domain ${\widetilde{\Omega}}_{\Gamma}$ and we relate the
harmonic measure for ${\widetilde{\Omega}}$ to the one for
${\widetilde{\Omega}}_{\Gamma}$ using the maximum principle. Lastly we take
advantage of the fact that the behavior of harmonic measure is well understood
on Lipschitz domains.

###  1.2. The Alt-Caffarelli-Friedman monotonicity formula and the Friedland-
Hayman inequality

Given a domain in the unit sphere,
$\Omega\subset\mathbb{S}^{n}\subset{\mathbb{R}}^{n+1}$ whose first Dirichlet
eigenvalue is $\lambda_{\Omega}$, the characteristic constant of $\Omega$ is
the positive number $\alpha_{\Omega}$ such that
$\lambda_{\Omega}=\alpha_{\Omega}(n-1+\alpha_{\Omega})$.

Recall that the Alt-Caffarelli-Friedman (ACF) monotonicity formula asserts the
following:

######  Theorem 1.4.

Let $x\in{\mathbb{R}}^{n+1}$ and $R>0$. Let $u_{1},u_{2}\in
W^{1,2}(B(x,R))\cap C(B(x,R))$ be nonnegative subharmonic functions such that
$u_{1}(x)=u_{2}(x)=0$ and $u_{1}\cdot u_{2}\equiv 0$. Set

(1.10) $J(x,r)=\left(\frac{1}{r^{2}}\int_{B(x,r)}\frac{|\nabla
u_{1}(y)|^{2}}{|y-x|^{n-1}}dy\right)\cdot\left(\frac{1}{r^{2}}\int_{B(x,r)}\frac{|\nabla
u_{2}(y)|^{2}}{|y-x|^{n-1}}dy\right)$

Then $J(x,r)$ is an absolutely continuous function of $r\in(0,R)$ and

(1.11)
$\frac{\partial_{r}J(x,r)}{J(x,r)}\geq\frac{2}{r}\bigl{(}\alpha_{1}+\alpha_{2}-2\bigr{)}.$

where $\alpha_{i}$ is the characteristic constant of the open subset
$\Omega_{i}\subset{\mathbb{S}}^{n}$ given by

$\Omega_{i}=\bigl{\\{}r^{-1}(y-x):y\in\partial B(x,r),\,u_{i}(y)>0\bigr{\\}}.$

Further, for $r\in(0,R/2)$ and $i=1,2$, we have

(1.12) $\frac{1}{r^{2}}\int_{B(x,r)}\frac{|\nabla
u_{i}(y)|^{2}}{|y-x|^{n-1}}dy\lesssim\frac{1}{r^{n+1}}\|\nabla
u_{i}\|_{L^{2}(B(x,2r))}^{2}.$

The Friedland-Hayman [FH] inequality ensures that, for any two disjoint open
subsets $\Omega_{1},\Omega_{2}\subset{\mathbb{S}}^{n}$,

$\alpha_{1}+\alpha_{2}-2\geq 0,$

so that $J(x,r)$ is non-decreasing on $r$, by (1.11). In fact, more is known.
By Sperner’s inequality [Sp], among all the open subsets with a fixed measure
${\mathcal{H}}^{n}$ on ${\mathbb{S}}^{n}$, the one that minimizes the
characteristic constant is a spherical ball with the same measure
${\mathcal{H}}^{n}$. That is to say, if $B_{i}$ is a spherical ball such that
${\mathcal{H}}^{n}(B_{i})={\mathcal{H}}^{n}(\Omega_{i})$ and
$\bar{\alpha}_{i}$ denotes its characteristic constant, then

$\alpha_{i}\geq\bar{\alpha}_{i}.$

Further, if one of the sets $\Omega_{i}$ differs from a hemisphere by a
surface measure $h$, that is,

$\Big{|}{\mathcal{H}}^{n}(\Omega_{i})-\frac{1}{2}{\mathcal{H}}^{n}({\mathbb{S}}^{n})\Big{|}\geq
h,$

then

$\alpha_{1}+\alpha_{2}-2\geq c\,h^{2}.$

In this paper we will deduce other more precise estimates for
$\alpha_{1}+\alpha_{2}-2$ using Theorem B and Theorem 1.2. To state the
precise result, we need some additional notation. Let
$\Omega_{1},\Omega_{2}\subset{\mathbb{S}}^{n}$ be open and disjoint and let
$H\subset{\mathbb{R}}^{n+1}$ be half-space such that $0\in\partial H$. Denote

$S_{H,1}={\mathbb{S}}^{n}\cap H,\qquad
S_{H,2}={\mathbb{S}}^{n}\setminus\overline{H}.$

We denote

$V_{c_{0}}(\Omega_{1},\Omega_{2},H,a)=T_{c_{0}}(\Omega_{1},S_{H,1},a)\cup
T_{c_{0}}(\Omega_{2},S_{H,2},a).$

Notice that if both $\Omega_{1},\Omega_{2}$ satisfy the the CDC and $c_{0}$ is
small enough, then
$V_{c_{0}}=(\Omega_{1},\Omega_{2},H,a)=T_{c_{0}}(S_{H,1}\setminus\Omega_{1})\cup
T_{c_{0}}(S_{H,2}\setminus\Omega_{2})$.

For fixed $c_{0}>0$ and $a\in(0,1)$, we denote for $0<s<n$,

${\varepsilon}_{s}(\Omega_{1},\Omega_{2})=\inf_{H}\int_{V_{c_{0}}(\Omega_{1},\Omega_{2},H,a)}{\rm
dist}(y,\partial H)^{n-s}\,d{\mathcal{H}}_{\infty}^{s}(y),$

with the infimum taken over all half-space $H$ such that $0\in\partial H$. On
the other hand, for $s=n$, we set

${\varepsilon}_{n}(\Omega_{1},\Omega_{2})=\inf_{H}{\mathcal{H}}^{n}\big{(}(S_{H,1}\setminus\Omega_{1})\cup(S_{H,2}\setminus\Omega_{2})\big{)}.$

We will prove the following:

######  Theorem C.

Let $n\geq 2$ and $0<s\leq n$. Let
$\Omega_{1},\Omega_{2}\subset{\mathbb{S}}^{n}$ be open and disjoint. For any
$c_{0}>0$ and $a\in(0,1)$ we have

${\varepsilon}_{s}(\Omega_{1},\Omega_{2})^{2}\lesssim_{s,c_{0},a}\min(1,\alpha_{1}+\alpha_{2}-2).$

Remark that when ${\mathcal{H}}^{n}(\Omega_{i})\to 0$, we have
$\alpha_{i}\to\infty$, and thus $\alpha_{1}+\alpha_{2}-2\to\infty$ too. On the
other hand, we always have ${\varepsilon}_{s}(\Omega_{1},\Omega_{2})\lesssim
1$ by definition. This is the reason for writing
$\min(1,\alpha_{1}+\alpha_{2}-2)$ instead of $\alpha_{1}+\alpha_{2}-2$ on the
right hand side of the inequality in Theorem C.

###  1.3. Carleson’s conjecture

Next we introduce the precise notion of a tangent point for a pair of disjoint
open sets in ${\mathbb{R}}^{n+1}$. For a point $x\in{\mathbb{R}}^{n+1}$, a
unit vector $u$, and an aperture parameter $a\in(0,1)$ we consider the two
sided cone with axis in the direction of $u$ defined by

$X_{a}(x,u)=\bigl{\\{}y\in{\mathbb{R}}^{n+1}:|(y-x)\cdot u|>a|y-x|\bigr{\\}}.$

Given disjoint open sets $\Omega_{1},\Omega_{2}\subset{\mathbb{R}}^{n+1}$ and
$x\in\partial\Omega_{1}\cap\partial\Omega_{2}$, we say that $x$ is a tangent
point for the pair $\Omega_{1},\Omega_{2}$ if
$x\in{\partial\Omega}_{1}\cap{\partial\Omega}_{2}$ and there exists a unit
vector $u$ such that, for all $a\in(0,1)$, there exists some $r>0$ such that

$(\partial\Omega_{1}\cup\partial\Omega_{2})\cap X_{a}(x,u)\cap
B(x,r)=\varnothing,$

and moreover, one component of $X_{a}(x,u)\cap B(x,r)$ is contained in
$\Omega_{1}$ and the other in $\Omega_{2}$. The hyperplane $L$ orthogonal to
$u$ through $x$ is called a tangent hyperplane at $x$. In case that
$\Omega_{2}={\mathbb{R}}^{n+1}\setminus\overline{\Omega_{1}}$, we say that $x$
is a tangent point for $\Omega_{1}$.

Let $\Omega_{1}$ be a Jordan domain in ${\mathbb{R}}^{2}$, and set
$\Gamma=\partial\Omega_{1}$ and
$\Omega_{2}={\mathbb{R}}^{2}\setminus\overline{\Omega_{1}}$. For
$x\in{\mathbb{R}}^{2}$ and $r>0$, denote by $I_{1}(x,r)$ and $I_{2}(x,r)$ the
longest open arcs of the circumference $\partial B(x,r)$ contained in
$\Omega_{1}$ and $\Omega_{2}$, respectively (they may be empty). Then we
define

(1.13) ${\varepsilon}(x,r)=\frac{1}{r}\,\max\big{(}\big{|}\pi
r-{\mathcal{H}}^{1}(I_{1}(x,r))\big{|},\,\big{|}\pi
r-{\mathcal{H}}^{1}(I_{2}(x,r))\big{|}\big{)}.$

The Carleson ${\varepsilon}^{2}$-square function is given by

(1.14)
${\mathcal{E}}(x)^{2}:=\int_{0}^{1}{\varepsilon}(x,r)^{2}\,\frac{dr}{r}.$

Carleson’s conjecture, now a theorem, asserts the following.

######  Theorem 1.5.

Let $\Omega_{1}\subset{\mathbb{R}}^{2}$ be a Jordan domain, let
$\Gamma=\partial\Omega_{1}$, and let ${\mathcal{E}}$ be the associated square
function defined in (1.14). Then the set of tangent points for $\Omega_{1}$
coincides with the subset of those points $x\in\Gamma$ such that
${\mathcal{E}}(x)<\infty$, up to a set of zero measure ${\mathcal{H}}^{1}$. In
particular, the set $G=\\{x\in\Gamma:{\mathcal{E}}(x)<\infty\\}$ is
$1$-rectifiable.

Recall here that a set $E\subset{\mathbb{R}}^{n+1}$ is called $d$-rectifiable
if there are Lipschitz maps $f_{i}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{n+1}$,
$i\in{\mathbb{N}}$, such that

(1.15)
${\mathcal{H}}^{d}\Big{(}E\setminus\textstyle\bigcup_{i=1}^{\infty}f_{i}({\mathbb{R}}^{d})\Big{)}=0.$

The fact that ${\mathcal{E}}(x)<\infty$ for ${\mathcal{H}}^{1}$-a.e. tangent
point in a Jordan curve was proved by Bishop in [Bi1] (see also [BCGJ]). The
most difficult implication of Theorem 1.5, i.e, the fact that the set $G$ is
$1$-rectifiable and tangents to $\Gamma$ exist for ${\mathcal{H}}^{1}$-a.e.
$x\in G$, was proved more recently by Ben Jaye and the last two authors of
this paper [JTV].

It is natural to wonder about the existence of a suitable version of
Carleson’s conjecture in higher dimensions. There are two natural questions:
which coefficients should replace the coefficients ${\varepsilon}(x,r)$
defined in (1.13)? and second, for which open sets
$\Omega_{1}\subset{\mathbb{R}}^{n+1}$ should we expect to obtain a
characterization such as the one in Theorem 1.5? In this theorem, the fact
that $\Omega_{1}$ is a Jordan domain ensures that its boundary is connected,
which plays an essential role in the proof. Indeed, the arguments in [JTV] are
based on the connectivity of the boundary and they do not extend to higher
dimensions (this should not be a surprise, since typically the role of
connectivity in the plane is much more relevant than in higher dimensions for
many geometric problems).

Regarding the coefficients ${\varepsilon}(x,r)$, in [AKN2] the authors show a
very interesting connection with the characteristic constants of spherical
domains and the Friedland-Hayman inequality, which we proceed to describe.
Given two disjoint open sets $\Omega_{1},\Omega_{2}\subset{\mathbb{R}}^{n+1}$,
consider the open subsets $\Sigma_{1},\Sigma_{2}\subset{\mathbb{S}}^{n}$
defined by

(1.16) $\Sigma_{i}=\bigl{\\{}r^{-1}(y-x):y\in\Omega_{i}\cap\partial
B(x,r)\bigr{\\}},$

and let $\alpha_{i}(x,r)=\alpha_{\Sigma_{i}}$, the characteristic constant of
$\Sigma_{i}$. Analogously, set $\lambda_{i}(x,r)=\lambda_{\Sigma_{i}}$. In
[AKN2] it is remarked that, in the planar case $n=1$,

(1.17) $\alpha_{1}(x,r)+\alpha_{2}(x,r)-2\gtrsim{\varepsilon}(x,r)^{2}.$

This is easy to check. Indeed, without loss of generality, assume $x=0$,
$r=1$. Taking into account that the characteristic of a domain decreases as
its size increases, we have $\alpha_{I_{i}}=\alpha_{\Sigma_{i}}$ and also
$\lambda_{I_{i}}=\lambda_{\Sigma_{i}}$, for $I_{i}:=I_{i}(x,r)$ as in (1.13).
Let $\gamma_{i}={\mathcal{H}}^{1}(I_{i})/(2\pi)$. Since the first
eigenfunction for $I_{i}$ is the function
$u_{i}(\theta)=\sin((2\gamma_{i})^{-1}\theta)$ (modulo a translation in the
torus), we have $\alpha_{i}=\lambda_{i}^{1/2}=(2\gamma_{i})^{-1}$. Suppose,
for example, that
${\varepsilon}(x,r)=\big{|}\pi-{\mathcal{H}}^{1}(I_{1})\big{|}$ and write
$\alpha_{i}=\alpha_{i}(x,r)$. Let $\tilde{\alpha}_{2}$ the characteristic of
${\mathbb{S}}^{1}\setminus\overline{I_{1}}$. Since
$I_{2}\subset{\mathbb{S}}^{1}\setminus\overline{I_{1}}$, we have
$\alpha_{2}\geq\tilde{\alpha}_{2}$. Thus,

$\alpha_{1}+\alpha_{2}-2\geq\alpha_{1}+\tilde{\alpha}_{2}-2=\frac{1}{2\gamma_{1}}+\frac{1}{2(1-\gamma_{1})}-2=\frac{1-4\gamma_{1}(1-\gamma_{1})}{2\gamma_{1}(1-\gamma_{1})}=\frac{2(\frac{1}{2}-\gamma_{1})^{2}}{\gamma_{1}(1-\gamma_{1})}\approx\frac{{\varepsilon}(x,r)^{2}}{\gamma_{1}(1-\gamma_{1})},$

which completes the proof of (1.17), since $\gamma_{1}\in(0,1)$. Further, in
case that $I_{1}(x,r)$ and $I_{2}(x,r)$ are complementary arcs, arguing as
above, one can deduce

$\min\big{(}1,\alpha_{1}(x,r)+\alpha_{2}(x,r)-2\big{)}\approx{\varepsilon}(x,r)^{2}.$

See also [Bi2] for a very related discussion.

In view of the preceding discussion, a possible generalization of Carleson’s
conjecture to higher dimensions may consist in showing that, for a suitable
domain $\Omega_{1}\subset{\mathbb{R}}^{n+1}$ and
$\Omega_{2}={\mathbb{R}}^{n+1}\setminus\overline{\Omega_{1}}$, it holds

(1.18)
$\int_{0}^{1}\frac{\min(1,\alpha_{1}(x,r)+\alpha_{2}(x,r)-2)}{r}\,dr<\infty\qquad\Leftrightarrow\qquad\text{$x$
is a tangent point of $\Omega_{1}$,}$

for every $x\in\partial\Omega_{1}$, up to a set of zero measure
${\mathcal{H}}^{n}$.

In this paper we prove the implication $\Rightarrow$ in the above equivalence
(1.18) for a very large class of domains. Further, we do not ask the domains
$\Omega_{1},\Omega_{2}$ to be complementary. The precise result is the
following.

######  Theorem D.

Let $\Omega_{1},\Omega_{2}\subset{\mathbb{R}}^{n+1}$ be disjoint Wiener
regular domains in ${\mathbb{R}}^{n+1}$. Then, for ${\mathcal{H}}^{n}$-a.e.
tangent point $x$ for the pair $\Omega_{1},\Omega_{2}$, it holds

(1.19)
$\int_{0}^{1}\frac{\min(1,\alpha_{1}(x,r)+\alpha_{2}(x,r)-2)}{r}\,dr<\infty.$

Recall that a domain is called Wiener regular if the Dirichlet problem for the
Laplacian with continuous data is solvable in that domain. This is a property
which is implied by the CDC, which is a strictly stronger condition (see
Section 2.3). We will prove Theorem D by integrating the Alt-Caffarelli-
Friedman inequality (1.11) applied to the Green functions $u_{i}$ of
$\Omega_{i}$ and using the fact that the harmonic measure of $\Omega_{i}$
satisfies $\omega_{\Omega_{i}}(B(x,r))\gtrsim r^{n}$ as $r\to 0$ at
${\mathcal{H}}^{n}$-a.e. tangent point $x\in\partial\Omega_{i}$, which in turn
implies that $J(x,0)>0$:

$\int_{0}^{r_{0}}\frac{2}{r}\bigl{(}\alpha_{1}(x,r)+\alpha_{2}(x,r)-2\bigr{)}\,dr\leq\int_{0}^{r_{0}}\frac{\partial_{r}J(x,r)}{J(x,r)}\,dr=\log\frac{J(x,r_{0})}{J(x,0)}<\infty.$

Putting together Theorem C and Theorem D, for $\Omega_{1},\Omega_{2}$ as above
and for $0<s\leq n$, denoting
${\varepsilon}_{s}(x,r)={\varepsilon}_{s}(\Sigma_{1},\Sigma_{2})$, with
$\Sigma_{i}$ as in (1.16), we deduce that

$\int_{0}^{1}{\varepsilon}_{s}(x,r)^{2}\,\frac{dr}{r}<\infty$

for ${\mathcal{H}}^{n}$-a.e. tangent point $x$ for the pair
$\Omega_{1},\Omega_{2}$.

Finally, we remark that in the companion paper [FTV] we will prove the
converse implication in (1.18) when $\Omega_{1}\cup\Omega_{2}$ satisfies the
CDC. The precise result is the following.

######  Theorem E.

For $n\geq 1$, let $\Omega_{1},\Omega_{2}\subset{\mathbb{R}}^{n+1}$ be
disjoint open sets. The set of points $x\in{\mathbb{R}}^{n+1}$ such that

$\int_{0}^{1}{\varepsilon}_{n}(x,r)^{2}\,\frac{dr}{r}<\infty$

is $n$-rectifiable. In case that the set $\Omega_{1}\cup\Omega_{2}$ satisfies
the CDC, then ${\mathcal{H}}^{n}$-a.e. $x\in{\mathbb{R}}^{n+1}$ such that

$\int_{0}^{1}\frac{\min(1,\alpha_{1}(x,r)+\alpha_{2}(x,r)-2)}{r}\,dr<\infty$

is a tangent point for the pair $\Omega_{1},\Omega_{2}$.

We insist on the fact that the CDC for $\Omega_{1}\cup\Omega_{2}$ is a quite
mild lower regularity condition. For example, it holds if111In fact, in [Le]
it is shown that this is also a necessary condition for the CDC. there exist
$s\in(n-1,n+1]$ and $c>0$ such that

${\mathcal{H}}^{s}_{\infty}(B(x,r)\cap(\partial\Omega_{1}\cup\partial\Omega_{2}))\geq
c\,r^{s}\quad\mbox{ for all $x\in\partial\Omega_{1}\cup\partial\Omega_{2}$,
$0<r\leq 1$.}$

Because of the discussion above, we think that this result can be considered
as a natural extension of Carleson’s conjecture to higher dimensions. Observe
that a pair $\Omega_{1},\Omega_{2}$ consisting of a Jordan domain in the plane
and its complementary, as in [JTV], satisfies the assumptions in Theorem E. In
fact, in the case of the plane, the above theorem applies to domains much more
general than Jordan domains and so it is also new.

We remark that in [AKN2] the authors propose another possible extension of
Carleson’s conjecture to higher dimensions. Let

$a(x,r)^{2}:=|\lambda_{1}(x,r)-n|^{2}+|\lambda_{2}(x,r)-n|^{2}.$

Assuming that $\Omega_{2}={\mathbb{R}}^{n+1}\setminus\overline{\Omega_{1}}$,
they ask under what minimal assumptions on $\partial\Omega_{1}$ the set of
those points $x\in\partial\Omega_{1}$ such that

(1.20) $\int_{0}^{1}\frac{a(x,r)^{2}}{r}\,dr<\infty$

coincides with the rectifiable part of $\partial\Omega_{1}$, up to a set of
${\mathcal{H}}^{n}$ measure zero. As shown in [AKN1], one has

(1.21) $\min(1,a(x,r))^{2}\lesssim\min(1,\alpha_{1}(x,r)+\alpha_{2}(x,r)-2).$

Thus the condition (1.19) implies that

(1.22) $\int_{0}^{1}\frac{\min(1,a(x,r))^{2}}{r}\,dr<\infty,$

which is equivalent to (1.20) for “reasonable” domains, for example when both
$\Omega_{1}$ and ${\mathbb{R}}^{n+1}\setminus\Omega_{1}$ are connected and
have diameter larger than $1$. By (1.21) and Theorem D, the condition (1.22)
holds for tangent points $x\in\partial\Omega_{1}$ in the case when both
$\Omega_{1}$ and ${\mathbb{R}}^{n+1}\setminus\Omega_{1}$ are Wiener regular.
However, the inequality converse to the one in (1.21) does not hold in
general, and so it is not clear to us if the condition (1.22) implies the
existence of tangents for domains $\Omega_{1},\Omega_{2}$ such as the ones in
Theorem E.

###  1.4. A counterexample

For $n\geq 2$, let $u,v:B(0,10)\to{\mathbb{R}}$ be two nonnegative continuous
functions satisfying

$-\Delta u\leq 0$ in $\\{u>0\\}$, $-\Delta v\leq 0$ in $\\{v>0\\}$ and $uv=0$
in $B(0,10)$.

Define $\Gamma^{*}:=\\{x\,:\,J(x,0^{+})>0\\}$, where $J(0^{+}):=\lim_{r\to
0^{+}}J(x,r)$ and $J(x,r)$ is the ACF monotonicity formula defined in (1.10)
relative to $u,v$. In [AKN2], Allen, Kriventsov ans Neumayer prove the
following theorem.

######  Theorem 1.6 ([AKN2, Theorem 1.2]).

For $n\geq 2$, let $u,v$ as above. Then $\Gamma^{*}\cap B(0,1)$ is
$(n-1)$-rectifiable.

In the same work, see [AKN2, Problem 2.8], the authors pose the converse
question: suppose that $E=\partial\\{u>0\\}\cap\partial\\{v>0\\}$ is
$(n-1)$-rectifiable. Is it true, then, that $J(x,0^{+})>0$ for
${\mathcal{H}}^{n-1}$-almost every $x\in E$ (under some minimal assumptions on
$E$ and $u,v$)? In the last section of this paper, we provide an example which
shows that, without further structural assumptions on the domains, the natural
converse to Theorem 1.6 is false. More precisely we construct two Wiener
regular domains $\Omega_{1},\Omega_{2}$ in ${\mathbb{R}}^{2}$ with
$1$-rectifiable boundary and such that, if
$E=\partial\Omega_{1}\cap\partial\Omega_{2}$, then ${\mathcal{H}}^{1}(E)>0$
and the limit $J(x,0^{+})$ of the ACF functional associated to the Green
functions of $\Omega_{1},\Omega_{2}$ equals to $0$ for
${\mathcal{H}}^{1}$-almost every point $x\in E$. See Section 9 for the
detailed construction. On the other hand, it is implicit in the proof of
Theorem D that $J(x,0^{+})>0$ whenever $x$ is a (true) tangent point of $E$
and $u$ and $v$ are the Green functions of disjoint Wiener regular domains (up
to a set of ${\mathcal{H}}^{n-1}$ measure zero).

Acknowledgements. This work was initiated while the authors were in residence
at the Hausdorff Institute of Mathematics in Spring 2022 during the program
“Interactions between geometric measure theory, singular integrals, and PDEs”.
Other parts of this work took place during a two weeks visit of M. Villa to
Princeton University and during another one month visit of I. Fleschler to the
Universitat Autònoma de Barcelona. I. Fleschler and M. Villa would like to
thank G. De Philippis for some inspiring conversations in some early stages of
this work. We also thank G. David for providing us some information in
connection with the Friedland-Hayman inequality.

##  2\. Preliminaries

###  2.1. Miscellaneous notation

In the paper, constants denoted by $C$ or $c$ depend just on the dimension and
perhaps other fixed parameters, such as $a,\beta$ in Theorem A, for example.
We will write $a\lesssim b$ if there is $C>0$ such that $a\leq Cb$ . We write
$a\approx b$ if $a\lesssim b\lesssim a$.

Open balls in ${\mathbb{R}}^{n+1}$ centered in $x$ with radius $r>0$ are
denoted by $B(x,r)$, and closed balls by $\bar{B}(x,r)$. For an open or closed
ball $B\subset{\mathbb{R}}^{n+1}$ with radius $r$, we write
${\operatorname{rad}}(B)=r$. Similarly, open and closed balls in
${\mathbb{M}}^{n}$ are denoted by $B_{{\mathbb{M}}^{n}}(x,r)$ and
$\bar{B}_{{\mathbb{M}}^{n}}(x,r)$ respectively. For an open or closed ball
$B\subset{\mathbb{M}}^{n}$ with radius $r$, we write
${\operatorname{rad}}_{{\mathbb{M}}^{n}}(B)=r$. An open annulus in
${\mathbb{R}}^{n+1}$ centered in $x$ with inner radius $r_{1}$ and outer
radius $r_{2}$ is denoted by $A(x,r_{1},r_{2})$, and the corresponding closed
annulus by $\bar{A}(x,r_{1},r_{2})$. Similarly, the analogous annuli in
${\mathbb{M}}^{n}$ are denoted by $A_{{\mathbb{M}}^{n}}(x,r_{1},r_{2})$ and
$\bar{A}_{{\mathbb{M}}^{n}}(x,r_{1},r_{2})$, respectively.

We use the two notations $S(x,r)\equiv\partial B(x,r)$ for spheres in
${\mathbb{R}}^{n+1}$ centered in $x$ with radius $r$, so that
${\mathbb{S}}^{n}=S(0,1)$.

We call open sets in ${\mathbb{S}}^{n}$ spherical domains, and the geodesic
distance in ${\mathbb{M}}^{n}$ is written as ${\rm dist}_{{\mathbb{M}}^{n}}$.
Notice that, assuming ${\mathbb{S}}^{n}$ to be embedded in
${\mathbb{R}}^{n+1}$, the geodesic distance ${\rm dist}_{{\mathbb{S}}^{n}}$ is
comparable to the Euclidean distance in the ambient space
${\mathbb{R}}^{n+1}$.

Given a set $F\subset{\mathbb{R}}^{n+1}$, the notation $M_{+}(F)$ stands for
the set of (positive) Radon measures supported on $F$. For $s>0$, the
$s$-dimensional Hausdorff measure is denoted by ${\mathcal{H}}^{s}$, and the
$s$-dimensional Hausdorff content by ${\mathcal{H}}^{s}_{\infty}$. Recall
that, for any set $E\subset{\mathbb{R}}^{n+1}$,

${\mathcal{H}}^{s}_{\infty}(E)=\inf\Big{\\{}\sum_{i}{\rm
diam}(A_{i})^{s}:E\subset\bigcup_{i}A_{i}\Big{\\}}.$

We say that $E$ is lower $s$-content regular if there exists some constant
$c>0$ such that

${\mathcal{H}}^{s}_{\infty}(B(x,r)\cap E)\geq c\,r^{s}\quad\text{ for all
$x\in E$ and $0<r\leq{\rm diam}(E)$.}$

The barycenter of a set $E\subset{\mathbb{R}}^{n}$ is defined by

$x_{E}=\int_{E}x\,d{\mathcal{H}}^{n}(x).$

On the other hand, the ${\mathbb{S}}^{n}$-barycenter of
$E\subset{\mathbb{S}}^{n}$ is defined by $x_{E}/|x_{E}|$, with $x_{E}$ as
above. So this belongs to ${\mathbb{S}}^{n}$ and it is defined only when
$x_{E}\neq 0$.

###  2.2. Capacities

The fundamental solution of the minus Laplacian in ${\mathbb{R}}^{n}$, for
$n\geq 3$, equals

${\mathcal{E}}_{n}(x)=\frac{c_{n}}{|x|^{n-2}},$

where $c_{n}=(n-2){\mathcal{H}}^{n-1}(\mathbb{S}^{n-1})$. In the plane, the
fundamental solution is

${\mathcal{E}}_{2}(x)=\frac{1}{2\pi}\,\log\frac{1}{|x|}.$

For a Radon measure $\mu$, we consider the potential defined by

$U_{n-2}\mu(x)={\mathcal{E}}_{n}*\mu(x).$

Given a set $F\subset{\mathbb{R}}^{n}$ (or, more generally,
$F\subset{\mathbb{R}}^{m}$) we define the capacity
$\operatorname{Cap}_{n-2}(F)$ by the identity

(2.1) $\operatorname{Cap}_{n-2}(F)=\frac{1}{\inf_{\mu\in
M_{1}(F)}I_{n-2}(\mu)},$

where the infimum is taken over all probability measures $\mu$ supported on
$F$ and $I_{n-2}(\mu)$ is the energy

(2.2) $I_{n-2}(\mu)=\iint{\mathcal{E}}_{n}(x-y)\,d\mu(x)\,d\mu(y)=\int
U_{n-2}\mu(x)\,d\mu(x).$

For $n\geq 3$ and $F\subset{\mathbb{R}}^{n}$, $\operatorname{Cap}_{n-2}(F)$ is
the Newtonian capacity of $F$, and for $n=2$ and $F\subset{\mathbb{R}}^{2}$,
$\operatorname{Cap}_{0}(F)$ is the Wiener capacity of $F$. Remark that for
$n\geq 3$, one also has

(2.3) $\operatorname{Cap}_{n-2}(F)=\sup\big{\\{}\mu(F):\mu\in
M_{+}(F),\,\|U_{n-2}\mu\|_{\infty,F}\leq 1\big{\\}},$

where $M_{+}(F)$ is the family of all Radon measures supported in $F$. In the
plane, the analogous identity with $n=2$ holds when ${\rm diam}F<1$.

It is easy to check that the Newtonian capacity $\operatorname{Cap}_{n-2}$ is
homogeneous of degree $n-2$ when $n\geq 3$. The Wiener capacity is not
homogeneous. However, the logarithmic capacity $\operatorname{Cap}_{L}$,
defined by

$\operatorname{Cap}_{L}(F)=e^{-\frac{2\pi}{\operatorname{Cap}_{0}(F)}},$

is $1$-homogeneous.

For $n\geq 2$, let $\Omega\subset{\mathbb{R}}^{n}$ be an open set. We say that
$\Omega$ satisfies the $n$-dimensional capacity density condition (CDC) if
there exists some constant $c>0$ such that, for every $x\in\partial\Omega$ and
$r\in(0,{\rm diam}(\Omega))$,

$\operatorname{Cap}_{n-2}(B(x,r)\setminus\Omega)\geq c\,r^{n-2}\quad\mbox{in
the case $n\geq 3$,}$

and

$\operatorname{Cap}_{L}(B(x,r)\setminus\Omega)\geq c\,r\quad\mbox{in the case
$n=2$.}$

For subsets $F\subset{\mathbb{S}}^{n}$ and domains
$\Omega\subset{\mathbb{S}}^{n}$, we define $\operatorname{Cap}_{n-2}(F)$ in
the same way as for $F\subset{\mathbb{R}}^{n}$ and domains
$\Omega\subset{\mathbb{R}}^{n}$ (using the fundamental solution of the
Laplacian in ${\mathbb{R}}^{n}$). We define the $n$-dimensional CDC in
${\mathbb{S}}^{n}$ by asking the following condition for every
$x\in\partial\Omega$ and $r\in(0,{\rm diam}(\Omega))$:

$\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(x,r)\setminus\Omega)\geq
c\,r^{n-2}\quad\mbox{in the case $n\geq 3$,}$

and

$\operatorname{Cap}_{L}(B_{{\mathbb{S}}^{n}}(x,r)\setminus\Omega)\geq
c\,r\quad\mbox{in the case $n=2$.}$

We will need the following auxiliary results.

######  Lemma 2.1.

Let $E\subset{\mathbb{R}}^{n}$ or $E\subset{\mathbb{S}}^{n}$ be compact and
$n-2<s\leq n$. In the case $n>2$, we have

$\operatorname{Cap}_{n-2}(E)\gtrsim_{s,n}{\mathcal{H}}_{\infty}^{s}(E)^{\frac{n-2}{s}}.$

In the case $n=2$, we have

$\operatorname{Cap}_{L}(E)\gtrsim_{s}{\mathcal{H}}_{\infty}^{s}(E)^{\frac{1}{s}}.$

The proof of this result is an immediate consequence of Frostman’s Lemma. See
[Ma, Chapter 8] for the case $n>2$, and [CTV, Lemma 4] for the case $n=2$, for
example.

######  Lemma 2.2.

Let $\ell>0$, consider a compact set $F\subset{\mathbb{R}}^{n}$, and denote
${\widetilde{F}}=F\times[0,\ell]\subset{\mathbb{R}}^{n+1}$. In the case $n\geq
3$, we have

$\operatorname{Cap}_{n-1}({\widetilde{F}})\gtrsim\ell\,\operatorname{Cap}_{n-2}(F).$

In the case $n=2$, given $C_{1}\geq 1$, if ${\rm diam}(F)\leq C_{1}\,\ell$, we
have

$\operatorname{Cap}_{1}({\widetilde{F}})\gtrsim\frac{\ell}{\log\dfrac{2C_{1}\ell}{\operatorname{Cap}_{L}(F)}},$

with the implicit constant depending on $C_{1}$.

###### Proof.

By the $n-1$ homogeneity of $\operatorname{Cap}_{n+1}(F)$ and of the
inequalities stated in the lemma we can assume ${\rm diam}(F)<1$, so that
(2.3) also holds in the case $n=2$. Let $\mu\in M_{+}(F)$ be such that
$U_{n-2}\mu(x)\leq 1$ for all $x\in F$ and
$\mu(F)=\operatorname{Cap}_{n-2}(F)$ (it is well known that such measure,
called equilibrium measure, exists for any compact set $F$). Denote
$I_{\ell}=[0,\ell]$. Consider the product measure
${\widetilde{\mu}}=\mu\times{\mathcal{H}}^{1}|_{I_{\ell}}$, which is supported
on ${\widetilde{F}}$. For $x^{\prime}=(x,x_{n+1})\in{\widetilde{F}}$, we have

(2.4)
$U_{n-1}{\widetilde{\mu}}(x^{\prime})=\\!\int\\!\frac{c_{n}}{|x^{\prime}-y^{\prime}|^{n-1}}\,d{\widetilde{\mu}}(y^{\prime})\approx\\!\int_{y_{n+1}\in
I_{\ell}}\int_{y\in
F}\frac{1}{|x-y|^{n-1}+|x_{n+1}-y_{n+1}|^{n-1}}\,d{\mathcal{H}}^{1}(y_{n+1})d\mu(y).$

For each $x,y\in F$, we have

$\int_{y_{n+1}\in
I_{\ell}}\frac{1}{|x-y|^{n-1}+|x_{n+1}-y_{n+1}|^{n-1}}\,d{\mathcal{H}}^{1}(y_{n+1})\\\
=\int_{|x_{n+1}-y_{n+1}|\leq|x-y|}\\!\\!\\!\cdots\,d{\mathcal{H}}^{1}|_{I_{\ell}}(y_{n+1})+\int_{|x-y|<|x_{n+1}-y_{n+1}|\leq\ell}\\!\\!\\!\cdots\,d{\mathcal{H}}^{1}|_{I_{\ell}}(y_{n+1})=:T_{1}(x,y)+T_{2}(x,y).$

Concerning $T_{1}(x,y)$, we have

$T_{1}(x,y)\leq\int_{|x_{n+1}-y_{n+1}|\leq|x-y|}\frac{1}{|x-y|^{n-1}}\,d{\mathcal{H}}^{1}|_{I_{\ell}}(y_{n+1})\leq\frac{1}{|x-y|^{n-2}}.$

Also,

$T_{2}(x,y)\leq\int_{|x-y|<|x_{n+1}-y_{n+1}|\leq\ell}\frac{1}{|x_{n+1}-y_{n+1}|^{n-1}}\,d{\mathcal{H}}^{1}|_{I_{\ell}}(y_{n+1}).$

In the case $n\geq 3$, it easily follows also that

$T_{2}(x,y)\lesssim\frac{1}{|x-y|^{n-2}}.$

Thus,

$U_{n-1}{\widetilde{\mu}}(x^{\prime})\lesssim\int_{y\in
F}\frac{1}{|x-y|^{n-2}}\,d\mu(y)\approx U_{n-2}\mu(x)\leq 1.$

Consequently,

$\operatorname{Cap}_{n-1}({\widetilde{F}})\geq\frac{{\widetilde{\mu}}(F)}{\|U_{n-1}{\widetilde{\mu}}\|_{\infty,{\widetilde{F}}}}\gtrsim{\widetilde{\mu}}({\widetilde{F}})=\ell\,\mu(F)\approx\ell\,\operatorname{Cap}_{n-2}(F)\quad\mbox{
for $n\geq 3$}.$

In the case $n=2$, if $T_{2}(x,y)\neq 0$, we have

$T_{2}(x,y)\leq\log\frac{\ell}{|x-y|}.$

Since $|x-y|\leq C_{1}\ell$ for any $x,y\in F$, we can always write
$T_{2}(x,y)\leq\log\frac{C_{1}\ell}{|x-y|},$ and so

$T_{1}(x,y)+T_{2}(x,y)\leq
1+\log\frac{C_{1}\ell}{|x-y|}\lesssim\log\frac{2C_{1}\ell}{|x-y|}.$

Thus,

$\displaystyle U_{1}{\widetilde{\mu}}(x^{\prime})$
$\displaystyle\lesssim\int_{y\in
F}\log\frac{2C_{1}\ell}{|x-y|}\,d\mu(y)=\log(2C_{1}\ell)\,\mu(F)+\int_{y\in
F}\log\frac{1}{|x-y|}\,d\mu(y)$
$\displaystyle=\log(2C_{1}\ell)\,\mu(F)+2\pi\,U_{0}\mu(x)\leq\log(2C_{1}\ell)\,\mu(F)+2\pi.$

Therefore,

$\operatorname{Cap}_{1}(F)\geq\frac{{\widetilde{\mu}}(F)}{\|U_{1}{\widetilde{\mu}}\|_{\infty,{\widetilde{F}}}}\gtrsim\frac{\ell\,\mu(F)}{\log(2C_{1}\ell)\,\mu(F)+2\pi}=\frac{\ell}{\log(2C_{1}\ell)+\frac{2\pi}{\mu(F)}}.$

Writing

$\frac{2\pi}{\mu(F)}=\frac{2\pi}{\operatorname{Cap}_{0}(F)}=\log\frac{1}{\operatorname{Cap}_{L}(F)},$

we obtain

$\operatorname{Cap}_{1}(F)\gtrsim\frac{\ell}{\log(2C_{1}\ell)+\log\frac{1}{\operatorname{Cap}_{L}(F)}}=\frac{\ell}{\log\frac{2C_{1}\ell}{\operatorname{Cap}_{L}(F)}}.$

∎

######  Lemma 2.3.

Let $\ell>0$, $F\subset{\mathbb{S}}^{n}$ compact, and let $I_{\ell}$ be an
interval of length $\ell$ contained in $[1/4,2]$. Denote

${\widetilde{F}}=\\{x^{\prime}\in{\mathbb{R}}^{n+1}:x^{\prime}=tx\mbox{ for
some $x\in F$ and $t\in I_{\ell}$}\\}.$

Then we have

$\operatorname{Cap}_{n-1}({\widetilde{F}})\gtrsim\ell\,\operatorname{Cap}_{n-2}(F)\quad\mbox{in
the case $n\geq 3$,}$

and, in the case $n=2$, if ${\rm diam}(F)\leq C_{1}\,\ell$,

$\operatorname{Cap}_{1}({\widetilde{F}})\gtrsim\frac{\ell}{\log\frac{2C_{1}\ell}{\operatorname{Cap}_{L}(F)}}.$

###### Proof.

The proof is very similar to the one of Lemma 2.2. We assume ${\rm
diam}(F)<1$, so that (2.3) holds in the case $n=2$. We take $\mu\in M_{+}(F)$
be such that $U_{n-2}\mu(x)\leq 1$ for all $x\in F$ and
$\mu(F)=\operatorname{Cap}_{n-2}(F)$. Then we consider the “product measure”
${\widetilde{\mu}}$ defined by

$\int f\,d{\widetilde{\mu}}=\int_{t\in I_{\ell}}\int_{x\in
F}f(tx)\,d\mu(x)\,dt,\quad\text{ for $f\in C({\mathbb{R}}^{n+1})$}.$

Notice that ${\widetilde{\mu}}$ is supported on ${\widetilde{F}}$.

For $x^{\prime}=sx\in{\widetilde{F}}$, with $x\in F$, $s\in I_{\ell}$, we have

$U_{n-1}{\widetilde{\mu}}(x^{\prime})=\int\frac{c_{n}}{|x^{\prime}-y^{\prime}|^{n-1}}\,d{\widetilde{\mu}}(y^{\prime})=\int_{t\in
I_{\ell}}\int_{y\in F}\frac{c_{n}}{|sx-ty|^{n-1}}\,d\mu(y)\,dt$

Now we claim that

$|sx-ty|\gtrsim|x-y|+|s-t|.$

Indeed, we have

$|s-t|=||sx|-|ty||\leq|sx-ty|.$

Also,

$|x-y|=\frac{1}{s}\,|sx-sy|\leq\frac{1}{s}\,\big{(}|sx-ty|+|ty-
sy|\big{)}\approx|sx-ty|+|t-s|\lesssim|sx-ty|.$

Adding the last two inequalities, the claim follows. Then we deduce

$U_{n-1}{\widetilde{\mu}}(x^{\prime})\lesssim\int_{t\in I_{\ell}}\int_{y\in
F}\frac{1}{|x-y|^{n-1}+|s-t|^{n-1}}\,d\mu(y)dt$

Notice now the similarity between this inequality and (2.4). Then by the same
arguments following (2.4) in the proof of Lemma 2.2 we obtain the desired
estimates. ∎

In the next lemma we show the equivalence of the statement of Theorem A with
different values of the parameter $a$ in the case $n\geq 3$.

######  Lemma 2.4.

Given $n\geq 3$, let ${\mathbb{M}}^{n}$ be either ${\mathbb{R}}^{n}$ or
${\mathbb{S}}^{n}$ and let $\beta>0$. Let $\Omega$ be a relatively open subset
of ${\mathbb{M}}^{n}$ and let $B$ be a geodesic ball in ${\mathbb{M}}^{n}$
such that ${\mathcal{H}}^{n}(B)={\mathcal{H}}^{n}(\Omega)$ with radius
$r_{B}$. In the case ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, suppose also that
$\beta\leq{\mathcal{H}}^{n}(\Omega)\leq{\mathcal{H}}^{n}({\mathbb{S}}^{n})-\beta$,
and in the case ${\mathbb{M}}^{n}={\mathbb{R}}^{n}$ just that
${\mathcal{H}}^{n}(\Omega)\leq\beta$. For $a,a^{\prime}\in(0,1)$, we have

$\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x)\\\
\approx_{a,a^{\prime}}\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,a^{\prime}tr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x).$

###### Proof.

To shorten notation, for any $a,t\in(0,1)$, write

$I_{a,t}:=\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x).$

Assume $a>a^{\prime}$. Then it is clear that, for every $t\in(0,1)$,
$I_{a,t}\geq I_{a^{\prime},t}$ since
$\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)\geq\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,a^{\prime}tr_{B})\setminus\Omega)$.
So is suffices to show that
$I_{a,t}\lesssim_{a,a^{\prime}}\sup_{s\in(0,1)}I_{a^{\prime},s}$.

Denote by $x_{B}$ the center of $B$. For a given $t\in(0,1)$ and
$r=(1-t)r_{B}$, we consider a covering of
$\partial_{{\mathbb{M}}^{n}}B_{{\mathbb{M}}^{n}}(x_{B},r)$ by a family of
${\mathbb{M}}^{n}$-balls $\\{B_{i}\\}_{i\in I_{r}}$ (where $I_{r}$ is just a
suitable set of indices) centered in
$\partial_{{\mathbb{M}}^{n}}B_{{\mathbb{M}}^{n}}(x_{B},r)$ with radius
$b(r_{B}-r)=btr_{B}$, with $b\ll a^{\prime}$ to be chosen below, so that the
family $\\{B_{i}\\}_{i\in I_{r}}$ has bounded overlap. Then, denoting by
$x_{i}$ the center of $B_{i}$, we split

$\displaystyle\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}$
$\displaystyle(x,atr_{B})\setminus\Omega)\,d{\mathcal{H}}^{n-1}(x)$
$\displaystyle\leq\sum_{i\in
I_{r}}\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)\cap
B_{i}}\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)\,d{\mathcal{H}}^{n-1}(x)$
$\displaystyle\lesssim\sum_{i\in I_{r}}\sup_{x\in
B_{i}}\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)\,{\operatorname{rad}}_{{\mathbb{M}}^{n}}(B_{i})^{n-1}$
$\displaystyle\leq\sum_{i\in
I_{r}}\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x_{i},(a+b)tr_{B})\setminus\Omega)\,(tr_{B})^{n-1}.$

Next we wish to cover the annulus
$A_{r}=A_{{\mathbb{M}}^{n}}(x_{B},r-atr_{B},r+atr_{B})$ by another suitable
family of balls. First we let $N$ be the least integer such that $N\geq 2a/b$,
and then we consider the radii

$r_{k}=r+k\,\frac{atr_{B}}{N}\quad\mbox{ for $-N\leq k\leq N$.}$

So we have $[r-atr_{B},r+atr_{B}]=\bigcup_{k=-N}^{N-1}[r_{k},r_{k+1}]$, and
$r_{k+1}-r_{k}=\frac{atr_{B}}{N}\leq\frac{btr_{B}}{2}$. It is easy to check
that there exist covering of $A_{r}$ by a family of ${\mathbb{M}}^{n}$-balls
$\Delta_{j}$, $j\in J_{r}$, with
${\operatorname{rad}}_{{\mathbb{M}}^{n}}(\Delta_{j})=btr_{B}$, which are
centered in $\bigcup_{k=-N}^{N}\partial_{{\mathbb{M}}^{n}}B(x_{B},r_{k})$, and
have bounded overlap. We write $j\in J_{r,k}$ if $\Delta_{j}$ is centered in
$\partial_{{\mathbb{M}}^{n}}B(x_{B},r_{k})$. For $i\in I_{r}$, we also write
${\widetilde{B}}_{i}=B_{{\mathbb{M}}^{n}}(x_{i},(a+b)tr_{B})$ to shorten
notation. By the subadditivity of $\operatorname{Cap}_{n-2}$, then we have

(2.5) $\displaystyle\sum_{i\in
I_{r}}\operatorname{Cap}_{n-2}({\widetilde{B}}_{i}\setminus\Omega)\,(tr_{B})^{n-1}$
$\displaystyle\leq\sum_{i\in
I_{r}}\sum_{j:\Delta_{j}\cap{\widetilde{B}}_{i}\neq\varnothing}\operatorname{Cap}_{n-2}(\Delta_{j}\setminus\Omega)\,(tr_{B})^{n-1}$
$\displaystyle\leq C(a,b)\sum_{j\in
J_{r}}\operatorname{Cap}_{n-2}(\Delta_{j}\setminus\Omega)\,(tr_{B})^{n-1},$

taking into account that for each $\Delta_{j}$ there is a bounded number
(depending on $a$ and $b$) of balls ${\widetilde{B}}_{i}$ which intersect it.
Using also that, for any $x\in\Delta_{j}$, $\Delta_{j}\subset
B_{{\mathbb{M}}^{n}}(x,2btr_{B})$, we get

$\displaystyle\sum_{j\in
J_{r}}\operatorname{Cap}_{n-2}(\Delta_{j}\setminus\Omega)\,(tr_{B})^{n-1}$
$\displaystyle\approx_{b}\sum_{k=-N}^{N}\sum_{j\in
J_{r,k}}\operatorname{Cap}_{n-2}(\Delta_{j}\setminus\Omega)\,{\operatorname{rad}}_{{\mathbb{M}}^{n}}(\Delta_{j})^{n-1}$
$\displaystyle\lesssim\sum_{k=-N}^{N}\sum_{j\in
J_{r,k}}\int_{\partial_{{\mathbb{M}}^{n}}B(x_{B},r_{k})\cap\Delta_{j}}\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,2btr_{B})\setminus\Omega)\,d{\mathcal{H}}^{n-1}(x)$
$\displaystyle\lesssim\sum_{k=-N}^{N}\int_{\partial_{{\mathbb{M}}^{n}}B(x_{B},r_{k})}\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,2btr_{B})\setminus\Omega)\,d{\mathcal{H}}^{n-1}(x)$

Writing $t_{k}=\frac{r_{B}-r_{k}}{r_{B}}$ and taking into account that
$t_{k}\approx_{a}t$, it easily follows that

$\displaystyle I_{a,t}$
$\displaystyle=\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t)B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,atr_{B})\setminus\Omega)}{(t\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x)$
$\displaystyle\lesssim
C(a,b)\sum_{k=-N}^{N}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{n}}((1-t_{k})B)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,Cbt_{k}r_{B})\setminus\Omega)}{(t_{k}\,r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(x)$
$\displaystyle\lesssim C^{\prime}(a,b)\sup_{s\in(0,1)}I_{Cb,s}.$

So choosing $b$ so that $Cb\leq a^{\prime}$, we deduce
$I_{a,t}\lesssim_{a,a^{\prime}}\sup_{s\in(0,1)}I_{a^{\prime},s}$, as wished. ∎

The following lemma is the analogue of the preceding result for the case
$n=2$.

######  Lemma 2.5.

Let ${\mathbb{M}}^{2}$ be either ${\mathbb{R}}^{2}$ or ${\mathbb{S}}^{2}$, and
let $\beta>0$. Let $\Omega$ be a relatively open subset of ${\mathbb{M}}^{2}$
and let $B$ be a geodesic ball in ${\mathbb{M}}^{2}$ such that
${\mathcal{H}}^{2}(B)={\mathcal{H}}^{2}(\Omega)$ with radius $r_{B}$. In the
case ${\mathbb{M}}^{2}={\mathbb{S}}^{2}$, suppose also that
$\beta\leq{\mathcal{H}}^{2}(\Omega)\leq{\mathcal{H}}^{2}({\mathbb{S}}^{2})-\beta$,
and in the case ${\mathbb{M}}^{2}={\mathbb{R}}^{2}$ just that
${\mathcal{H}}^{2}(\Omega)\leq\beta$. For $a,a^{\prime}\in(0,1)$, we have

$\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{2}}((1-t)B)}\frac{t\,r_{B}}{\log\frac{2tar_{B}}{\operatorname{Cap}_{L}(B_{{\mathbb{M}}^{2}}(x,atr_{B})\setminus\Omega)}}\,d{\mathcal{H}}^{1}(x)\\\
\approx\sup_{t\in(0,1)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{M}}^{2}}((1-t)B)}\frac{t\,r_{B}}{\log\frac{2ta^{\prime}r_{B}}{\operatorname{Cap}_{L}(B_{{\mathbb{M}}^{2}}(x,a^{\prime}tr_{B})\setminus\Omega)}}\,d{\mathcal{H}}^{1}(x).$

###### Proof.

The arguments are very similar to the ones for the preceding lemma with $n\geq
3$. The main change in the proof is that instead of using the subadditivity of
$\operatorname{Cap}_{n-2}$, we use the fact that, by Theorem 5.1.4 from [Ra],

$\displaystyle\frac{1}{\log\frac{2tar_{B}}{\operatorname{Cap}_{L}(B_{{\mathbb{M}}^{2}}(x,atr_{B})\setminus\Omega)}}\leq\sum_{j:\Delta_{j}\cap{\widetilde{B}}_{i}\neq\varnothing}\frac{1}{\log\frac{2tar_{B}}{\operatorname{Cap}_{L}(\Delta_{j}\setminus\Omega)}}\leq\sum_{j:\Delta_{j}\cap{\widetilde{B}}_{i}\neq\varnothing}\frac{1}{\log\frac{2\,{\operatorname{rad}}_{{\mathbb{S}}^{2}}(\Delta_{j})}{\operatorname{Cap}_{L}(\Delta_{j}\setminus\Omega)}}.$

We leave the details for the reader. ∎

###  2.3. Harmonic measure

A bounded open set $\Omega\subset{\mathbb{R}}^{n+1}$ is Wiener regular if for
any continuous $f:{\partial\Omega}\to{\mathbb{R}}$ there exists the solution
of the Dirichlet problem for the Laplace equation. The Wiener criterion
characterizes the Wiener regularity of any bounded open. For our purposes, it
suffices to know that, in the case $n\geq 2$, if for any $x\in\partial\Omega$
there exists some $r_{x}>0$ and $c=c(x)>0$ such that

(2.6) $\operatorname{Cap}_{n-1}(B(x,r)\setminus\Omega)\geq
c\,r^{n-1}\quad\mbox{ for $0<r\leq r_{x}$,}$

then $\Omega$ is Wiener regular. In the case $n=1$, instead a sufficient
condition for Wiener regularity is that
$\operatorname{Cap}_{L}(B(x,r)\setminus\Omega)\geq c\,r$ for $0<r\leq r_{x}$,
with $c=c(x)>0$.

Recall that if $\Omega\subset{\mathbb{R}}^{n+1}$ is a bounded Wiener regular
open set and $f:\overline{\Omega}\to{\mathbb{R}}$ is continuous and of class
$C^{2}$ in $\Omega$, then

$f(x)=\int_{\partial\Omega}f\,d\omega_{\Omega}^{x}-\int_{\Omega}\Delta
f(y)\,g_{\Omega}(x,y)\,dy,$

where $g_{\Omega}$ stands for the Green function for $\Omega$.

The following result is well known222In some references, it is called
“Bourgain’s lemma”.. See Section 2.1 from [To], for example.

######  Lemma 2.6.

For $n\geq 1$ there exists $c(n)>0$ such that, given an open set
$\Omega\subset{\mathbb{R}}^{n+1}$ and a closed ball $B$ intersecting
${\partial\Omega}$, it holds

$\omega^{x}(B)\geq
c(n)\,\frac{\operatorname{Cap}_{n-1}(\tfrac{1}{4}B\setminus\Omega)}{{\operatorname{rad}}(B)^{n-1}}\quad\mbox{
for all $x\in\tfrac{1}{4}B\cap\Omega$ when $n\geq 2$},$

and

$\omega^{x}(B)\geq
c(n)\,\frac{1}{\log\dfrac{{\operatorname{rad}}(B)}{\operatorname{Cap}_{L}(\frac{1}{4}B\setminus\Omega)}}\qquad\mbox{
for all $x\in\tfrac{1}{4}B\cap\Omega$ when $n=1$.}$

The following two lemmas are valid for the so called NTA domains. However, we
only state them for Lipschitz domains, which suffices for the purposes of this
paper. For the proof, see [JK].

######  Lemma 2.7.

Let $n\geq 1$ and $\Omega\subset{\mathbb{R}}^{n+1}$ be a Lipschitz domain. Let
$B$ be a closed ball centered in $\partial\Omega$. Then

(2.7) $\omega^{p}(B)\approx r(B)^{n-1}\,g_{\Omega}(p,y)\quad\mbox{ for all
$p\in\Omega\backslash 2B$ and $y\in B\cap\Omega$,}$

with the implicit constant just depending on $n$ and the Lipschitz character
of $\Omega$.

The following theorem states the so called “change of pole formula”. Again,
this holds for NTA domains (see [JK]) but we only state for Lipschitz domains.

######  Lemma 2.8 (Change of pole formula).

For $n\geq 1$, let $\Omega\subset{\mathbb{R}}^{n+1}$ be a Lipschitz domain and
let $B$ be a ball centered in $\partial\Omega$. Let $p_{1},p_{2}\in\Omega$
such that ${\rm dist}(p_{i},B\cap\partial\Omega)\geq c_{1}^{-1}\,r(B)$ for
$i=1,2$. Then, for any Borel set $E\subset B\cap\partial\Omega$,

$\frac{\omega^{p_{1}}(E)}{\omega^{p_{1}}(B)}\approx\frac{\omega^{p_{2}}(E)}{\omega^{p_{2}}(B)},$

with the implicit constant depending only on $n$, $c_{1}$, and the Lipschitz
character of $\Omega$.

##  3\. Proof of Theorem A and beginning of the proof of Theorem B in the
Wiener regular case for ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$

In this section we assume that we are under the conditions of Theorems A and B
and that ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$, and we fix $\beta>0$ as in both
theorems. In the proof we will allow all the implicit constants in the
notation “$\lesssim$” to depend on $\beta$. We may assume that the barycenter
of $\Omega$ is well defined, because otherwise, by Theorem 1.2,
$\lambda_{\Omega}-\lambda_{B}\gtrsim{\mathcal{H}}^{n}(\Omega\triangle{\mathbb{S}}^{n})^{2}\gtrsim
1$ and then Theorems A and B are trivial. We let $B$ be a ball with the same
barycenter as $\Omega$ such that
${\mathcal{H}}^{n}(B)={\mathcal{H}}^{n}(\Omega)$.

As in Theorem 1.2, we let $u_{\Omega}$ and $u_{B}$ be the corresponding
eigenfunctions normalized so that they are positive and
$\|u_{\Omega}\|_{L^{2}({\mathbb{S}}^{n})}=\|u_{B}\|_{L^{2}({\mathbb{S}}^{n})}=1$.
Let $\alpha_{\Omega},\alpha_{B}$ be the respective characteristic constants of
$\Omega$ and $B$, so that
$\lambda_{\Omega}=\alpha_{\Omega}\,(\alpha_{\Omega}+n-1)$ and
$\lambda_{B}=\alpha_{B}\,(\alpha_{B}+n-1)$. Let ${\widetilde{u}}_{\Omega}$ and
${\widetilde{u}}_{B}$ be the $\alpha_{B}$-homogeneous extensions of
$u_{\Omega}$ and $u_{B}$, respectively. That is,

${\widetilde{u}}_{\Omega}(y)=|y|^{\alpha_{B}}\,u_{\Omega}(|y|^{-1}y),\qquad{\widetilde{u}}_{B}(y)=|y|^{\alpha_{B}}\,u_{B}(|y|^{-1}y).$

Also, denote by ${\widetilde{\Omega}}$ and ${\widetilde{B}}$ the following
truncated conical domains generated by $\Omega$ and $B$:

${\widetilde{\Omega}}=\\{y\in
A(0,1/4,1):|y|^{-1}y\in\Omega\\},\qquad{\widetilde{B}}=\\{y\in
A(0,1/4,1):|y|^{-1}y\in B\\},$

and the enlarged versions

${\widetilde{\Omega}}^{\prime}=\\{y\in
A(0,1/8,2):|y|^{-1}y\in\Omega\\},\qquad{\widetilde{B}}^{\prime}=\\{y\in
A(0,1/8,2):|y|^{-1}y\in B\\}.$

We also set

$S=\partial A(0,1/4,1).$

Notice that $S$ contains some part of the boundary of ${\widetilde{B}}$. To
prove Theorem B, first we will assume that ${\widetilde{\Omega}}$ and
${\widetilde{\Omega}}^{\prime}$ are Wiener regular domains in
${\mathbb{R}}^{n+1}$. Later on we will deduce the general result from the
Wiener regular case.

From the identity in spherical coordinates

$\Delta
f=\partial_{rr}f+\frac{n}{r}\,\partial_{r}f+\frac{1}{r^{2}}\,\Delta_{{\mathbb{S}}^{n}}f$

for a homogeneous function $f$ defined in ${\mathbb{R}}^{n+1}$, it is
immediate that $\Delta{\widetilde{u}}_{B}=0$ in ${\widetilde{B}}^{\prime}$,
and so in ${\widetilde{B}}$. On the other hand,
$\Delta{\widetilde{u}}_{\Omega}\neq 0$ in ${\widetilde{\Omega}}^{\prime}$ in
general, unless $\alpha_{\Omega}=\alpha_{B}$. Instead, we have

(3.1) $\displaystyle\Delta{\widetilde{u}}_{\Omega}(r,\theta)$
$\displaystyle=\partial_{rr}{\widetilde{u}}_{\Omega}(r,\theta)+\frac{n}{r}\,\partial_{r}{\widetilde{u}}_{\Omega}(r,\theta)+\frac{1}{r^{2}}\,\Delta_{{\mathbb{S}}^{n}}u_{\Omega}(\theta)$
$\displaystyle=r^{\alpha_{B}-2}\big{[}\big{(}\alpha_{B}(\alpha_{B}-1)+n\alpha_{B}\big{)}u_{\Omega}(\theta)+\Delta_{{\mathbb{S}}^{n}}u(\theta)$
$\displaystyle=r^{\alpha_{B}-2}\,(\lambda_{B}-\lambda_{\Omega})\,u_{\Omega}(\theta)=r^{-2}\,(\lambda_{B}-\lambda_{\Omega})\,u_{{\widetilde{\Omega}}}(r,\theta).$

We will need to use the following auxiliary result below. This may be known to
experts but we provide a detailed proof for completeness.

######  Lemma 3.1 (Basic estimates).

We have

(3.2) ${\widetilde{u}}_{B}(y)\approx{\rm
dist}(y,\partial{\widetilde{B}}^{\prime}\cap A(0,1/8,2))\quad\mbox{ for all
$y\in{\widetilde{B}}^{\prime}$}$

and

(3.3) ${\widetilde{u}}_{\Omega}(y)\lesssim 1\quad\mbox{ for all
$y\in{\widetilde{\Omega}}^{\prime}$.}$

Further, $\omega_{{\widetilde{B}}^{\prime}}$ and $\omega_{{\widetilde{B}}}$
are mutually absolutely continuous with
${\mathcal{H}}^{n}|_{\partial{\widetilde{B}}^{\prime}}$ and
${\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}$ respectively, and for $x\in
A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ such that $|x|^{-1}x\in\frac{1}{2}B$, we
have

(3.4)
$\frac{d\omega_{{\widetilde{B}}^{\prime}}^{x}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}^{\prime}}}\lesssim
1\quad\mbox{ in $\partial{\widetilde{B}}^{\prime}$}$

and

(3.5)
$\frac{d\omega_{{\widetilde{B}}}^{x}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}}\approx
1\quad\mbox{ in $\partial{\widetilde{B}}\cap A(0,1/3,4/5)$.}$

Remark that the implicit constants involved in the estimates of this lemma
depend on the constant $\beta$.

###### Proof.

First we will show that

(3.6) $\|{\widetilde{u}}_{B}\|_{\infty,{\widetilde{B}}^{\prime}}\approx 1.$

By homogeneity and Cauchy-Schwarz, we have

$\|{\widetilde{u}}_{B}\|_{\infty,{\widetilde{B}}^{\prime}}\approx\|u_{B}\|_{\infty,B}\gtrsim\|u_{B}\|_{L^{2}(B)}=1.$

To prove $\|u_{B}\|_{\infty,B}\lesssim 1$, let $y\in B$, consider the
(Euclidean) ball $B_{y}=B(y,1/2)$ and extend ${\widetilde{u}}_{B}$ by $0$ in
${\mathbb{R}}^{n+1}\setminus{\widetilde{B}}^{\prime}$. Notice that $B_{y}$ may
intersect $\partial{\widetilde{B}}^{\prime}\cap A(0,1/8,2)$. It is immediate
to check that ${\widetilde{u}}_{B}$ is subharmonic in $B_{y}$. Therefore,
using also the homogeneity of ${\widetilde{u}}_{B}$, we get

$u_{B}(y)={\widetilde{u}}_{B}(y)\leq\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{y}}{\widetilde{u}}_{B}\,dx\leq\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{y}}|{\widetilde{u}}_{B}|^{2}\,dx\right)^{1/2}\lesssim\|u_{B}\|_{L^{2}(B)}\lesssim
1,$

which completes the proof of (3.6).

Remark that essentially the same argument used to prove that
$\|u_{B}\|_{\infty,B}\lesssim 1$ shows that

$\|{\widetilde{u}}_{\Omega}\|_{\infty,{\widetilde{\Omega}}^{\prime}}\lesssim
1,$

proving (3.3) (in this case we have to argue with the
$\alpha_{\Omega}$-homogeneous extension of $u_{\Omega}$ in place of
${\widetilde{u}}_{\Omega}$).

Next we deal with (3.2). Again by homogeneity it suffices to show that

(3.7) $u_{B}(y)\approx{\rm dist}(y,\partial_{{\mathbb{S}}^{n}}B)\quad\mbox{
for all $y\in B$}.$

First we will prove that $u_{B}(y)\lesssim{\rm
dist}(y,\partial_{{\mathbb{S}}^{n}}B)$. Since $\|u_{B}\|_{\infty,B}\lesssim
1,$ we can assume that ${\rm
dist}(y,\partial_{{\mathbb{S}}^{n}}B)\leq\frac{1}{10}\,{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)$.
For a fixed $y_{0}\in B$ satisfying this condition, let
$\xi\in\partial_{{\mathbb{S}}^{n}}B$ be such that ${\rm
dist}(y_{0},\partial_{{\mathbb{S}}^{n}}B)={\rm dist}(y_{0},\xi)$ and consider
a ball $V_{\xi}\subset{\mathbb{R}}^{n+1}\setminus{\widetilde{B}}^{\prime}$
such that $\xi\in\partial V_{\xi}$ with
$c\leq{\operatorname{rad}}(V_{\xi})\leq 1/10$, so that $V_{\xi}$ is outer
tangent to ${\widetilde{B}}^{\prime}$ in $\xi$. Let $g_{V}$ be the Green
function of the open set ${\mathbb{R}}^{n+1}\setminus\overline{V_{\xi}}$.
Notice that the functions ${\widetilde{u}}_{B}$ and $g_{V}(x_{B},\cdot)$ are
both harmonic in the domain
${\widetilde{B}}_{0}={\widetilde{B}}^{\prime}\setminus
B(x_{B},\frac{1}{10}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B))$. It is easy
to check that $g_{V}(x_{B},y)\approx 1$ for
$y\in\partial{\widetilde{B}}_{0}\cap\partial A(0,1/8,2)$ and $y\in\partial
B(x_{B},\frac{1}{10}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B))$. So using
also (3.6) we deduce that

${\widetilde{u}}_{B}(y)\lesssim
g_{V}(x_{B},y)\,\|{\widetilde{u}}_{B}\|_{\infty,{\widetilde{B}}^{\prime}}\lesssim
g_{V}(x_{B},y)\quad\mbox{ for all $y\in\partial{\widetilde{B}}_{0}$.}$

By the maximum principle, the same estimate holds for all
$y\in{\widetilde{B}}_{0}$, and so

${\widetilde{u}}_{B}(y)\lesssim g_{V}(x_{B},y)\approx|y-\xi|\quad\mbox{ for
all $y\in{\widetilde{B}}_{0}$,}$

by the smoothness of $g_{V}$ in $\xi$. In particular,

(3.8) ${\widetilde{u}}_{B}(y_{0})\lesssim|y_{0}-\xi|={\rm
dist}(y_{0},\partial_{{\mathbb{S}}^{n}}B).$

Now we will show that

(3.9) $u_{B}(y)\gtrsim{\rm dist}(y,\partial_{{\mathbb{S}}^{n}}B)\quad\mbox{
for all $y\in B$}.$

To this end, observe first that the condition (3.8) and the fact that
$\|u_{B}\|_{\infty,B}\approx 1$ imply that the $\max_{B}u_{B}$ is attained at
a point $x_{0}\in B$ such that ${\rm
dist}(x_{0},\partial_{{\mathbb{S}}^{n}}B)\approx 1$. By a Harnack chain
argument, this implies that

(3.10) ${\widetilde{u}}_{B}(y)\approx 1\quad\mbox{ for all
$y\in{\widetilde{B}}^{\prime}$ such that ${\rm
dist}(y,\partial{\widetilde{B}}^{\prime})\approx 1$.}$

To prove (3.9), let $y_{0}$ and $\xi$ be as above, and consider a ball
$U_{\xi}\subset{\widetilde{B}}^{\prime}$ such that $\xi\in\partial U_{\xi}$
with $c\leq{\operatorname{rad}}(U_{\xi})\leq 1/10$, so that $U_{\xi}$ is inner
tangent to ${\widetilde{B}}^{\prime}$ in $\xi$. Let $g_{U}$ be the Green
function of the ball $U_{\xi}$ and let $x_{U}$ be the center of $U_{\xi}$.
Notice that the functions ${\widetilde{u}}_{B}$ and $g_{U}(x_{U},\cdot)$ are
both harmonic in
${\widetilde{B}}_{1}=U_{\xi}\setminus\bar{B}(x_{U},\frac{1}{10}{\operatorname{rad}}(U_{\xi}))$.
It is easy to check that, for $y\in\partial
B(x_{U},\frac{1}{10}{\operatorname{rad}}(U_{\xi}))$, $g_{U}(x_{U},y)\approx 1$
while ${\widetilde{u}}_{B}(y)\approx 1$, by (3.10). So we deduce that

${\widetilde{u}}_{B}(y)\gtrsim g_{U}(x_{U},y)\quad\mbox{ for all
$y\in\partial{\widetilde{B}}_{1}$.}$

By the maximum principle, the same estimate holds for all
$y\in{\widetilde{B}}_{1}$, and so

${\widetilde{u}}_{B}(y)\gtrsim g_{U}(x_{U},y)\approx|y-\xi|\quad\mbox{ for all
$y\in{\widetilde{B}}_{1}$,}$

by the smoothness of $g_{U}$ in $\xi$. In particular,
${\widetilde{u}}_{B}(y_{0})\gtrsim|y_{0}-\xi|={\rm
dist}(y_{0},\partial_{{\mathbb{S}}^{n}}B)$, which concludes the proof of (3.9)
and (3.7).

Let us turn our attention to the behavior of the harmonic measures
$\omega_{{\widetilde{B}}^{\prime}}$ and $\omega_{{\widetilde{B}}}$. The fact
that they are mutually absolutely continuous with
${\mathcal{H}}^{n}|_{\partial{\widetilde{B}}^{\prime}}$ and
${\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}$, respectively, is due to the
fact that both ${\widetilde{B}}^{\prime}$ and ${\widetilde{B}}$ are Lipschitz
domains, and in fact, piecewise $C^{\infty}$ domains. To prove (3.4) for $x$
as in the lemma (i.e., $x\in A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ such that
$|x|^{-1}x\in\frac{1}{2}B$), we consider an arbitrary point
$\xi\in\partial{\widetilde{B}}^{\prime}$ and we assume that
$\partial{\widetilde{B}}^{\prime}$ is smooth in a neighborhood of $\xi$ (this
happens for ${\mathcal{H}}^{n}$-a.e.
$\xi\in\partial{\widetilde{B}}^{\prime}$). As above, let
$V_{\xi}\subset{\mathbb{R}}^{n+1}\setminus{\widetilde{B}}^{\prime}$ be a ball
such that $\xi\in\partial V_{\xi}$ with
$c\leq{\operatorname{rad}}(V_{\xi})\leq 1/10$, so that $V_{\xi}$ is (outer)
tangent to ${\widetilde{B}}^{\prime}$ in $\xi$, and let $g_{V}$ be the Green
function of ${\mathbb{R}}^{n+1}\setminus\overline{V_{\xi}}$. Also, let
$g_{{\widetilde{B}}^{\prime}}$ be the Green function of
${\widetilde{B}}^{\prime}$. Then the function
$g_{V}(x,\cdot)-g_{{\widetilde{B}}^{\prime}}(x,\cdot)$ is harmonic in
${\widetilde{B}}^{\prime}$ and it is non-negative in
$\partial{\widetilde{B}}^{\prime}$. So

$g_{{\widetilde{B}}^{\prime}}(x,y)\leq g_{V}(x,y)\quad\mbox{ for all
$y\in{\widetilde{B}}^{\prime}$,}$

by the maximum principle. Consequently,

$\frac{d\omega_{{\widetilde{B}}^{\prime}}^{x}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}^{\prime}}}(\xi)=\partial_{\nu}g_{{\widetilde{B}}^{\prime}}(x,\xi)\leq\partial_{\nu}g_{V}(x,\xi)=\frac{d\omega_{{\mathbb{R}}^{n+1}\setminus\overline{V_{\xi}}}^{x}}{d{\mathcal{H}}^{n}|_{\partial
V_{\xi}}}(\xi)\approx 1,$

where $\partial_{\nu}$ stands for the normal derivative in the inner
direction.

The same argument as above shows that

$\frac{d\omega_{{\widetilde{B}}}^{x}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}}\lesssim
1\quad\mbox{ in $\partial{\widetilde{B}}$.}$

To prove the converse estimate for $\xi\in\partial{\widetilde{B}}\cap
A(0,1/3,4/5)$, consider again a ball $U_{\xi}\subset{\widetilde{B}}^{\prime}$
such that $\xi\in\partial U_{\xi}$ with
$c\leq{\operatorname{rad}}(U_{\xi})\leq 1/10$, so that $U_{\xi}$ is (inner)
tangent to ${\widetilde{B}}$ in $\xi$. Let $g_{{\widetilde{B}}}$ be the Green
function of ${\widetilde{B}}$ and $g_{U}$ be the Green function of $U_{\xi}$,
and $x_{U}$ the center of $U_{\xi}$. Then the function
$g_{{\widetilde{B}}}(x_{U},\cdot)-g_{U}(x_{U},\cdot)$ is harmonic in $U_{\xi}$
and it is non-negative in $\partial U_{\xi}$. So

$g_{{\widetilde{B}}}(x_{U},y)\geq g_{U}(x_{U},y)\quad\mbox{ for all $y\in
U_{\xi}$,}$

by the maximum principle. Consequently,

$\frac{d\omega_{{\widetilde{B}}}^{x_{U}}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}}(\xi)=\partial_{\nu}g_{{\widetilde{B}}}(x_{U},\xi)\geq\partial_{\nu}g_{U}(x_{U},\xi)=\frac{d\omega_{U_{\xi}}^{x_{U}}}{d{\mathcal{H}}^{n}|_{\partial
U_{\xi}}}(\xi)\approx 1.$

By a Harnack chain argument, it follows that

$\frac{d\omega_{{\widetilde{B}}}^{x_{U}}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}}(\xi)\approx\frac{d\omega_{{\widetilde{B}}}^{x}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}}}(\xi),$

and so the proof of (3.5) is concluded. ∎

######  Claim 1 (Extension of $u_{B}$ and ${\widetilde{u}}_{B}$).

There exists an extension $u_{B}^{ex}$ of $u_{B}|_{B}$ to ${\mathbb{S}}^{n}$
which belongs to $C^{2}({\mathbb{S}}^{n})$ and such that
$\|\nabla^{j}u_{B}^{ex}\|_{\infty,{\mathbb{S}}^{n}}\lesssim 1$ for $j=0,1,2$
and $\|\Delta{\widetilde{u}}_{B}^{ex}\|_{\infty,A(0,1/4,2)}\lesssim 1$, where
${\widetilde{u}}_{B}^{ex}$ is the $\alpha_{B}$-homogeneous extension of
$u_{B}^{ex}$ to ${\mathbb{R}}^{n+1}$. Further, we may construct $u_{B}^{ex}$
so that it is supported in an ${\mathbb{S}}^{n}$-ball $B_{0}$ concentric with
$B$ such that ${\mathcal{H}}^{n}({\mathbb{S}}^{n}\setminus B_{0})\geq\beta/2$.

###### Proof.

The arguments are quite standard. Since the function ${\widetilde{u}}_{B}$ is
harmonic in ${\widetilde{B}}^{\prime}$ and it vanishes identically in
$\partial{\widetilde{B}}^{\prime}\setminus\partial A(0,1/8,2)$, which is a
$C^{\infty}$ portion of the boundary, it follows that ${\widetilde{u}}_{B}\in
C^{2}(\overline{V})$, for $V={\widetilde{B}}^{\prime}\cap A(0,1/4,3/2)$ and
that
$\|{\widetilde{u}}_{B}\|_{C^{2}(\overline{V})}\lesssim\|{\widetilde{u}}_{B}\|_{C(\overline{{\widetilde{B}}^{\prime}})}$.
See for example Corollary 6.7 from [GT]. By Lemma 3.1,
$\|{\widetilde{u}}_{B}\|_{C(\overline{{\widetilde{B}}^{\prime}})}\lesssim 1$.
By a suitable reflection, one can construct a function $f\in
C^{2}(\bar{A}(0,1/2,5/4))$ which coincides with ${\widetilde{u}}_{B}$ in $V$
and satisfies
$\|f\|_{C^{2}(\bar{A}(0,1/2,5/4))}\lesssim\|{\widetilde{u}}_{B}\|_{C^{2}(\overline{V})}$.
Multiplying $f$ by a suitable bump $C^{\infty}$ function, we can assume that
$f|_{{\mathbb{S}}^{n}}$ is supported in an ${\mathbb{S}}^{n}$-ball $B_{0}$
concentric with $B$ such that ${\mathcal{H}}^{n}({\mathbb{S}}^{n}\setminus
B_{0})\geq\beta/2$. Then we set $u_{B}^{ex}:=f|_{{\mathbb{S}}^{n}}$ and we let
${\widetilde{u}}_{B}^{ex}$ be its $\alpha_{B}$-homogeneous extension. ∎

Recall that we assume that $u_{B}$ and $u_{\Omega}$ vanish respectively in
${\mathbb{S}}^{n}\setminus B$ and ${\mathbb{S}}^{n}\setminus\Omega$. Notice
that the estimate (1.2) from Theorem 1.2 may not hold for the extension
$u_{B}^{ex}$. Instead, we have

(3.11) $\lambda_{\Omega}-\lambda_{B}\geq
C(\beta)\,\Big{(}{\mathcal{H}}^{n}(\Omega\triangle
B)^{2}+\int_{B}|u_{\Omega}-u_{B}^{ex}|^{2}\,d{\mathcal{H}}^{n}\Big{)}.$

Consider the function $v:{\mathbb{S}}^{n}\to{\mathbb{R}}$ defined by
$v=u_{B}^{ex}-u_{\Omega}$ and let ${\widetilde{v}}$ be its
$\alpha_{B}$-homogeneous extension, so that
${\widetilde{v}}={\widetilde{u}}_{B}^{ex}-{\widetilde{u}}_{\Omega}$. Notice
that in ${\widetilde{\Omega}}^{\prime}$, by (3.1) we have

(3.12)
$\Delta{\widetilde{v}}=\Delta{\widetilde{u}}_{B}^{ex}-\Delta{\widetilde{u}}_{\Omega}=\Delta{\widetilde{u}}_{B}^{ex}-(\lambda_{B}-\lambda_{\Omega})\,|y|^{-2}\,{\widetilde{u}}_{\Omega}.$

Notice also that
$\operatorname{supp}\Delta{\widetilde{u}}_{B}^{ex}\subset({\widetilde{B}}^{\prime})^{c}$.

Recall that $S=\partial A(0,1/4,1)$. We can write, for
$x\in{\widetilde{\Omega}}$,

(3.13) $\displaystyle{\widetilde{v}}(x)$
$\displaystyle=\int_{\partial{\widetilde{\Omega}}}{\widetilde{v}}\,d\omega_{{\widetilde{\Omega}}}^{x}-\int_{{\widetilde{\Omega}}}\Delta{\widetilde{v}}(y)\,g_{{\widetilde{\Omega}}}(x,y)\,dy$
$\displaystyle=\int_{\partial{\widetilde{\Omega}}\cap{\widetilde{B}}\setminus
S}{\widetilde{v}}\,d\omega_{{\widetilde{\Omega}}}^{x}+\int_{\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S)}{\widetilde{v}}\,d\omega_{{\widetilde{\Omega}}}^{x}+\int_{\partial{\widetilde{\Omega}}\cap
S}{\widetilde{v}}\,d\omega_{{\widetilde{\Omega}}}^{x}-\int_{{\widetilde{\Omega}}}\Delta{\widetilde{v}}(y)\,g_{{\widetilde{\Omega}}}(x,y)\,dy$
$\displaystyle=:{\widetilde{v}}_{1}(x)+{\widetilde{v}}_{2}(x)+{\widetilde{v}}_{3}(x)+{\widetilde{v}}_{4}(x),$

where $g_{{\widetilde{\Omega}}}(x,y)$ is the Green function of
${\widetilde{\Omega}}$. For the points $x\in
A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ such that $|x|^{-1}x\in\frac{9}{10}B$,
we intend to estimate ${\widetilde{v}}(x)$ from below and to relate this to
the terms appearing on the right hand side of the inequalities (1.3), (1.4),
and (1.9), so that then we can apply the inequality (3.11) to prove Theorem A
and Theorem B. To this end, we will estimate
$|{\widetilde{v}}_{2}(x)+{\widetilde{v}}_{3}(x)+{\widetilde{v}}_{4}(x)|$ from
above and ${\widetilde{v}}_{1}(x)$ from below.

######  Lemma 3.2.

Let $x\in A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ and also that
$|x|^{-1}x\in\frac{9}{10}B$, and let ${\widetilde{v}}_{i}$, for $1\leq i\leq
4$, be the functions in (3.13). Suppose that
$|\lambda_{B}-\lambda_{\Omega}|\leq 1$. Then we have

$\big{|}{\widetilde{v}}_{2}(x)+{\widetilde{v}}_{3}(x)+{\widetilde{v}}_{4}(x)\big{|}\lesssim(\lambda_{\Omega}-\lambda_{B})^{1/2}.$

###### Proof.

Estimate of $|{\widetilde{v}}_{3}(x)|$. By the maximum principle and taking
into account that $\frac{d\omega_{A(0,1/4,1)}^{x}}{d{\mathcal{H}}^{n}}\approx
1$ on $S$, for any set $F\subset\partial{\widetilde{\Omega}}\cap S$, we have

$\omega_{{\widetilde{\Omega}}}^{x}(F)\leq\omega_{A(0,1/4,1)}^{x}(F)\approx{\mathcal{H}}^{n}(F).$

Therefore, $\omega_{{\widetilde{\Omega}}}^{x}|_{S}$ is absolutely continuous
with respect to surface measure on $S$, and

$\frac{d\omega_{{\widetilde{\Omega}}}^{x}}{d{\mathcal{H}}^{n}}(y)\lesssim
1\quad\mbox{ for ${\mathcal{H}}^{n}$-a.e.\
$y\in\partial{\widetilde{\Omega}}\cap S$.}$

Hence, using also the $\alpha_{B}$-homogeneity of ${\widetilde{v}}$,

$|{\widetilde{v}}_{3}(x)|\lesssim\int_{\partial{\widetilde{\Omega}}\cap
S}|{\widetilde{v}}|\,d{\mathcal{H}}^{n}\lesssim\int_{\Omega}|u_{B}^{ex}-u_{\Omega}|\,d{\mathcal{H}}^{n}.$

Then, by Cauchy-Schwarz and (3.11), we deduce

$\int_{\Omega\cap
B}|u_{B}^{ex}-u_{\Omega}|\,d{\mathcal{H}}^{n}\leq\bigg{(}\int_{\Omega\cap
B}|u_{B}-u_{\Omega}|^{2}\,d{\mathcal{H}}^{n}\bigg{)}^{1/2}\lesssim(\lambda_{\Omega}-\lambda_{B})^{1/2}.$

On the other hand, from (3.2), (3.3), and again (3.11), we have

$\int_{\Omega\setminus
B}|u_{B}^{ex}-u_{\Omega}|\,d{\mathcal{H}}^{n}\leq\big{(}\|u_{B}^{ex}\|_{\infty}+\|u_{\Omega}\|_{\infty}\big{)}\,{\mathcal{H}}^{n}(\Omega\setminus
B)\lesssim{\mathcal{H}}^{n}(\Omega\setminus
B)\lesssim(\lambda_{\Omega}-\lambda_{B})^{1/2}.$

Adding the two preceding estimates we derive

$|{\widetilde{v}}_{3}(x)|\lesssim(\lambda_{\Omega}-\lambda_{B})^{1/2}.$

Estimate of $|{\widetilde{v}}_{4}(x)|$. To this end, we use the trivial
estimate $g_{{\widetilde{\Omega}}}(x,y)\leq|x-y|^{1-n}$ and the fact that by
(3.1) and Claim 1,

$\|\Delta{\widetilde{u}}_{\Omega}\|_{\infty,{\widetilde{\Omega}}}\lesssim|\lambda_{\Omega}-\lambda_{B}|\leq
1\quad\text{ and
}\quad\|\Delta{\widetilde{u}}_{B}^{ex}\|_{\infty,{\widetilde{\Omega}}\setminus
B}\lesssim 1.$

Taking also into account that
$\operatorname{supp}\Delta{\widetilde{u}}_{B}^{ex}\subset({\widetilde{B}})^{c}$,
we get

$\displaystyle|{\widetilde{v}}_{4}(x)|$
$\displaystyle\leq\int_{{\widetilde{\Omega}}\cap{\widetilde{B}}}|\lambda_{B}-\lambda_{\Omega}|\,|{\widetilde{u}}_{\Omega}(y)|\,\frac{1}{|y|^{2}\,|x-y|^{n-1}}\,dy+\int_{{\widetilde{\Omega}}\setminus{\widetilde{B}}}\frac{|\Delta{\widetilde{u}}_{B}^{ex}(y)|+|\Delta{\widetilde{u}}_{{\widetilde{\Omega}}}(y)|}{|x-y|^{n-1}}\,dy$
$\displaystyle\lesssim|\lambda_{B}-\lambda_{\Omega}|\,\|{\widetilde{u}}_{\Omega}\|_{\infty}\int_{{\widetilde{\Omega}}\cap{\widetilde{B}}}\frac{1}{|x-y|^{n-1}}\,dy+\int_{{\widetilde{\Omega}}\setminus{\widetilde{B}}}\,\frac{1}{|x-y|^{n-1}}\,dy.$

To estimate the first summand we use the fact that
$\|{\widetilde{u}}_{\Omega}\|_{\infty}\lesssim 1$ and a trivial bound for the
integral, while for second one we take into account that
${\mathcal{H}}^{n+1}({\widetilde{\Omega}}\setminus{\widetilde{B}})\lesssim{\mathcal{H}}^{n}(\Omega\setminus
B)\lesssim|\lambda_{B}-\lambda_{\Omega}|^{1/2}$ and the fact that ${\rm
dist}(x,{\widetilde{B}}^{c})\gtrsim 1$, since $|x|^{-1}x\in\frac{9}{10}B$.
Using also the assumption that $|\lambda_{B}-\lambda_{\Omega}|\leq 1$, we
obtain

$|{\widetilde{v}}_{4}(x)|\lesssim|\lambda_{B}-\lambda_{\Omega}|+|\lambda_{B}-\lambda_{\Omega}|^{1/2}\lesssim|\lambda_{B}-\lambda_{\Omega}|^{1/2}.$

Estimate of $|{\widetilde{v}}_{2}(x)|$. Recall that

${\widetilde{v}}_{2}(x)=\int_{\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S)}{\widetilde{v}}\,d\omega_{{\widetilde{\Omega}}}^{x}=\int_{\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S)}({\widetilde{u}}_{B}^{ex}-{\widetilde{u}}_{\Omega})\,\,d\omega_{{\widetilde{\Omega}}}^{x}.$

Since $u_{\Omega}$ vanishes on ${\partial_{{\mathbb{S}}^{n}}\Omega}$, we can
write

$|{\widetilde{v}}_{2}(x)|\leq\int_{0}^{\infty}\omega_{{\widetilde{\Omega}}}^{x}\big{(}\big{\\{}y\in\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S):|{\widetilde{u}}_{B}^{ex}(y)|>t\big{\\}}\big{)}\,dt.$

Since $\|\nabla{\widetilde{u}}_{B}^{ex}\|_{\infty,A(0,1/4,1)}\approx\|\nabla
u_{B}^{ex}\|_{\infty,{\mathbb{S}}^{n}}+\|u_{B}^{ex}\|_{\infty,{\mathbb{S}}^{n}}\lesssim
1$ and ${\widetilde{u}}_{B}^{ex}$ vanishes on
$\partial{\widetilde{B}}\setminus S$, we have

$|{\widetilde{u}}_{B}^{ex}(y)|\lesssim{\rm
dist}(y,\partial{\widetilde{B}}\setminus S)\approx{\rm
dist}_{{\mathbb{S}}^{n}}(|y|^{-1}y,\partial B)=\delta_{B}(|y|^{-1}y).$

Therefore, recalling also that ${\widetilde{u}}_{B}^{ex}$ vanishes in
${\mathbb{S}}^{n}\setminus B_{0}$ (with $B_{0}$ defined in Claim 1), for some
$c>0$ we have

$\displaystyle|{\widetilde{v}}_{2}(x)|$
$\displaystyle\leq\int_{0}^{\infty}\omega_{{\widetilde{\Omega}}}^{x}\big{(}\big{\\{}y\in\partial{\widetilde{\Omega}}\cap{\widetilde{B}}_{0}\setminus({\widetilde{B}}\cup
S):\delta_{B}(|y|^{-1}y)\geq ct\big{\\}}\big{)}\,dt$
$\displaystyle\approx\int_{0}^{t_{0}}\omega_{{\widetilde{\Omega}}}^{x}\big{(}\big{\\{}y\in\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S):\delta_{B}(|y|^{-1}y)\geq t\big{\\}}\big{)}\,dt,$

where $t_{0}={\rm dist}_{{\mathbb{S}}^{n}}(\partial B,\partial B_{0})$. For
each $t\in(0,t_{0})$, take the ${\mathbb{S}}^{n}$-ball
$B_{t}=\\{z\in{\mathbb{S}}^{n}:{\rm dist}_{{\mathbb{S}}^{n}}(z,B)<t\\}$ and
the associated truncated conical regions

${\widetilde{B}}_{t}=\\{y\in A(0,1/4,1):|y|^{-1}y\in
B_{t}\\},\qquad{\widetilde{B}}^{\prime}_{t}=\\{y\in A(0,1/8,2):|y|^{-1}y\in
B_{t}\\}.$

Notice that $B\subset B_{t}\subset B_{0}$ and that

$\big{\\{}y\in\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S):{\rm dist}_{{\mathbb{S}}^{n}}(|y|^{-1}y,\partial B)\geq
t\big{\\}}=\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}_{t}\cup S).$

By the maximum principle, since
${\widetilde{\Omega}}\cap{\widetilde{B}}_{t}\subset{\widetilde{\Omega}}$ and
${\widetilde{\Omega}}\cap\partial({\widetilde{\Omega}}\cap{\widetilde{B}}_{t})={\widetilde{\Omega}}\cap\partial{\widetilde{B}}_{t}$,
for $0<t\leq t_{0}$ we have

$\omega_{{\widetilde{\Omega}}}^{x}\big{(}\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}_{t}\cup
S)\big{)}\leq\omega_{{\widetilde{\Omega}}\cap{\widetilde{B}}_{t}}^{x}\big{(}\partial{\widetilde{B}}_{t}\cap{\widetilde{\Omega}}\big{)}.$

By the maximum principle and standard estimates (taking into account that the
ball $B_{0}$ in Claim 1 does not degenerate) we deduce that, for any set
$F\subset\partial{\widetilde{B}}_{t}\cap{\widetilde{\Omega}}\setminus S$,

$\omega_{{\widetilde{\Omega}}\cap{\widetilde{B}}_{t}}^{x}(F)\leq\omega_{{\widetilde{B}}_{t}^{\prime}}^{x}(F)\approx{\mathcal{H}}^{n}(F).$

Therefore, for each $t\in(0,t_{0})$,

$\omega_{{\widetilde{\Omega}}}^{x}\big{(}\big{\\{}y\in\partial{\widetilde{\Omega}}\setminus({\widetilde{B}}\cup
S):\delta_{B}(|y|^{-1}y)\geq
t\big{\\}}\big{)}\leq\omega_{{\widetilde{\Omega}}\cap{\widetilde{B}}_{t}}^{x}\big{(}\partial{\widetilde{B}}_{t}\cap{\widetilde{\Omega}}\big{)}\lesssim{\mathcal{H}}^{n}\big{(}\partial{\widetilde{B}}_{t}\cap{\widetilde{\Omega}}\big{)}.$

Consequently, applying Fubini and using the conical property of
${\widetilde{\Omega}}$, we obtain

$|{\widetilde{v}}_{2}(x)|\lesssim\int_{0}^{t_{0}}{\mathcal{H}}^{n}\big{(}\partial{\widetilde{B}}_{t}\cap{\widetilde{\Omega}}\big{)}\,dt\approx{\mathcal{H}}^{n+1}({\widetilde{\Omega}}\cap{\widetilde{B}}_{0}\setminus{\widetilde{B}})\lesssim{\mathcal{H}}^{n}(\Omega\setminus
B)\lesssim|\lambda_{B}-\lambda_{\Omega}|^{1/2},$

using also (3.11) for the last estimate. ∎

Estimate of ${\widetilde{v}}_{1}(x)$. Next we turn our attention to the
function ${\widetilde{v}}_{1}$. Since ${\widetilde{u}}_{\Omega}$ vanishes in
$\partial{\widetilde{\Omega}}\setminus S$ and
${\widetilde{u}}_{B}^{ex}={\widetilde{u}}_{B}$ in ${\widetilde{B}}$, for $x\in
A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ we have

${\widetilde{v}}_{1}(x)=\int_{\partial{\widetilde{\Omega}}\cap({\widetilde{B}}\setminus
S)}\big{(}{\widetilde{u}}_{B}^{ex}-{\widetilde{u}}_{\Omega})\,d\omega_{{\widetilde{\Omega}}}^{x}=\int_{\partial{\widetilde{\Omega}}\setminus
S}{\widetilde{u}}_{B}\,d\omega_{{\widetilde{\Omega}}}^{x}.$

Observe first that, by the maximum principle, if we let $\Omega_{0}=\Omega\cap
B$ and ${\widetilde{\Omega}}_{0}={\widetilde{\Omega}}\cap{\widetilde{B}}$,
then it holds

(3.14)
${\widetilde{v}}_{1}(x)\geq\int_{\partial{\widetilde{\Omega}}_{0}\setminus
S}{\widetilde{u}}_{B}\,d\omega_{{\widetilde{\Omega}}_{0}}^{x}=:f_{0}(x)\quad\mbox{
for all $x\in{\widetilde{\Omega}}_{0}$}.$

We extend $f_{0}$ to the whole ${\widetilde{B}}$ by letting

(3.15) $f_{0}(x)={\widetilde{u}}_{B}(x)\quad\mbox{ for all
$x\in{\widetilde{B}}\setminus{\widetilde{\Omega}}_{0}$}.$

We claim that $f_{0}$ is superharmonic in ${\widetilde{B}}_{0}$. Indeed, first
notice that, by the maximum principle,

$f_{0}(x)\leq{\widetilde{u}}_{B}(x)\quad\mbox{ for all
$x\in{\widetilde{\Omega}}_{0}$}.$

So $f_{0}-{\widetilde{u}}_{B}$ is continuous in ${\widetilde{B}}$ (although it
may happen that $f_{0}$ does not extend continuously to
$\partial{\widetilde{B}}\cap S$), it vanishes in
${\widetilde{B}}\setminus{\widetilde{\Omega}}$, and it is harmonic and
negative in
${\widetilde{\Omega}}\cap{\widetilde{B}}={\widetilde{\Omega}}_{0}$. This
implies that $f_{0}-{\widetilde{u}}_{B}$ is superharmonic in
${\widetilde{B}}$, and so the same happens for
$f_{0}=(f_{0}-{\widetilde{u}}_{B})+{\widetilde{u}}_{B}$.

######  Lemma 3.3.

Suppose that $|\lambda_{B}-\lambda_{\Omega}|\leq 1$ and that
${\widetilde{\Omega}}$ is Wiener regular. Denote
${\widetilde{\frac{9}{10}B}}=\\{x\in{\widetilde{B}}:|x|^{-1}x\in\frac{9}{10}B\\}.$
Then we have

$\lambda_{\Omega}-\lambda_{B}\geq
C(\beta)\left(\int_{{\widetilde{\frac{9}{10}B}}\cap
A(0,1/2,3/4)}f_{0}(x)\,d{\mathcal{H}}^{n+1}(x)\right)^{2}.$

###### Proof.

By the Allen-Kriventsov-Neumayer theorem and the $\alpha_{B}$-homogeneity of
${\widetilde{v}}$, we have

$\lambda_{\Omega}-\lambda_{B}\gtrsim\int_{\frac{9}{10}B}|u_{B}-u_{\Omega}|^{2}\,d{\mathcal{H}}^{n}\approx_{\beta}\int_{{\widetilde{\frac{9}{10}B}}}|{\widetilde{u}}_{B}-{\widetilde{u}}_{\Omega}|^{2}\,d{\mathcal{H}}^{n+1},$

since $\alpha_{B}\approx_{\beta}1$. By Lemma 3.2 and (3.14), for all
$x\in{\widetilde{\frac{9}{10}B}}\cap A(0,1/2,3/4)\cap{\widetilde{\Omega}}$, we
have

$\displaystyle{\widetilde{u}}_{B}(x)-{\widetilde{u}}_{\Omega}(x)$
$\displaystyle={\widetilde{u}}_{B}^{ex}(x)-{\widetilde{u}}_{\Omega}(x)={\widetilde{v}}(x)$
$\displaystyle\geq{\widetilde{v}}_{1}(x)-|{\widetilde{v}}_{2}(x)+{\widetilde{v}}_{3}(x)+{\widetilde{v}}_{4}(x)|\geq
f_{0}(x)-C(\lambda_{\Omega}-\lambda_{B})^{1/2}.$

On the other hand, for $x\in{\widetilde{\frac{9}{10}B}}\cap
A(0,1/2,3/4)\setminus{\widetilde{\Omega}}$, by definition

${\widetilde{u}}_{B}(x)-{\widetilde{u}}_{\Omega}(x)={\widetilde{u}}_{B}(x)=f_{0}(x).$

Therefore, by Cauchy-Schwarz,

$\displaystyle\lambda_{\Omega}-\lambda_{B}$
$\displaystyle\gtrsim_{\beta}\int_{{\widetilde{\frac{9}{10}B}}\cap
A(0,1/2,3/4)}|f_{0}(x)-C(\lambda_{\Omega}-\lambda_{B})^{1/2}|^{2}\,d{\mathcal{H}}^{n+1}(x)$
$\displaystyle\gtrsim_{\beta}\left(\int_{{\widetilde{\frac{9}{10}B}}\cap
A(0,1/2,3/4)}|f_{0}(x)-C(\lambda_{\Omega}-\lambda_{B})^{1/2}|\,d{\mathcal{H}}^{n+1}(x)\right)^{2}$
$\displaystyle\geq\left(\int_{{\widetilde{\frac{9}{10}B}}\cap
A(0,1/2,3/4)}f_{0}(x)\,d{\mathcal{H}}^{n+1}(x)\right)^{2}-C|\lambda_{\Omega}-\lambda_{B}|,$

which proves the lemma. ∎

######  Lemma 3.4.

There exists some constant $c_{2}\in(0,1/4)$ depending only on $n$ such that
the following holds. Let $\Delta_{z}$ be an ${\mathbb{S}}^{n}$-ball centered
in $z\in{\mathbb{S}}^{n}$ such that $\Delta_{z}\subset B$, with ${\rm
dist}_{{\mathbb{S}}^{n}}(\Delta_{z},\partial_{{\mathbb{S}}^{n}}B)\geq\rho$.
Let $B_{z}$ be an Euclidean ball centered in the segment $L_{z}=[0,z]\cap
A(0,1/2,3/4)$ with
$\frac{1}{8}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})\leq{\operatorname{rad}}(B_{z})\leq{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})$
such that $S\cap B_{z}\neq\varnothing$. Then, in the case $n\geq 3$ we have

(3.16)
$f_{0}(y)\gtrsim\frac{\rho\,\operatorname{Cap}_{n-2}(c_{2}\Delta_{z}\\!\setminus\Omega)}{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})^{n-2}}\quad\text{
for all $y\in\tfrac{1}{4}B_{z}\cap{\widetilde{\Omega}}$,}$

and in the case $n=2$,

(3.17)
$f_{0}(y)\gtrsim\frac{\rho}{\log\dfrac{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})}{\operatorname{Cap}_{L}(c_{2}\,\Delta_{z}\\!\setminus\Omega)}}\quad\text{
for all $y\in c_{2}B_{z}\cap{\widetilde{\Omega}}$.}$

###### Proof.

Suppose $n\geq 3$. By Lemma 2.6 and Lemma 2.2, for all
$y\in\frac{1}{4}B_{z}\cap{\widetilde{\Omega}}_{0}$, we have

$\omega_{{\widetilde{\Omega}}_{0}}^{y}(B_{z})\gtrsim\frac{\operatorname{Cap}_{n-1}(\tfrac{1}{4}B_{z}\setminus{\widetilde{\Omega}}_{0})}{{\operatorname{rad}}(B_{z})^{n-1}}\gtrsim\frac{\operatorname{Cap}_{n-2}(c_{2}\Delta_{z}\setminus\Omega_{0})}{{\operatorname{rad}}(B_{z})^{n-2}},$

for a suitable constant $c_{2}$ depending just on $n$. Then, taking into
account that ${\widetilde{u}}_{B}(z)\approx{\rm
dist}(z,\partial{\widetilde{B}})\gtrsim\rho$ for all $z\in B_{z}$ and that
$B_{z}\cap S=\varnothing$ (recall that $S=\partial A(0,1/4,1)$), from the
definition of $f_{0}$ in (3.14) we infer

$f_{0}(y)\geq\int_{\partial{\widetilde{\Omega}}_{0}\cap
B_{z}}{\widetilde{u}}_{B}\,d\omega_{{\widetilde{\Omega}}_{0}}^{y}\gtrsim\rho\,\omega_{{\widetilde{\Omega}}_{0}}^{y}(B_{z})\gtrsim\frac{\rho\,\operatorname{Cap}_{n-2}(c_{2}\Delta_{z}\setminus\Omega_{0})}{{\operatorname{rad}}(B_{z})^{n-2}}\quad\mbox{
for all $y\in\frac{1}{4}B_{z}\cap{\widetilde{\Omega}}_{0}$.}$

On the other hand, for
$y\in\tfrac{1}{4}B_{z}\setminus{\widetilde{\Omega}}_{0}$, we also have
$f_{0}(y)={\widetilde{u}}_{B}(y)\approx{\rm dist}(z,{\widetilde{B}})\geq\rho$.

The arguments for the case $n=2$ are analogous. ∎

###### Proof of Theorem A assuming ${\widetilde{\Omega}}$ to be Wiener
regular, for ${\mathbb{M}}^{n}={\mathbb{S}}^{n}$.

Suppose $n\geq 3$. We can assume that $|\lambda_{B}-\lambda_{\Omega}|\leq 1$,
because otherwise the inequality (1.3) is immediate, since the right hand side
of (1.3) is bounded above by some absolute constant. Recall that we denote
$r_{B}={\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)$. We will prove the theorem
with $a=c_{2}/4$, with $c_{2}$ as in (3.16) and (3.17).

We will show first that

(3.18) $\lambda_{\Omega}-\lambda_{B}\geq
C(\beta)\,\bigg{(}\sup_{s\in(0,c_{2}/4)}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{S}}^{n}}(sB)}\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(x,\tfrac{c_{2}}{4}r_{B})\setminus\Omega)\,d{\mathcal{H}}^{n-1}(y)\bigg{)}^{2}.$

To this end, observe that, from (3.16) in Lemma 3.4 applied to
$\Delta_{z}=\frac{1}{2}\,B$ and to a ball $B_{z}$ as in that lemma (so that
${\operatorname{rad}}(B_{z})\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)$),
we get

$f_{0}(y)\gtrsim\frac{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)\,\operatorname{Cap}_{n-2}(\tfrac{c_{2}}{2}B\setminus\Omega)}{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)^{n-2}}=\frac{\operatorname{Cap}_{n-2}(\tfrac{c_{2}}{2}B\setminus\Omega)}{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)^{n-3}}\quad\text{
for all $y\in\tfrac{1}{4}B_{z}\cap{\widetilde{\Omega}}$.}$

The same inequality holds for
$y\in\tfrac{1}{4}B_{z}\setminus{\widetilde{\Omega}}$, since for these points
it holds
$f_{0}(y)={\widetilde{u}}_{B}(y)\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)$,
by (3.2). Consequently, by Lemma 3.3, choosing $B_{z}$ so that
${\mathcal{H}}^{n+1}({\widetilde{\tfrac{9}{10}B}}\cap A(0,1/2,3/4)\cap
B_{z})\approx{\mathcal{H}}^{n+1}({\widetilde{B}})\approx
r_{B}^{n+1}\approx_{\beta}1,$ we deduce

(3.19) $\lambda_{\Omega}-\lambda_{B}\geq
C(\beta)\left(\int_{{\widetilde{\frac{9}{10}B}}\cap A(0,1/2,3/4)\cap
B_{z}}f_{0}(x)\,d{\mathcal{H}}^{n+1}(x)\right)^{2}\gtrsim\left(r_{B}^{\;4}\operatorname{Cap}_{n-2}(\tfrac{c_{2}}{2}B\setminus\Omega)\right)^{2}.$

Notice that, for any $s\in(0,c_{2}/4)$ and
$y\in\partial_{{\mathbb{S}}^{n}}(sB)$, it holds
$\operatorname{Cap}_{n-2}(\tfrac{c_{2}}{2}B\setminus\Omega)\geq\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(y,\tfrac{c_{2}}{4}r_{B})\setminus\Omega)$.
Thus,

$\operatorname{Cap}_{n-2}(\tfrac{c_{2}}{2}B\setminus\Omega)\geq\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{\partial_{{\mathbb{S}}^{n}}(sB)}\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(y,\tfrac{c_{2}}{4}r_{B})\setminus\Omega)\,d{\mathcal{H}}^{n-1}(y).$

Together with (3.19), this gives (3.18).

We consider now the case $s=1-t\in(c_{2}/4,1)$ (here $t$ is the parameter
appearing in (1.3) and (1.4)). Let $B_{s}=sB$ and

${\widetilde{B}}_{s}=\\{y\in A(0,1/4,1):|y|^{-1}y\in sB\\}$

and denote

$\Gamma_{s}=\partial{\widetilde{B}}_{s}\cap A(0,1/4,1).$

Notice that ${\widetilde{B}}_{s}$ is a Lipschitz domain with Lipschitz
character depending at most on $n$ and $\beta$, since $s\geq c_{2}/4$.

Let ${\varphi}_{0}$ be a smooth bump function which equals $1$ in
$A(0,1/3,4/5)$ is supported on $A(0,1/4,1)$. We define
$f_{1}:{\widetilde{B}}_{s}\to{\mathbb{R}}$ by

(3.20) $f_{1}(x)=\int_{\partial{\widetilde{B}}_{s}\setminus
S}{\varphi}_{0}\,f_{0}\,d\omega_{{\widetilde{B}}_{s}}^{x}=\int_{\Gamma_{s}}f_{0}\,d\omega_{{\widetilde{B}}_{s}}^{x}.$

Since the boundary data ${\varphi}_{0}f_{0}$ is continuous in
$\partial{\widetilde{B}}_{s}$, the function $f_{1}$ is harmonic in
${\widetilde{B}}_{s}$ and continuous in $\overline{{\widetilde{B}}_{s}}$.
These are the advantages of $f_{1}$ over $f_{0}$. Moreover, since $f_{0}$ is
superharmonic in ${\widetilde{B}}_{s}$ and $f_{1}$ harmonic in
${\widetilde{B}}_{s}$, $f_{0}\geq f_{1}$ in $\Gamma_{s}$, and

$\liminf_{y\to z}(f_{0}-f_{1})(z)=\liminf_{y\to z}f_{0}(z)\geq 0\quad\mbox{
for all $z\in\partial{\widetilde{B}}_{s}\setminus\Gamma_{s}$},$

by the maximum principle then it follows that $f_{0}(x)\geq f_{1}(x)$ for all
$x\in{\widetilde{B}}_{s}$ (see Theorem 3.1.5 from [AG], for example).

By the same arguments as the ones for (3.5), we deduce that for $x\in
A(0,1/2,3/4)\cap{\widetilde{B}}_{s/2}$

$\frac{d\omega_{{\widetilde{B}}_{s}}^{x}}{d{\mathcal{H}}^{n}|_{\partial{\widetilde{B}}_{s}}}\approx
1\quad\mbox{ in $\partial{\widetilde{B}}_{s}\cap A(0,1/3,4/5)$.}$

Therefore, by (3.20), for these points $x$,

$f_{1}(x)\gtrsim\int_{\partial{\widetilde{B}}_{s}\cap
A(0,1/3,4/5)}f_{0}(y)\,d{\mathcal{H}}^{n}(y).$

On the other hand, for every $y\in\partial{\widetilde{B}}_{s}\cap
A(0,1/3,4/5)$, by Lemma 3.4 applied to an ${\mathbb{S}}^{n}$-ball $\Delta_{z}$
centered in $z=|y|^{-1}y$ with geodesic radius
$\frac{1}{2}\delta_{B}(z)=\frac{1-s}{2}\,r_{B}=\rho$ , we have

$f_{0}(y)\gtrsim\frac{\rho\,\operatorname{Cap}_{n-2}(c_{2}\Delta_{z}\\!\setminus\Omega)}{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})^{n-2}}\approx\frac{\operatorname{Cap}_{n-2}(c_{2}\Delta_{z}\\!\setminus\Omega)}{((1-s)r_{B})^{n-3}}=\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(|y|^{-1}y,c_{2}(1-s)r_{B})\setminus\Omega)}{((1-s)r_{B})^{n-3}}.$

Thus,

$\displaystyle\int_{\partial{\widetilde{B}}_{s}\cap
A(0,1/3,4/5)}f_{0}(y)\,d{\mathcal{H}}^{n}(y)$
$\displaystyle\gtrsim\int_{\partial{\widetilde{B}}_{s}\cap
A(0,1/3,4/5)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(|y|^{-1}y,c_{2}(1-s)r_{B})\setminus\Omega)}{((1-s)r_{B})^{n-3}}\,d{\mathcal{H}}^{n}(y)$
$\displaystyle\approx\int_{\partial_{{\mathbb{S}}^{n}}(sB)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(y,c_{2}(1-s)r_{B})\setminus\Omega)}{((1-s)r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(y).$

So, for all $x\in A(0,1/2,3/4)\cap{\widetilde{B}}_{s/2}$,

$f_{0}(x)\geq
f_{1}(x)\gtrsim\int_{\partial_{{\mathbb{S}}^{n}}(sB)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(y,c_{2}(1-s)r_{B})\setminus\Omega)}{((1-s)r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(y).$

By Lemma 3.3, this implies

$\lambda_{\Omega}-\lambda_{B}\gtrsim
C(\beta)\left(\int_{\partial_{{\mathbb{S}}^{n}}(sB)}\frac{\operatorname{Cap}_{n-2}(B_{{\mathbb{S}}^{n}}(y,c_{2}(1-s)r_{B})\setminus\Omega)}{((1-s)r_{B})^{n-3}}\,d{\mathcal{H}}^{n-1}(y)\right)^{2},$

for all $s\geq c_{2}/4$. Together with (3.18), this proves the theorem with
$a=c_{2}/4$ and $n\geq 3$. The arguments for the case $n=2$ are almost the
same. The only essential change is that above we have to use the estimate
(3.17) instead of (3.16). Thanks to Lemmas 2.4 and 2.5, this implies that the
theorem is valid for any $a\in(0,1)$. ∎

##  4\. End of the proof of Theorem B in the Wiener regular case for
${\mathbb{M}}^{n}={\mathbb{S}}^{n}$

Notice that the case $s=n$ in Theorem B is an immediate consequence of Theorem
1.2. So we assume that $0<s<n$. Recall that, for $n\geq 3$,

$T_{c_{0}}(\Omega,B,a)=\big{\\{}x\in
B\setminus\Omega:\operatorname{Cap}_{n-2}(B_{{\mathbb{M}}^{n}}(x,a\,\delta_{B}(x))\setminus\Omega)\geq
c_{0}\,\delta_{B}(x)^{n-2}\big{\\}},$

and, in the case $n=2$,

$T_{c_{0}}(\Omega,B,a)=\big{\\{}x\in
B\setminus\Omega:\operatorname{Cap}_{L}(B_{{\mathbb{M}}^{n}}(x,a\,\delta_{B}(x))\setminus\Omega)\geq
c_{0}\,\delta_{B}(x)\big{\\}}.$

To shorten notation we write

$T_{c_{0}}=T_{c_{0}}(\Omega,B,a).$

We will assume $a=1/2$, but all the arguments below work with arbitrary
$a\in(0,1)$. We denote

$\gamma_{s}(\Omega)=\int_{T_{c_{0}}}\delta_{B}(y)^{n-s}\,d{\mathcal{H}}^{s}_{\infty}(y).$

So to prove Theorem B we have to show that, for $0<s\leq n$,

(4.1) $\lambda_{\Omega}-\lambda_{B}\geq
C(a,s,\beta,c_{0})\,\gamma_{s}(\Omega)^{2}.$

To this end, we distinguish two cases:

  * •

Case 1. There exists some point $y_{0}\in T_{c_{0}}$ such that
$\delta_{B}(y_{0})\geq\frac{1}{4}\,{\operatorname{rad}}_{{\mathbb{S}}^{n}}(B)$.

  * •

Case 2. Such point $y_{0}$ does not exist.

###  4.1. Case 1.

Let $y_{0}\in T_{c_{0}}$ be as described above. By covering
$B_{{\mathbb{S}}^{n}}(y_{0},\frac{1}{2}\delta_{B}(y_{0}))$ with a finite
number of ${\mathbb{S}}^{n}$-balls with radius
$\frac{1}{2c_{2}}\delta_{B}(y_{0})$, we find an ${\mathbb{S}}^{n}$-ball
$\Delta_{z}=B_{{\mathbb{S}}^{n}}(z,\frac{1}{2c_{2}}\delta_{B}(y_{0}))$ such
that
$\operatorname{Cap}_{n-2}(\Delta_{z}\setminus\Omega)\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})^{n-2}$
in the case $n\geq 3$ and
$\operatorname{Cap}_{L}(\Delta_{z}\setminus\Omega)\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})^{n-2}$
in the case $n=2$. Then, by Lemma 3.4, in the case $n\geq 3$ the function
$f_{0}$ (defined in (3.14) and (3.15)) satisfies

$f_{0}(y)\gtrsim\frac{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})\,\operatorname{Cap}_{n-2}(c_{2}\Delta_{z}\\!\setminus\Omega)}{{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})^{n-2}}\gtrsim
c_{0}\quad\text{ for all $y\in\tfrac{1}{4}B_{z}\cap{\widetilde{\Omega}}$,}$

for an Euclidean ball $B_{z}\subset{\widetilde{\frac{9}{10}B}}$ centered in
the segment $L_{z}=[0,z]\cap A(0,1/2,3/4)$ with
$\approx{\operatorname{rad}}(B_{z})\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})\approx
1$. The same estimate holds in the case $n=2$, using that
$\operatorname{Cap}_{L}(\Delta_{z}\setminus\Omega)\approx{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{z})$.
Then, by Lemma 3.3, we deduce that either $|\lambda_{B}-\lambda_{\Omega}|>1$
or

$\lambda_{\Omega}-\lambda_{B}\geq
C(\beta)\left(\int_{{\widetilde{\frac{9}{10}B}}\cap
A(0,1/2,3/4)}f_{0}(x)\,d{\mathcal{H}}^{n+1}(x)\right)^{2}\geq
C^{\prime}(\beta)\left(\int_{B_{z}}c_{0}\,d{\mathcal{H}}^{n+1}(x)\right)^{2}\gtrsim_{\beta}c_{0}^{2},$

which implies (4.1) in any case, as $\gamma_{s}(\Omega)\lesssim 1$.

###  4.2. Case 2.

From now on we will assume that $x\in A(0,1/2,3/4)\cap{\widetilde{\Omega}}$
and also that $|x|^{-1}x\in\frac{1}{2}B$. The two conditions and the fact we
are in Case 2 ensure that

${\rm dist}(x,{\mathbb{R}}^{n+1}\setminus{\widetilde{B}})\gtrsim 1\quad\text{
and }\quad{\rm dist}(x,{\widetilde{T}}_{c_{0}})\gtrsim 1,$

where

${\widetilde{T}}_{c_{0}}=\\{y\in A(0,1/4,1):|y|^{-1}y\in T_{c_{0}}\\}.$

######  Main Lemma 4.1.

Under the assumptions of Case 2, for all $x\in
A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ such that $|x|^{-1}x\in\frac{1}{2}B$, we
have

$f_{0}(x)\gtrsim\gamma_{s}(\Omega).$

By Lemma 3.3 and the preceding result, then we deduce that (4.1) also holds in
Case 2 and Theorem B follows. So to conclude the proof of Theorem B it just
remains to prove Main Lemma 4.1.

Let $M\geq 10$ be some constant to be fixed below. For each $y\in T_{c_{0}}$,
let $y^{\prime}\in\partial B$ be such that $\delta_{B}(y)={\rm
dist}_{{\mathbb{S}}^{n}}(y,y^{\prime})$ and let $\Delta_{y}$ be an
${\mathbb{S}}^{n}$-ball centered in $y^{\prime}$ with
${\mathbb{S}}^{n}$-radius $2M\delta_{B}(y)$ (in case that
$2M\delta_{B}(y)>\pi$, then $\Delta_{y}={\mathbb{S}}^{n}$). By Vitali’s
covering theorem, there exists a subfamily $\\{\Delta_{i}^{\prime}\\}_{i\in
I_{0}}\subset\\{\Delta_{y}\\}_{y\in T_{c_{0}}}$ such that
$T_{c_{0}}\subset\bigcup_{i\in I_{0}}5\Delta_{i}^{\prime}$ and the balls
$\Delta_{i}^{\prime}$, $i\in I_{0}$, are pairwise disjoint. We denote
$\Delta_{i}=\frac{1}{M}\Delta_{i}^{\prime}$. Notice that the
${\mathbb{S}}^{n}$-balls $\Delta_{i}$, $i\in I_{0}$, satisfy $M\Delta_{i}\cap
M\Delta_{j}=\varnothing$ for all $i\neq j$. Moreover, for each ball
$\Delta_{i}$ there exists some $y_{i}\in
T_{c_{0}}\cap\frac{1}{2}\overline{\Delta}_{i}$ such that
$\delta_{B}(y_{i})=\frac{1}{2}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i})$.

######  Lemma 4.2.

For $0<s<n$, we have

$\gamma_{s}(\Omega)\lesssim_{M}\sum_{i\in
I_{0}}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i})^{n}.$

###### Proof.

Changing variables, we get

(4.2) $\displaystyle\gamma_{s}(\Omega)$
$\displaystyle=\int_{T_{c_{0}}}\delta_{B}(y)^{n-s}\,d{\mathcal{H}}^{s}_{\infty}(y)=\int_{0}^{\infty}{\mathcal{H}}^{s}_{\infty}(\\{y\in
T_{c_{0}}:\delta_{B}(y)>t^{\frac{1}{n-s}}\\})\,dt$
$\displaystyle=(n-s)\int_{0}^{\infty}t^{n-s-1}\,{\mathcal{H}}^{s}_{\infty}(\\{y\in
T_{c_{0}}:\delta_{B}(y)>t\\})\,dt.$

Since the balls $5\Delta_{i}^{\prime}$ are centered in $\partial B$ and they
cover $T_{c_{0}}$, the condition $\delta_{B}(y)>t$ for $y\in T_{c_{0}}$
implies that $y$ belongs to some ball $5\Delta_{i}^{\prime}$ with
${\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})>t$. Therefore,

${\mathcal{H}}^{s}_{\infty}(\\{y\in
T_{c_{0}}:\delta_{B}(y)>t\\})\lesssim\sum_{i\in
I_{0}:{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})>t}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})^{s}.$

Thus, by Fubini,

$\displaystyle\gamma_{s}(\Omega)$
$\displaystyle\lesssim\int_{0}^{\infty}t^{n-s-1}\\!\sum_{i\in
I_{0}:{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})>t}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})^{s}\,dt$
$\displaystyle=\sum_{i\in
I_{0}}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})^{s}\int_{t<{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})}t^{n-s-1}\,dt\approx\sum_{i\in
I_{0}}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(5\Delta_{i}^{\prime})^{n}\approx_{M}\sum_{i\in
I_{0}}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i})^{n}.$

∎

Now we need to define a family of separated balls centered in

$L:=\partial{\widetilde{B}}\setminus S.$

For each $i\in I_{0}$, let $z_{i}\in{\mathbb{S}}^{n}$ be the center of
$\Delta_{i}$ and consider the segment

$L_{i}=[0,z_{i}]\cap A(0,1/2,3/4).$

Next we choose a maximal family of points $z_{i,j}\in L_{i}$, $j\in J_{i}$,
which are $(M{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i}))$-separated,
so that
$\\#J_{i}\approx(M{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i}))^{-1}$
for each $i\in I_{0}$. It is easy to check that the balls
$B_{i,j}:=B(z_{i,j},{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i}))$,
with $i\in I_{0}$, $j\in J_{i}$, satisfy $c_{3}MB_{i,j}\cap
c_{3}MB_{i^{\prime},j^{\prime}}=\varnothing$ for a suitable absolute constant
$c_{3}>0$ if $(i,j)\neq(i^{\prime},j^{\prime})$. Notice that

(4.3) $\sum_{i\in I_{0},j\in
J_{i}}{\operatorname{rad}}(B_{i,j})^{n+1}=\sum_{i\in
I_{0}}\\#J_{i}\,{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i})^{n+1}\approx_{M}\sum_{i\in
I_{0}}{\operatorname{rad}}_{{\mathbb{S}}^{n}}(\Delta_{i})^{n}\gtrsim_{M}\gamma_{s}(\Omega),$

by Lemma 4.2.

To simplify notation, we denote

$\\{B_{k}\\}_{k\in K}=\\{B_{i,j}\\}_{i\in I_{0},j\in J_{i}}.$

We need now to construct an auxiliary Lipschitz graph $\Gamma$ which will play
an important role in the proof of Main Lemma 4.1. First observe that, from the
fact that
$\operatorname{Cap}_{n-2}(B(y_{i},\frac{1}{2}\delta_{B}(y_{i}))\cap{\mathbb{S}}^{n}\setminus\Omega)\geq
c_{0}r^{n-2}$ for each $i\in I_{0}$ in the case $n\geq 3$ and the analogous
estimate for the logarithmic capacity in the case $n=2$ (since $y_{i}\in
T_{c_{0}}$), by Lemma 2.3 we deduce that, for all $k\in K$,

$\operatorname{Cap}_{n-1}(\\{y\in B_{k}\setminus{\widetilde{\Omega}}:{\rm
dist}(y,\partial{\widetilde{B}})\geq
c_{4}\,{\operatorname{rad}}(B_{k})\\}\big{)}\gtrsim{\operatorname{rad}}(B_{k})^{n-1},$

for a suitable constant $c_{4}\in(0,1/4)$. By covering
$\\{y\in\partial{\widetilde{\Omega}}\cap B_{k}:{\rm
dist}(y,\partial{\widetilde{B}})\geq c_{4}\,{\operatorname{rad}}(B_{k})\\}$
with balls with radius $c_{4}\,{\operatorname{rad}}(B_{k})/10$, we infer that
there exists some point $z_{k}\in B_{k}\setminus{\widetilde{\Omega}}$ such
that ${\rm dist}(z_{k},\partial{\widetilde{B}})\geq
c_{4}\,{\operatorname{rad}}(B_{k})$ and so that the ball
$B_{k}^{\prime}:=B(z_{k},c_{4}{\operatorname{rad}}(B_{k})/4)$ satisfies

(4.4) $4B_{k}^{\prime}\subset{\widetilde{B}}\quad\text{ and
}\quad\operatorname{Cap}_{n-1}\big{(}\tfrac{1}{4}B_{k}^{\prime}\setminus{\widetilde{\Omega}}\big{)}\gtrsim{\operatorname{rad}}(B_{k})^{n-1}.$

For each $k\in K$, we consider a Lipschitz function
${\varphi}_{k}:L\to{\widetilde{B}}$ supported in $\frac{1}{2}c_{2}MB_{k}$ such
that ${\rm Lip}({\varphi}_{k})\lesssim(M\,{\operatorname{rad}}(B_{k}))^{-1}$
and so that $z_{k}$ belongs to the graph of ${\varphi}_{k}$. It is easy to
check that such a function exists. Then we define
${\varphi}:L\to{\widetilde{B}}$ as follows:

${\varphi}(y)=\left\\{\begin{array}[]{ll}{\varphi}_{k}(y)&\quad\mbox{if $y\in
c_{2}MB_{k}\cap L$ fo some $k\in K$,}\\\ &\\\
0&\quad\mbox{otherwise.}\end{array}\right.$

We assume that the balls $c_{2}MB_{k}$, $k\in K$, are pairwise disjoint
(otherwise we could replace $c_{2}$ by a smaller constant in the construction
above), and then it is clear that the definition is correct. It is also clear
that ${\varphi}$ is a Lipschitz function whose graph is contained in
$\overline{{\widetilde{B}}}$. Further, because the functions ${\varphi}_{k}$
are $cM$-Lipschitz and because of the smoothness of $L$, it follows easily
that

(4.5) ${\varphi}|_{F}$ is $c^{\prime}M^{-1}$-Lipschitz for any set $F\subset
L$ with ${\rm diam}(F)$ small enough (depending on $M$).

We denote by $\Gamma$ the graph of ${\varphi}$, and we let $F_{\Gamma}$ be the
closed set comprised between $L$ and $\Gamma$. Next we consider the domain

${\widetilde{\Omega}}_{\Gamma}={\widetilde{B}}\setminus F_{\Gamma}.$

Notice that ${\widetilde{\Omega}}_{\Gamma}$ is a Lipschitz domain contained in
${\widetilde{B}}$ whose boundary equals
$\Gamma\cup(S\cap\partial{\widetilde{B}})$.

Recall that

$f_{0}(y)=\int_{\partial{\widetilde{\Omega}}_{0}\setminus
S}{\widetilde{u}}_{B}\,d\omega_{{\widetilde{\Omega}}_{0}}^{y}\quad\mbox{ for
all $y\in{\widetilde{\Omega}}_{0}$}$

and

$f_{0}(x)={\widetilde{u}}_{B}(x)\quad\mbox{ for all
$x\in{\widetilde{B}}\setminus{\widetilde{\Omega}}_{0}$},$

where $\Omega_{0}=\Omega\cap B$ and
${\widetilde{\Omega}}_{0}={\widetilde{\Omega}}\cap{\widetilde{B}}$. Now we
define $f_{1}:{\widetilde{\Omega}}_{\Gamma}\to{\mathbb{R}}$ by

(4.6) $f_{1}(x)=\int_{\partial{\widetilde{\Omega}}_{\Gamma}\setminus
S}f_{0}\,d\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}=\int_{\Gamma}f_{0}\,d\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}.$

Remark that $f_{1}$ is harmonic in ${\widetilde{\Omega}}_{\Gamma}$ and
continuous in $\overline{{\widetilde{\Omega}}_{\Gamma}}$, except perhaps in
$\Gamma\cap S$, because the boundary data $f_{0}\,\chi_{\Gamma}$ is continuous
in $\partial{\widetilde{\Omega}}_{\Gamma}$. Since $f_{0}$ is superharmonic in
${\widetilde{\Omega}}_{\Gamma}$ and $f_{1}$ harmonic in
${\widetilde{\Omega}}_{\Gamma}$, $f_{0}$ and $f_{1}$ have the same boundary
values in $\Gamma$, and

$\liminf_{y\to z}(f_{0}-f_{1})(z)=\liminf_{y\to z}f_{0}(z)\geq 0\quad\mbox{
for all $z\in\partial{\widetilde{\Omega}}_{\Gamma}\setminus\Gamma$},$

by the maximum principle then it follows that

(4.7) $f_{0}(x)\geq f_{1}(x)\quad\mbox{ for all
$x\in{\widetilde{\Omega}}_{\Gamma}$}$

(see Theorem 3.1.5 from [AG]).

######  Lemma 4.3.

For each $k\in K$, let $\Gamma_{k}$ be the following subset of $\Gamma$:

$\Gamma_{k}=\big{\\{}y\in{\varphi}_{k}(c_{2}MB_{k}\cap L):{\rm
dist}(y,L)\geq\tfrac{1}{10}{\operatorname{rad}}(B_{k})\big{\\}}.$

Then

$f_{0}(y)=f_{1}(y)\gtrsim{\operatorname{rad}}(B_{k})\quad\text{ for all
$y\in\Gamma\cap\tfrac{1}{4}B_{k}^{\prime}$.}$

###### Proof.

The fact that $f_{0}(y)=f_{1}(y)$ for
$y\in\Gamma\cap\tfrac{1}{4}B_{k}^{\prime}$ is an immediate consequence of the
definition of $f_{1}(y)$.

By (4.4) and Lemma 2.6, for all
$y\in\frac{1}{4}B_{k}^{\prime}\cap{\widetilde{\Omega}}_{0}$, we have

$\omega_{{\widetilde{\Omega}}_{0}}^{y}(2B_{k}^{\prime})\gtrsim\frac{\operatorname{Cap}_{n-1}(\tfrac{1}{4}B_{k}^{\prime}\setminus{\widetilde{\Omega}}_{0})}{{\operatorname{rad}}(B_{k}^{\prime})^{n-1}}\gtrsim
1.$

Then, taking into account that ${\widetilde{u}}_{B}(z)\approx{\rm
dist}(z,\partial{\widetilde{B}})\approx{\operatorname{rad}}(B_{k}^{\prime})\approx{\operatorname{rad}}(B_{k})$
for all $z\in 2B_{k}^{\prime}$, we infer

$f_{0}(y)\geq\int_{\partial{\widetilde{\Omega}}_{0}\cap
2B_{k}^{\prime}}{\widetilde{u}}_{B}\,d\omega_{{\widetilde{\Omega}}_{0}}^{y}\gtrsim{\operatorname{rad}}(B_{k}^{\prime})\,\omega_{{\widetilde{\Omega}}_{0}}^{y}(2B_{k}^{\prime})\gtrsim{\operatorname{rad}}(B_{k})\quad\mbox{
for all $y\in\frac{1}{4}B_{k}^{\prime}\cap{\widetilde{\Omega}}_{0}$.}$

On the other hand, for
$y\in\tfrac{1}{4}B_{k}^{\prime}\setminus{\widetilde{\Omega}}_{0}$, we also
have $f_{0}(y)={\widetilde{u}}_{B}(y)\approx{\rm
dist}(z,{\widetilde{B}})\approx{\operatorname{rad}}(B_{k})$. ∎

From (4.7), the definition of $f_{1}$ in (4.6), the preceding lemma, and the
doubling property of $\omega_{{\widetilde{\Omega}}_{\Gamma}}$ (since
${\widetilde{\Omega}}_{\Gamma}$ is a Lipschitz domain) we deduce that

$f_{0}(x)\geq f_{1}(x)\gtrsim\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(\tfrac{1}{4}B_{k}^{\prime})\approx\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k}\cap\Gamma)\quad\mbox{
for all $x\in{\widetilde{\Omega}}_{\Gamma}$}.$

Hence the Main Lemma 4.1 is a consequence of the following result and (4.3).

######  Lemma 4.4.

For all $x\in A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ such that
$|x|^{-1}x\in\frac{1}{2}B$, we have

$\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k}\cap\Gamma)\gtrsim_{M}\sum_{k\in
K}{\operatorname{rad}}(B_{k})^{n+1}.$

To prove this, we will need the following auxiliary lemma.

######  Lemma 4.5.

Given any $\alpha\in(0,1/2)$, suppose that the constant $M$ above is chosen
large enough, depending on $\alpha$. Then there exists some constant
$C=C(\alpha)$ such that, for all $x\in A(0,1/2,3/4)\cap{\widetilde{\Omega}}$
such that $|x|^{-1}x\in\frac{1}{2}B$, all $y\in\Gamma\cap A(0,1/3,4/5)$, and
$0<r\leq R\leq 2$, we have

(4.8)
$C^{-1}\bigg{(}\frac{R}{r}\bigg{)}^{n-\alpha}\leq\frac{\omega_{{\widetilde{\Omega}}_{\Gamma}^{x}}(B(y,R))}{\omega_{{\widetilde{\Omega}}_{\Gamma}^{x}}(B(y,r))}\leq
C\bigg{(}\frac{R}{r}\bigg{)}^{n+\alpha}.$

This lemma follows from the local Reifenberg flatness of
$\partial{\widetilde{\Omega}}_{\Gamma}$ far away from $S$, by (4.5) and taking
$M$ large enough. The arguments are essentially the same as the ones for
Theorem 4.1 from [KT] and Lemma 3.7 from [PT1], and so we omit the proof. From
now on, we assume $M$ large enough so that we can take $a=1/4$ in (4.8).

###### Proof of Lemma 4.4.

Consider the function $h_{0}:\partial{\widetilde{B}}\to{\mathbb{R}}$ given by

$h_{0}(y)=\sum_{k\in K}{\operatorname{rad}}(B_{k})\,\chi_{B_{k}\cap L}(y)$

and let $h:{\widetilde{B}}\to{\mathbb{R}}$ be its harmonic extension to
${\widetilde{B}}$:

$h(y)=\int h_{0}(z)\,d\omega_{{\widetilde{B}}}^{y}(z)=\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{B}}}^{y}(B_{k}).$

By Lemma 3.1, we have
$\frac{d\omega_{{\widetilde{B}}}^{x}}{d{\mathcal{H}}^{n}}\approx 1$ on
${\widetilde{B}}\cap B_{k}$ for each $k$ and for $x\in
A(0,1/2,3/4)\cap{\widetilde{\Omega}}$ such that $|x|^{-1}x\in\frac{1}{2}B$.
Using also the fact that $h$ is harmonic in ${\widetilde{B}}$, that
$x\in{\widetilde{B}}\cap{\widetilde{\Omega}}_{\Gamma}$, and that $h$ vanishes
in $L\setminus\bigcup_{k\in
K}\frac{1}{2}c_{2}B_{k}=\Gamma\setminus\bigcup_{k\in K}\frac{1}{2}c_{2}B_{k}$,
we derive

$\displaystyle\sum_{k\in K}{\operatorname{rad}}(B_{k})^{n+1}$
$\displaystyle\approx\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{B}}}^{x}(B_{k}\cap
L)=\int_{L}h\,d\omega_{{\widetilde{B}}}^{x}=\int_{\Gamma}h\,d\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}$
$\displaystyle=\sum_{k\in
K}\int_{\frac{1}{2}c_{2}B_{k}\cap\Gamma}h(y)\,d\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(y)=\sum_{j,k\in
K}{\operatorname{rad}}(B_{j})\,\int_{\frac{1}{2}c_{2}B_{k}\cap\Gamma}\omega_{{\widetilde{B}}}^{y}(B_{j})\,d\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(y)$
$\displaystyle\lesssim\sum_{j,k\in
K}{\operatorname{rad}}(B_{j})\,\sup_{y\in\frac{1}{2}c_{2}B_{k}\cap\Gamma}\omega_{{\widetilde{B}}}^{y}(B_{j})\,\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k}).$

For all $j,k\in K$, we denote

$D(B_{j},B_{k})={\rm
dist}(B_{j},B_{k})+{\operatorname{rad}}(B_{j})+{\operatorname{rad}}(B_{k}).$

######  Claim 2.

For all $j,k\in K$, we have

(4.9)
$\sup_{y\in\frac{1}{2}c_{2}B_{k}\cap\Gamma}\omega_{{\widetilde{B}}}^{y}(B_{j})\lesssim\frac{{\operatorname{rad}}(B_{j})^{n}\,{\operatorname{rad}}(B_{k})}{D(B_{j},B_{k})^{n+1}}.$

We defer the proof of the claim to the end of the proof of the lemma. Assuming
this for the moment, we get

(4.10) $\sum_{k\in K}{\operatorname{rad}}(B_{k})^{n+1}\lesssim\sum_{j,k\in
K}\,\frac{{\operatorname{rad}}(B_{j})^{n+1}\,{\operatorname{rad}}(B_{k})}{D(B_{j},B_{k})^{n+1}}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k})=:S_{0}.$

To conclude the proof of the lemma we will bound the sum $S_{0}$ on right hand
side above by

$\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k}).$

To this end, we split the sum $S_{0}$ as follows:

$S_{0}=\Bigg{(}\sum_{\begin{subarray}{c}j,k\in K:\\\
{\operatorname{rad}}(B_{j})\leq{\operatorname{rad}}(B_{k})\end{subarray}}\\!+\sum_{\begin{subarray}{c}j,k\in
K:\\\
{\operatorname{rad}}(B_{j})>{\operatorname{rad}}(B_{k})\end{subarray}}\Bigg{)}\,\,\frac{{\operatorname{rad}}(B_{j})^{n+1}\,{\operatorname{rad}}(B_{k})}{D(B_{j},B_{k})^{n+1}}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k})=:S_{0,1}+S_{0,2}.$

#### Estimate of $S_{0,1}$

Using that ${\operatorname{rad}}(B_{j})\leq{\operatorname{rad}}(B_{k})$ in
this sum, we write

$\displaystyle S_{0,1}$ $\displaystyle\leq\sum_{\begin{subarray}{c}j,k\in
K:\\\
{\operatorname{rad}}(B_{j})\leq{\operatorname{rad}}(B_{k})\end{subarray}}\frac{{\operatorname{rad}}(B_{j})^{n}\,{\operatorname{rad}}(B_{k})^{2}}{D(B_{j},B_{k})^{n+1}}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k})$
$\displaystyle=\sum_{k\in
K}{\operatorname{rad}}(B_{k})^{2}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k})\,\,\sum_{i\geq
0}\sum_{\begin{subarray}{c}j\in
K:\,{\operatorname{rad}}(B_{j})\leq{\operatorname{rad}}(B_{k}),\\\
2^{i}{\operatorname{rad}}(B_{k})\leq
D(B_{j},B_{k})<2^{i+1}{\operatorname{rad}}(B_{k})\end{subarray}}\\!\\!\frac{{\operatorname{rad}}(B_{j})^{n}\,}{D(B_{j},B_{k})^{n+1}}.$

Notice that, for a fixed $i\geq 0$, the conditions
${\operatorname{rad}}(B_{j})\leq{\operatorname{rad}}(B_{k})$,
$2^{i}{\operatorname{rad}}(B_{k})\leq
D(B_{j},B_{k})<2^{i+1}{\operatorname{rad}}(B_{k})$ imply that $B_{j}\subset
2^{i+2}B_{k}$. Then we get

(4.11) $\displaystyle\sum_{\begin{subarray}{c}j\in
K:\,{\operatorname{rad}}(B_{j})\leq{\operatorname{rad}}(B_{k}),\\\
2^{i}{\operatorname{rad}}(B_{k})\leq
D(B_{j},B_{k})<2^{i+1}{\operatorname{rad}}(B_{k})\end{subarray}}\\!\\!\\!\frac{{\operatorname{rad}}(B_{j})^{n}}{D(B_{j},B_{k})^{n+1}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\in K:\,B_{j}\subset
2^{i+2}B_{k}\end{subarray}}\\!\\!\frac{{\mathcal{H}}^{n}(L\cap
B_{j})}{(2^{i}{\operatorname{rad}}(B_{k}))^{n+1}}$
$\displaystyle\lesssim\frac{{\mathcal{H}}^{n}(L\cap
2^{i+2}B_{k})}{(2^{i}{\operatorname{rad}}(B_{k}))^{n+1}}\approx\frac{1}{2^{i}{\operatorname{rad}}(B_{k})}.$

Therefore,

$S_{0,1}\lesssim\sum_{k\in
K}{\operatorname{rad}}(B_{k})^{2}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k})\,\,\sum_{i\geq
0}\frac{1}{2^{i}{\operatorname{rad}}(B_{k})}\approx\sum_{k\in
K}{\operatorname{rad}}(B_{k})\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k}).$

#### Estimate of $S_{0,2}$

Now we assume that ${\operatorname{rad}}(B_{j})>{\operatorname{rad}}(B_{k})$.
Let $A\geq 1$ be the minimal constant such that $B_{k}\subset AB_{j}$. Notice
that ${\operatorname{rad}}(AB_{j})\approx D(B_{j},B_{k})$. By Lemma 4.5, using
also the doubling property of $\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}$
(since ${\widetilde{\Omega}}_{\Gamma}$ is a Lipschitz domain), we have

$\displaystyle\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{j})$
$\displaystyle\gtrsim\bigg{(}\frac{{\operatorname{rad}}(B_{j})}{{\operatorname{rad}}(AB_{j})}\bigg{)}^{n+a}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(AB_{j})$
$\displaystyle\gtrsim\bigg{(}\frac{{\operatorname{rad}}(B_{j})}{{\operatorname{rad}}(AB_{j})}\bigg{)}^{n+\alpha}\bigg{(}\frac{{\operatorname{rad}}(AB_{j})}{{\operatorname{rad}}(B_{k})}\bigg{)}^{n-\alpha}\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k})\approx\frac{{\operatorname{rad}}(B_{j})^{n+\alpha}}{{\operatorname{rad}}(B_{k})^{n-\alpha}\,D(B_{j},B_{k})^{2\alpha}}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{k}).$

Plugging this estimate into the definition of $S_{0,2}$, we obtain

$\displaystyle S_{0,2}$ $\displaystyle\lesssim\sum_{\begin{subarray}{c}j,k\in
K:\\\
{\operatorname{rad}}(B_{j})>{\operatorname{rad}}(B_{k})\end{subarray}}\\!\\!\frac{{\operatorname{rad}}(B_{j})^{n+1}\,{\operatorname{rad}}(B_{k})}{D(B_{j},B_{k})^{n+1}}\,\frac{{\operatorname{rad}}(B_{k})^{n-\alpha}\,D(B_{j},B_{k})^{2\alpha}}{{\operatorname{rad}}(B_{j})^{n+\alpha}}\,\omega_{{\widetilde{\Omega}}_{\Gamma}}^{x}(B_{j})$
$\displaystyle\approx\sum_{\begin{subarray}{c}j,k\in K:\\\