# Density estimates and the fractional Sobolev inequality for sets of zero
$s$-mean curvature

Jack Thompson  The University of Western Australia (M019)35 Stirling Highway
PERTH WA 6009 Australia<EMAIL_ADDRESS>
###### Abstract.

We prove that measurable sets $E\subset\mathbb{R}^{n}$ with locally finite
perimeter and zero $s$-mean curvature satisfy the surface density estimates:

$\displaystyle\operatorname{Per}(E;B_{R}(x))\geq CR^{n-1}$

for all $R>0$, $x\in\partial^{\ast}E$. The $C$ depends only on $n$ and $s$,
and remains bounded as $s\to 1^{-}$. As an application, we prove that the
fractional Sobolev inequality holds on the boundary of sets with zero $s$-mean
curvature.

###### 2010 Mathematics Subject Classification:

Primary 35R11, 53A10

##  1\. Introduction and main results

A fundamental result in the regularity of minimal surfaces is that a
measurable set $E\subset\mathbb{R}^{n}$ that is is locally perimeter
minimising satisfies the thick density estimates:

(1.1) $\displaystyle|E\cap B_{R}(x)|\geq CR^{n}\text{ and }|B_{R}(x)\setminus
E|\geq CR^{n}$

for all $R>0$, $x\in\partial^{\ast}E$ where $B_{R}(x)$ is a ball in
$\mathbb{R}^{n}$ centred at $x$ with radius $R$ and $|F|$ denotes the Lebesgue
measure of $F$ in $\mathbb{R}^{n}$, see for example [Mag12, Section 16.2].
Here the constant $C>0$ depends only on $n$. Since any sufficiently regular
set $E$ satisfies (1.1) for $R$ sufficiently small and with the constant
depending on the $C^{2}$ regularity of $E$ at $x$, the non-triviality of (1.1)
is that it holds for all $R>0$ and that the constant is universal. The
assumption that $E$ is locally minimising is essential to the proof of (1.1)
since the proof takes advantage of the fact that $E\cap\partial B_{R}(x)$ is a
competitor for $\partial^{\ast}E\cap B_{R}(x)$.

If one only assumes that $E$ has zero mean curvature, that is, it is locally
stationary with respect to the perimeter functional then one can obtain the
surface density estimate:

(1.2) $\displaystyle\operatorname{Per}(E;B_{R}(x))\geq CR^{n-1}$

for all $R>0$, $x\in\partial^{\ast}E$, and with $C$ depending only on $n$.
Indeed, the monotonicity formula for sets of zero mean curvature immediately
implies that

$\displaystyle\frac{\operatorname{Per}(E;B_{R}(x))}{R^{n-1}}\geq\lim_{r\to
0^{+}}\frac{\operatorname{Per}(E;B_{r}(x))}{r^{n-1}}=n\omega_{n}$

where $\omega_{n}=|B_{1}|$, see [Mag12, Theorem 17.16]. Observe that the thick
density estimates (1.1) directly imply the surface density estimates (1.2) via
the restricted isoperimetric inequality

$\displaystyle\operatorname{Per}(E;B_{R}(x))\geq C\big{(}\min\\{|E\cap
B_{R}(x)|,|B_{R}(x)\setminus E\\}\big{)}^{\frac{n-1}{n}},$

but sets with zero mean curvature do not necessarily satisfy (1.1), for
example, if $E=\mathbb{R}^{n-1}\times(0,1)$ then $\partial E$ has zero mean
curvature, but $|E\cap B_{R}|\leq|B^{n-1}_{R}\times(0,1)|\leq CR^{n-1}$.

In the celebrated paper [CRS10], a nonlocal (or fractional) concept of
perimeter, known as the $s$-perimeter, was introduced that was a natural
generalisation of the classical perimeter. As well as motivating and defining
the $s$-perimeter, Caffarelli, Roquejoffre, and Savin in [CRS10] establish
several foundational results for the regularity theory of minimisers111For a
precise definition, see Section 2 of the $s$-perimeter including the existence
of minimisers, improvement of flatness, a monotonicity formula, blow-up limits
to $s$-minimal cones, and Federer’s dimension reduction. In particular, they
also established minimisers of the $s$-perimeter satisfy the thick density
estimates (1.1). Since their paper, the regularity theory of critical points
of the $s$-perimeter (stationary, stable, and minimisers) has become an
incredibly active area of research. Some important results include: interior
higher regularity [BFV14]; boundary regularity and stickiness [DSV17, DSV20];
classification of cones for $n=2$ [SV13]; classification of cones for $s$
close to $1$ [CV13]; regularity of $s$-minimal graphs [CC19]; explicit
examples [CV20, DPW18]; and Yau’s conjecture [CFS24a]. For a nice survey of
the current literature, see [DV18].

In analogy to the classical case, if $E$ is $s$-stationary, that is
stationary1 with respect to the $s$-perimeter, and $\partial E$ is
sufficiently smooth then $E$ satisfies

$\displaystyle\mathrm{H}_{s,E}(x):=\lim_{\varepsilon\to
0^{+}}\int_{\mathbb{R}^{n}\setminus
B_{\varepsilon}(x)}\frac{\chi_{\mathbb{R}^{n}\setminus
E}(y)-\chi_{E}(y)}{|x-y|^{n+s}}\mathop{}\\!dy=0$

for all $x\in\partial E$, see [Fig+15]. It is conventional to call
$\mathrm{H}_{s,E}$ the $s$-mean curvature of $E$. Much of the literature
focuses on the regularity of sets that minimise the $s$-perimeter and
relatively little is known about the regularity of sets with zero of $s$-mean
curvature. Indeed, although (1.1) was proven in [CRS10] for minimisers of the
$s$-perimeter, it is still an open problem to prove that sets of zero $s$-mean
curvature satisfy the surface density estimate (1.2). The main result of this
note establishes such an estimate.

######  Theorem 1.1.

jG1dQXin Let $s\in(0,1)$ and $E\subset\mathbb{R}^{n}$ be a measurable set with
locally finite perimeter satisfying $\mathrm{H}_{s,E}=0$ on
$\partial^{\ast}E$. Then

$\displaystyle\operatorname{Per}(E;B_{R})\geq CR^{n-1}\text{ for all }R>0.$

The constant $C>0$ depends only on $n$ and $s$, and remains bounded as $s\to
1^{-}$.

The precise definition $\mathrm{H}_{s,E}=0$ on $\partial^{\ast}E$ for a set of
locally finite perimeter is given in Section 2. Several articles in the
literature have explored related density estimates to jG1dQXin including
[CSV19] and [CFS24a, CFS24]. In [CSV19], the authors establish upper surface
density estimates of the form

(1.3) $\displaystyle\operatorname{Per}(E;B_{R}(x))\leq CR^{n-1}$

for all $x\in\partial^{\ast}E$, $R>0$ for stationary, stable sets $E$ with
respect to the $s$-perimeter and other nonlocal perimeters with more general
kernels. It is interesting to note that in the classical case of stable
surfaces with zero mean curvature, (1.3) is known only for $n=2$ and is
famously open for $n\geq 3$. In [CFS24a, CFS24], the main focus is on
establishing a nonlocal Yau’s conjecture, that is, in any closed Riemannian
manifold there exist infinitely many sets of zero $s$-mean curvature. In
particular, they prove the estimate (1.3) for sets with zero $s$-mean
curvature in closed Riemannian manifolds. We should mention that, though most
of the techniques present in this paper are also present in [CSV19, CFS24a,
CFS24], particularly [CFS24a, CFS24]; however, the result jG1dQXin is never
explicitly stated nor is its connection to the fractional Sobolev inequality
on hypersurfaces, see below.

A nice application of jG1dQXin is that the fractional Sobolev inequality holds
on hypersurfaces in $\mathbb{R}^{n}$ with zero $s$-mean curvature. The
classical Sobolev inequality in $\mathbb{R}^{n}$ states that if $1\leq p<n$
and $p^{\ast}:=\frac{np}{n-p}$ then

(1.4) $\displaystyle\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}\leq C\|\nabla
u\|_{L^{p}(\mathbb{R}^{n})}$

for all $u\in C^{\infty}_{0}(\mathbb{R}^{n})$ with $C>0$ depending only on $n$
and $p$. It is also interesting to ask if the Sobolev inequality can hold on
hypersurfaces $M^{n}\hookrightarrow\mathbb{R}^{n+1}$. It is easy to see that
(1.4) (with $\mathbb{R}^{n}$ replaced with $M$ and $\nabla=\nabla_{M}$ the
gradient with respect to the induced metric from $\mathbb{R}^{n+1}$) cannot
hold verbatim since if $M$ is compact then $u=1\in C^{\infty}_{0}(M)$;
however, a Sobolev-type inequality does hold if one adds an extra $L^{p}$
error term involving the mean curvature $\mathrm{H}_{M}$:

(1.5) $\displaystyle\|u\|_{L^{p^{\ast}}(M)}\leq
C\big{(}\|\nabla_{M}u\|_{L^{p}(M)}+\|\mathrm{H}_{M}u\|_{L^{p}(M)}\big{)}$

for all $u\in C^{\infty}_{0}(M)$. The inequality (1.5) is sometimes known as
the Michael-Simon and Allard inequality after [All72] and [MS73]. Also, see
[Bre21] where the optimal constant in (1.5) is obtained and [CM22] for a
simple proof of (1.5).

Naturally, one would like to know if (1.5) extends to the nonlocal case. If
$M=\mathbb{R}^{n}$, $1\leq p<n/s$, and $p^{\ast}=\frac{np}{n-sp}$ then it is
well known that

$\displaystyle\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}\leq
C[u]_{W^{s,p}(\mathbb{R}^{n})}$

where

(1.6) $\displaystyle[v]_{W^{s,p}(\mathbb{R}^{n})}$
$\displaystyle=\bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\mathop{}\\!dy\mathop{}\\!dx\bigg{)}^{\frac{1}{p}},$

see [DPV12] and references therein. If $E\subset\mathbb{R}^{n}$ is an open set
with locally finite perimeter (or sufficiently smooth boundary) and

$\displaystyle[v]_{W^{s,p}(\partial^{\ast}E)}$
$\displaystyle=\bigg{(}\int_{\partial^{\ast}E}\int_{\partial^{\ast}E}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\mathop{}\\!d\mathcal{H}^{n-1}_{y}\mathop{}\\!d\mathcal{H}^{n-1}_{x}\bigg{)}^{\frac{1}{p}}$

then it is currently an open problem whether the inequality

(1.7) $\displaystyle\|u\|_{L^{p^{\ast}}(\partial^{\ast}E)}\leq
C\big{(}[u]_{W^{s,p}(\partial^{\ast}E)}+\|\mathrm{H}_{s,E}u\|_{L^{p}(\partial^{\ast}E)}\big{)}$

holds for all $u\in C^{\infty}_{0}(\partial^{\ast}E)$. Recently, it was shown
in [CCC23] that (1.7) does hold provided that $E$ is convex. Their argument
relies on first establishing a pointwise lower bound on the $s$-mean curvature
in terms of the perimeter. Unfortunately, as they point out in their paper,
this pointwise inequality clearly cannot hold in the non-convex case, so the
argument doesn’t easily generalise to the case of arbitrary $E$. Furthermore,
in [CC19, Proposition 5.2], it was proven via an extension of an (unpublished)
argument due to Brezis (see [BV16, Theorem 2.2.1] for this argument), that the
boundary density estimate (1.2) implies (1.7) without the term involving the
$s$-mean curvature. Hence, via [CC19, Proposition 5.2], jG1dQXin immediately
implies that (1.7) holds for sets of zero $s$-mean curvature:

######  Corollary 1.2.

Let $s\in(0,1)$, $1\leq p<n/s$, $p^{\ast}=\frac{np}{n-sp}$, and
$E\subset\mathbb{R}^{n+1}$ be a measurable set with locally finite perimeter
satisfying $\mathrm{H}_{s,E}=0$ on $\partial^{\ast}E$. Then the Sobolev
inequality

$\displaystyle\|u\|_{L^{p^{\ast}}(\partial^{\ast}E)}\leq
C[u]_{W^{s,p}(\partial^{\ast}E)}$

holds for all $u\in C^{\infty}_{0}(\partial^{\ast}E)$. The constant $C>0$
depends only on $n$ and $s$.

###  1.1. Organisation of paper

The paper is organised as follows. In Section 2, we fix notation and give
standard definitions of the classical theory of BV functions and minimal
surfaces, and definitions from the theory of nonlocal minimal surfaces. In
Section 3, we prove an interpolation inequality between restricted
$s$-perimeter and the classical perimeter. In Section 4, we survey some
results related to the Caffarelli-Silvestre extension in the context of
nonlocal minimal surfaces that will be essential to the proof of jG1dQXin.
Finally, in Section 5, we give the proof of jG1dQXin.

## Acknowledgements

Jack Thompson is supported by an Australian Government Research Training
Program Scholarship. JT would like to thank Serena Dipierro, Giovanni
Giacomin, and Enrico Valdinoci for their valuable conversations and their
comments on the first draft of this note. He would also like to thank João
Gonçalves da Silva for the example at the end of the first page and Tommaso Di
Ubaldo for his deep insights.

##  2\. Definitions and notation

First, we recall some notation and definitions of functions of bounded
variation and the classical theory of minimal surfaces. Let
$\Omega\subset\mathbb{R}^{n}$ be an open set. Given a function $u\in
L^{1}(\Omega)$ the _total variation_ of $u$ in $\Omega$ is given by

$\displaystyle|\nabla
u|(\Omega)=\sup\bigg{\\{}\int_{\Omega}u\operatorname{div}\phi\mathop{}\\!dx\text{
s.t. }\phi\in
C^{1}_{0}(\Omega;\mathbb{R}^{n}),\|\phi\|_{L^{\infty}(\Omega;\mathbb{R}^{n})}\leq
1\bigg{\\}}.$

The space $\operatorname{BV}(\Omega)$ is defined to be the set of $u\in
L^{1}(\Omega)$ such that $|\nabla u|(\Omega)<+\infty$ and the space
$\operatorname{BV}_{\mathrm{loc}}(\Omega)$ is the set of functions in
$\operatorname{BV}(\Omega^{\prime})$ for all
$\Omega^{\prime}\subset\subset\Omega$. Moreover, we say a measurable set
$E\subset\mathbb{R}^{n}$ has _finite perimeter_ in $\Omega$ if
$\chi_{E}\in\operatorname{BV}(\Omega)$ where $\chi_{A}$ denotes the
characteristic function of $A$. In this case, we define the _perimeter of_ $E$
in $\Omega$ by

$\displaystyle\operatorname{Per}(E;\Omega)=|\nabla\chi_{E}|(\Omega).$

We also say $E$ has _locally finite perimeter_ in $\Omega$ if
$\chi_{E}\in\operatorname{BV}_{\mathrm{loc}}(\Omega)$ and simply $E$ has
locally finite perimeter if
$\chi_{E}\in\operatorname{BV}_{\mathrm{loc}}(\mathbb{R}^{n})$. Finally, given
$E$ with finite perimeter in $\Omega$, the distributional gradient of
$\chi_{E}$, i.e. $\nabla\chi_{E}$, is a vector-valued Radon measure on
$\mathbb{R}^{n}$. Then we can define the _reduced boundary_ of $E$, denoted
$\partial^{\ast}E$, as the set

$\displaystyle\bigg{\\{}x\in\partial E\text{ s.t.
}|\nabla\chi_{E}|(B_{r}(x))>0\text{ for all }r>0,\text{ and }\lim_{r\to
0^{+}}\frac{\nabla\chi_{E}(B_{r}(x))}{|\nabla\chi_{E}|(B_{r}(x))}\text{ exists
and is in }\mathbb{S}^{n-1}\bigg{\\}}.$

Now, let us turn to the nonlocal case. Let $s\in(0,1)$ and

$\displaystyle\mathcal{I}_{s}(A,B)=\int_{A}\int_{B}\frac{\mathop{}\\!dy\mathop{}\\!dx}{|x-y|^{n+s}}.$

The $s$-perimeter of a measurable set $E\subset\mathbb{R}^{n}$ in $\Omega$ is
defined to be

$\displaystyle\operatorname{Per}_{s}(E;\Omega)=\mathcal{I}_{s}(E\cap\Omega,E^{c}\cap\Omega)+\mathcal{I}_{s}(E\cap\Omega,E^{c}\setminus\Omega)+\mathcal{I}_{s}(E\setminus\Omega,E^{c}\cap\Omega).$

We will also sometimes refer to $\operatorname{Per}_{s}(E;\Omega)$ as the
_restricted $s$-perimeter_. Note, we also have that

(2.1)
$\displaystyle\operatorname{Per}_{s}(E;\Omega)=\frac{1}{2}\iint_{\mathcal{Q}(\Omega)}\frac{|\chi_{E}(x)-\chi_{E}(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$

where we define

$\displaystyle\mathcal{Q}(\Omega)=(\Omega^{c}\times\Omega^{c})^{c}=(\Omega\times\Omega)\cup(\Omega\times\Omega^{c})\cup(\Omega^{c}\times\Omega)\subset\mathbb{R}^{2n}.$

A measurable set $E$ is a _minimiser_ of the $s$-perimeter in $\Omega$ or
$s$-minimal in $\Omega$ if

$\displaystyle\operatorname{Per}_{s}(E;\Omega)\leq\operatorname{Per}_{s}(F;\Omega)$

for all measurable set $F$ such that $E\setminus\Omega=F\setminus\Omega$.
Moreover, a measurable set $E$ is stationary with respect to the $s$-perimeter
in $\Omega$ or $s$-stationary if

(2.2)
$\displaystyle\frac{\mathop{}\\!d}{\mathop{}\\!dt}\bigg{|}_{t=0}\operatorname{Per}_{s}(E_{t,T};\Omega)=0$

for all $T\in C^{\infty}_{0}(\Omega;\mathbb{R}^{n})$ where

$\displaystyle E_{t,T}:=\psi_{T}(E,t)$

and $\psi_{T}:\mathbb{R}^{n}\times(-\varepsilon,\varepsilon)\to\mathbb{R}^{n}$
satisfies

$\displaystyle\left\\{\begin{array}[]{r@{ \ }l @{\quad\: \;}
l}\partial_{t}\psi_{T}&=T\circ\psi_{T},\hfil\quad\>&\text{in
}\mathbb{R}^{n}\times(-\varepsilon,\varepsilon)\\\
\psi_{T}&=T,\hfil\quad\>&\text{on
}\mathbb{R}^{n}\times\\{t=0\\}.\end{array}\right.$

One can also define $s$-stationary stable sets, but we do not require this
definition for the current paper.

##  3\. An interpolation inequality

In this section, we prove an interpolation inequality which will allow us to
estimate the restricted $s$-perimeter with the restricted perimeter plus an
arbitrarily small error. The precise statement is as follows.

######  Theorem 3.1.

7qaGfSn3 Let $R>0$ and
$u\in\operatorname{BV}_{\mathrm{loc}}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$. Then, for all
$\varepsilon\in(0,3^{-\frac{1}{s}})$,

$\displaystyle\iint_{\mathcal{Q}(B_{R})}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx\leq
C\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}R^{1-s}}{1-s}|\nabla
u|(B_{(1+\varepsilon^{-1/s})R})+\frac{\varepsilon
R^{n-s}}{s}\|u\|_{L^{\infty}(\mathbb{R}^{n})}\bigg{)}.$

The constant $C>0$ depends only on $n$.

For similar interpolation inequalities to 7qaGfSn3 and 60sDucQ0 below, see
[CSV19, Equation 3.] and [CFS24, Lemma 3.15]. A small, but important
difference between 7qaGfSn3 and these results is the inclusion of the
$\varepsilon$ term. This will be essential to the proof of jG1dQXin since it
will allow us to absorb an error term coming from 7qaGfSn3 into the left-hand
side of a chain of inequalities.

Taking $u=\chi_{E}$ in 7qaGfSn3, we immediately obtain the following
corollary.

######  Corollary 3.2.

60sDucQ0 Let $R>0$ and $E\subset\mathbb{R}^{n}$ be a measurable set with
locally finite perimeter. Then, for all $\varepsilon\in(0,3^{-\frac{1}{s}})$,

$\displaystyle\operatorname{Per}_{s}(E;B_{R})\leq
C\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}R^{1-s}}{1-s}\operatorname{Per}(E;B_{(1+\varepsilon^{-1/s})R})+\frac{\varepsilon
R^{n-s}}{s}\big{)}$

The constant $C>0$ depends only on $n$.

Before we give the proof of 7qaGfSn3, we require the following lemma.

######  Lemma 3.3.

aIDAJA5Z For all $x,y\in\mathbb{R}^{n}$ and $t\in[0,1]$,

$\displaystyle\chi_{\mathcal{Q}(B_{1})}(x-ty,x+(1-t)y)\leq\chi_{\\{|x|\leq|y|+1\\}}(x,y).$

###### Proof.

Since $\mathcal{Q}(B_{1})=\\{(x,y)\in\mathbb{R}^{n}\text{ s.t. }|x|<1\text{ or
}|y|<1\\}$, if $(x-ty,x+(1-t)y)\in\mathcal{Q}(B_{1})$ then either $|x-ty|<1$
or $|x+(1-t)y|<1$. If $|x-ty|<1$ then

$\displaystyle|x|\leq t|y|+1\leq|y|+1.$

Similarly, if $|x+(1-t)y|<1$ then

$\displaystyle|x|\leq 1+(1-t)|y|\leq 1+|y|.$

∎

Now we give the proof of 7qaGfSn3.

###### Proof of 7qaGfSn3.

First, assume that $u\in C^{\infty}_{0}(\mathbb{R}^{n})$ and $R=1$. Let
$\rho>0$ to be chosen later and write

$\displaystyle\iint_{\mathcal{Q}(B_{1})}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx=I_{\rho}+J_{\rho}$

where

$\displaystyle
I_{\rho}=\iint_{\mathcal{Q}(B_{1})\cap\\{|x-y|<\rho\\}}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx\text{
and
}J_{\rho}=\iint_{\mathcal{Q}(B_{1})\cap\\{|x-y|\geq\rho\\}}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx.$

We will estimate $I_{\rho}$ and $J_{\rho}$ separately.

For $I_{\rho}$, via the change of variables $y\to y+x$, we have

$\displaystyle I_{\rho}$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|\chi_{\mathcal{Q}(B_{1})}(x,y)\chi_{\\{|x-y|<\rho\\}}(x,y)}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x+y)-u(x)|\chi_{\mathcal{Q}(B_{1})}(x,x+y)\chi_{B_{\rho}}(y)}{|y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx.$

Since

$\displaystyle|u(x+y)-u(x)|$ $\displaystyle\leq|y|\int_{0}^{1}|\nabla
u(x+ty)|\mathop{}\\!dt,$

it follows that

$\displaystyle I_{\rho}$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{0}^{1}\frac{|\nabla
u(x+ty)|\chi_{\mathcal{Q}(B_{1})}(x,x+y)\chi_{B_{\rho}}(y)}{|y|^{n+s-1}}\mathop{}\\!dt\mathop{}\\!dy\mathop{}\\!dx$

which, via the change of variable $x\to x-ty$, becomes

$\displaystyle I_{\rho}$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\int_{0}^{1}\int_{\mathbb{R}^{n}}\frac{|\nabla
u(x)|\chi_{\mathcal{Q}(B_{1})}(x-ty,x+(1-t)y)\chi_{B_{\rho}}(y)}{|y|^{n+s-1}}\mathop{}\\!dx\mathop{}\\!dt\mathop{}\\!dy.$

Then, from aIDAJA5Z, it follows that

$\displaystyle\chi_{\mathcal{Q}(B_{1})}(x-ty,x+(1-t)y)\chi_{B_{\rho}}(y)\leq\chi_{\\{|x|\leq|y|+1\\}}(x,y)\chi_{B_{\rho}}(y)\leq\chi_{B_{\rho+1}}(x)\chi_{B_{\rho}}(y),$

so

$\displaystyle I_{\rho}$ $\displaystyle\leq\|\nabla
u\|_{L^{1}(B_{\rho+1})}\int_{B_{\rho}}\frac{\mathop{}\\!dy}{|y|^{n+s-1}}\leq\frac{C\rho^{1-s}}{1-s}\|\nabla
u\|_{L^{1}(B_{\rho+1})}$

with $C$ depending only on $n$.

For $J_{\rho}$,

$\displaystyle J_{\rho}\leq
2\|u\|_{L^{\infty}(\mathbb{R}^{n})}\iint_{\mathcal{Q}(B_{1})\cap\\{|x-y|\geq\rho\\}}\frac{\mathop{}\\!dy\mathop{}\\!dx}{|x-y|^{n+s}}.$

Assuming that $\rho>3$,
$\mathcal{Q}(B_{1})\cap\\{|x-y|\geq\rho\\}\subset\big{(}(B_{1}\times
B_{1}^{c})\cup(B_{1}^{c}\times B_{1})\big{)}\cap\\{|x-y|\geq\rho\\}$. Indeed,
if there exists $(x,y)\in(B_{1}\times B_{1})\cap\\{|x-y|\geq\rho\\}$ then
$3<\rho\leq|x-y|\leq|x|+|y|<2$, a contradiction. Hence,

$\displaystyle J_{\rho}\leq
4\|u\|_{L^{\infty}(\mathbb{R}^{n})}\iint_{(B_{1}\times
B_{1}^{c})\cap\\{|x-y|\geq\rho\\}}\frac{\mathop{}\\!dy\mathop{}\\!dx}{|x-y|^{n+s}}.$

Since $|x-y|\geq|y|-|x|\geq|y|-1$ and $(B_{1}\times
B_{1}^{c})\cap\\{|x-y|\geq\rho\\}\subset B_{1}\times B_{\rho-1}^{c}$, we find
that

$\displaystyle J_{\rho}$ $\displaystyle\leq
C\|u\|_{L^{\infty}(\mathbb{R}^{n})}\int_{B_{1}}\int_{\mathbb{R}^{n}\setminus
B_{\rho-1}}\frac{\mathop{}\\!dy\mathop{}\\!dx}{(|y|-1)^{n+s}}.$

Since $\rho>3$, we obtain

$\displaystyle J_{\rho}\leq
C\|u\|_{L^{\infty}(\mathbb{R}^{n})}\int_{B_{1}}\int_{\mathbb{R}^{n}\setminus
B_{\rho-1}}\frac{\mathop{}\\!dy\mathop{}\\!dx}{|y|^{n+s}}\leq\frac{C(\rho-1)^{-s}}{s}\|u\|_{L^{\infty}(\mathbb{R}^{n})}\leq\frac{C\rho^{-s}}{s}\|u\|_{L^{\infty}(\mathbb{R}^{n})}$

with $C$ depending only on $n$.

Thus, combining the above estimates, we have

$\displaystyle\iint_{\mathcal{Q}(B_{1})}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq C\bigg{(}\frac{\rho^{1-s}}{1-s}\|\nabla
u\|_{L^{1}(B_{\rho+1})}+\frac{\rho^{-s}}{s}\|u\|_{L^{\infty}(\mathbb{R}^{n})}\bigg{)}$

for all $\rho>3$. Choosing $\rho=\varepsilon^{-\frac{1}{s}}$, we prove the
result.

For a general element
$u\in\operatorname{BV}_{\mathrm{loc}}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$, let $u_{k}\in C^{\infty}_{0}(\mathbb{R}^{n})$,
defined as in the proof of Theorem 2 of §5.2.2 in [EG92]. Then

$\displaystyle\lim_{k\to\infty}\|\nabla
u_{k}\|_{L^{1}(B_{1+\varepsilon^{-1/s}})}=|\nabla
u|(B_{1+\varepsilon^{-1/s}})\text{ and
}\|u_{k}\|_{L^{\infty}(\mathbb{R}^{n})}\leq\|u\|_{L^{\infty}(\mathbb{R}^{n})},$

so

$\displaystyle\iint_{\mathcal{Q}(B_{1})}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq\liminf_{k\to+\infty}\iint_{\mathcal{Q}(B_{1})}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq
C\liminf_{k\to+\infty}\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}}{1-s}\|\nabla
u_{k}\|_{L^{1}(B_{\rho+1})}+\frac{\varepsilon}{s}\|u_{k}\|_{L^{\infty}(\mathbb{R}^{n})}\bigg{)}$
$\displaystyle\leq C\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}}{1-s}|\nabla
u|(B_{1+\varepsilon^{-1/s}})+\frac{\varepsilon}{s}\|u\|_{L^{\infty}(\mathbb{R}^{n})}\bigg{)}.$

Finally, to obtain the case for general $R>0$, let $u_{R}(x)=u(Rx)$. Then

$\displaystyle\iint_{\mathcal{Q}(B_{R})}\frac{|u(x)-u(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle=R^{n-s}\iint_{\mathcal{Q}(B_{1})}\frac{|u_{R}(x)-u_{R}(y)|}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq
CR^{n-s}\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}}{1-s}|\nabla
u_{R}|(B_{1+\varepsilon^{-1/s}})+\frac{\varepsilon}{s}\|u_{R}\|_{L^{\infty}(\mathbb{R}^{n})}\bigg{)}$
$\displaystyle=CR^{n-s}\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}R^{1-n}}{1-s}|\nabla
u|(B_{(1+\varepsilon^{-1/s})R})+\frac{\varepsilon}{s}\|u\|_{L^{\infty}(\mathbb{R}^{n})}\bigg{)}.$

∎

##  4\. The extension problem

In this section, we record several results regarding the Caffarelli-Silvestre
extension, in the context of the theory of nonlocal minimal surfaces, that we
require in the proof of jG1dQXin. Throughout this section, we will denote
points in $\mathbb{R}^{n+1}$ with capital letters $X$, $Y$, etc and write
$X=(x,x_{n+1})=(x_{1},\dots,x_{n+1})$ where
$x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ (and analogously $Y=(y,y_{n+1})$,
etc). Moreover, let

$\displaystyle\mathbb{R}^{n+1}_{+}$
$\displaystyle=\\{X\in\mathbb{R}^{n+1}\text{ s.t. }x_{n+1}>0\\},$
$\displaystyle\tilde{B}_{R}(X)$ $\displaystyle=\\{Y\in\mathbb{R}^{n+1}\text{
s.t. }|Y-X|<R\\},$ $\displaystyle\tilde{B}_{R}^{+}(X)$
$\displaystyle=\tilde{B}_{R}(X)\cap\mathbb{R}^{n+1}_{+},$
$\displaystyle\tilde{B}_{R}$ $\displaystyle=\tilde{B}_{R}(0),\text{ and}$
$\displaystyle\tilde{B}_{R}^{+}$ $\displaystyle=\tilde{B}_{R}^{+}(0).$

Let $s\in(0,1)$, $E\subset\mathbb{R}^{n}$ be a measurable set, and

$\displaystyle\tilde{\chi}_{E}(x)$
$\displaystyle=\chi_{\mathbb{R}^{n}\setminus E}(x)-\chi_{E}(x)\qquad
x\in\mathbb{R}^{n}$

where $\chi_{E}$ is the characteristic function of $E$. Now we define
$U_{E}:\mathbb{R}^{n+1}\to\mathbb{R}$

$\displaystyle U_{E}(X)$
$\displaystyle=\int_{\mathbb{R}^{n}}P_{s/2}(X,y)\tilde{\chi}_{E}(y)\mathop{}\\!dy,\qquad\text{for
all }X\in\mathbb{R}^{n+1}_{+}$

where

(4.1) $\displaystyle
P_{s/2}(X,y)=a(n,s)\frac{x_{n+1}^{s}}{\big{(}|x-y|^{2}+x_{n+1}^{2}\big{)}^{\frac{n+s}{2}}},\qquad
a(n,s)=\frac{\Gamma\big{(}\frac{n+s}{2}\big{)}}{\pi^{\frac{n}{2}}\Gamma\big{(}\frac{s}{2}\big{)}}.$

Since $|\tilde{\chi}_{E}(x)|\leq 1$ and
$\int_{\mathbb{R}^{n}}P_{s/2}(X,y)\mathop{}\\!dy=1$ for all
$X\in\mathbb{R}^{n+1}_{+}$, see [Gar19, Remark 10.2], $U_{E}$ is well-defined,
and, in fact, is a smooth function. Moreover, $U_{E}$ satisfies the degenerate
PDE

$\displaystyle\left\\{\begin{array}[]{r@{ \ }l @{\quad\: \;}
l}\operatorname{div}\big{(}x_{n+1}^{1-s}\nabla
V\big{)}&=0,\hfil\quad\>&\text{in }\mathbb{R}^{n+1}_{+}\\\
V&=\tilde{\chi}_{E},\hfil\quad\>&\text{on
}\mathbb{R}^{n}\times\\{0\\}.\end{array}\right.$

where here $\nabla$ and $\operatorname{div}$ denotes the gradient and
divergence in $\mathbb{R}^{n+1}$ respectively. This PDE was first studied in
the context of the fractional Laplacian in [CS07] and in the context of
nonlocal minimal surfaces in [CRS10]. In particular, it was proven that if $E$
is a minimiser of the $s$-perimeter then the function

(4.2)
$\displaystyle\Phi_{E,x}(R):=\frac{1}{R^{n-s}}\int_{\tilde{B}_{R}^{+}((x,0))}y_{n+1}^{1-s}|\nabla
U_{E}(Y)|^{2}\mathop{}\\!dY$

is monotone increasing. This function plays an identical role in the theory as
the function $R^{1-n}\operatorname{Per}(E;B_{R}(x))$ plays in the classical
setting. See also [Gar19] and references therein for more details. Recently,
in [CFS24], it was shown that the monotonicity of (4.2) also holds for sets
that are stationary with respect to the $s$-perimeter:

######  Proposition 4.1 ([CFS24, Theorem 3.4]).

uUBAsC54 Let $s\in(0,1)$ and $E\subset\mathbb{R}^{n}$ be a measurable set that
is stationary with respect to the $s$-perimeter. Then $\Phi_{E,x}$ is monotone
increasing for all $x\in\partial^{\ast}E$.

Moreover, in [CRS10] the following “extension–trace” energy estimate was
proven which gives a control on $\Phi_{E,x}$ in terms of the $s$-perimeter of
$E$ in a ball.

######  Proposition 4.2 ([CRS10, Propostion 7.1(a)]).

Qu1H6WRf Let $s\in(0,1)$ and $E\subset\mathbb{R}^{n}$ gave locally finite
perimeter. Then

$\displaystyle\Phi_{E,x}(R)\leq\frac{Cs^{-1}}{R^{n-s}}\operatorname{Per}_{s}(E;B_{2R})$

for all $R>0$ and $x\in\partial^{\ast}E$. The constant $C>0$ depends only on
$n$.

The proof of Qu1H6WRf is given [CRS10] (applied to
$\chi_{\mathbb{R}^{n}\setminus E}-\chi_{E}$ and given again with more details
in [CFS24, Lemma 3.13]. We will sketch the proof given [CFS24, Lemma 3.13] in
our particular case, just to be explicit about how the constant $C(n)s^{-1/2}$
was obtained.

###### Proof.

By translating and rescaling, it is sufficient to consider the case $R=1$ and
$x=0$. We follow the argument given in [CFS24, Lemma 3.13] applied to
$u=\chi_{\mathbb{R}^{n}\setminus E}-\chi_{E}-\big{(}|B_{2}\setminus E|-|E\cap
B_{2}|\big{)}$. Note, we have chosen $u$ such that
$\int_{B_{2}}u\mathop{}\\!dx=0$. Let $\zeta\in C^{\infty}_{0}(\mathbb{R}^{n})$
be such that $\zeta=1$ in $B_{3/2}$, $\zeta=0$ in $\mathbb{R}^{n}\setminus
B_{2}$, and $0\leq\zeta\leq 1$. Define $u_{1}=\zeta u$, $u_{2}=(1-\zeta)u$,
and $U_{1}$ and $U_{2}$ the Caffarelli-Silvestre extensions of $u_{1}$ and
$u_{2}$ respectively. We observe that our choice of normalisation constant
implies

(4.3) $\displaystyle\int_{\mathbb{R}^{n+1}_{+}}x_{n+1}^{1-s}|\nabla
U_{1}|^{2}\mathop{}\\!dX=\frac{s\Gamma\big{(}\frac{n+s}{2}\big{)}}{2\pi^{\frac{n}{2}}\Gamma\big{(}\frac{s}{2}\big{)}}[u_{1}]_{H^{s/2}(\mathbb{R}^{n})}^{2}$

where $[\cdot]_{H^{s/2}(\mathbb{R}^{n})}$ is given by (1.6). Note that the
constant in (4.3) is comparable to $s^{2}$. Indeed, $P_{s/2}$ is the same as
the one in [Gar19] (after replacing $s$ with $s/2$), so from [Gar19,
Proposition 10.1] (see also [Gar19, Remark 10.5]), we have via an integration
by parts

$\displaystyle\int_{\mathbb{R}^{n+1}_{+}}x_{n+1}^{1-s}|\nabla
U_{1}|^{2}\mathop{}\\!dX$
$\displaystyle=-\int_{\mathbb{R}^{n}}u_{1}(x)\lim_{t\to
0^{+}}t^{1-s}\partial_{n+1}U_{1}(x,t)\mathop{}\\!dx$
$\displaystyle=\frac{s\Gamma\big{(}\frac{n+s}{2}\big{)}}{\pi^{\frac{n}{2}}\Gamma\big{(}\frac{s}{2}\big{)}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{u_{1}(x)(u_{1}(x)-u_{1}(y)}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle=\frac{s\Gamma\big{(}\frac{n+s}{2}\big{)}}{2\pi^{\frac{n}{2}}\Gamma\big{(}\frac{s}{2}\big{)}}[u_{1}]_{H^{s/2}(\mathbb{R}^{n})}^{2}.$

Moreover, in [CFS24] they apply the fractional Poincaré-Wirtinger inequality

(4.4) $\displaystyle\|u\|_{L^{2}(B_{2})}^{2}\leq
C(n)\int_{B_{2}}\int_{B_{2}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx,$

see for example222Note, in [BGV21, Corollary 2.5], they actually prove
actually prove $\|u\|_{L^{2}(B_{1})}\leq C[u]_{H^{s/2}(\mathbb{R}^{n})}$ for
$u$ with $\int_{B_{1}}u\mathop{}\\!dx=0$. In KnSkEdQI, we briefly outline an
extension operator which allows us to obtain (4.4).  [BGV21, Corollary 2.5].
All together, the argument in [CFS24] and (2.1) gives

$\displaystyle\int_{\mathbb{R}^{n+1}_{+}}x_{n+1}^{1-s}|\nabla
U_{1}|^{2}\mathop{}\\!dX\leq C(n)s^{2}\operatorname{Per}_{s}(E;B_{2})\leq
C(n)\operatorname{Per}_{s}(E;B_{2}).$

Furthermore, rearranging the two inequalities in [CFS24] at the end of p.41
(using again (2.1)), they prove

$\displaystyle\int_{\tilde{B}^{+}_{1}}x_{n+1}^{1-s}|\nabla
U_{2}|^{2}\mathop{}\\!dX$ $\displaystyle\leq
C(n)\bigg{(}\int_{\tilde{B}^{+}_{1}}x_{n+1}^{s-1}\mathop{}\\!dX\bigg{)}\operatorname{Per}_{s}(E;B_{2})\leq\frac{C(n)}{s}\operatorname{Per}_{s}(E;B_{2}).$

Thus, we obtain

$\displaystyle\int_{\tilde{B}^{+}_{1}}x_{n+1}^{1-s}|\nabla
U_{E}|^{2}\mathop{}\\!dX$ $\displaystyle\leq
C\bigg{(}\int_{\mathbb{R}^{n}_{+}}x_{n+1}^{1-s}|\nabla
U_{1}|^{2}\mathop{}\\!dX+\int_{\tilde{B}^{+}_{1}}x_{n+1}^{1-s}|\nabla
U_{2}|^{2}\mathop{}\\!dX\bigg{)}$ $\displaystyle\leq
C(n)\bigg{(}1+\frac{1}{s}\bigg{)}\operatorname{Per}_{s}(E;B_{2})$
$\displaystyle\leq\frac{C(n)}{s}\operatorname{Per}_{s}(E;B_{2}).$

∎

######  Remark 4.3.

KnSkEdQI In [BGV21, Corollary 2.5], they actually prove
$\|u-u_{B_{1}}\|_{L^{p}(B_{1})}\leq
C(n,p)\alpha(1-\alpha)[u]_{W^{\alpha,p}(\mathbb{R}^{n})}$ for all
$0<\alpha<1$, $1\leq p<+\infty$ where
$u_{B_{1}}=\frac{1}{\omega_{n}}\int_{B_{1}}u\mathop{}\\!dx$. To obtain (4.4),
we define $Eu:\mathbb{R}^{n}\to\mathbb{R}$ by

$\displaystyle Eu(x)$ $\displaystyle=\begin{cases}u(x),&\text{for }x\in
B_{1}\\\ u(x/|x|^{2}),&\text{for }x\in\mathbb{R}^{n}\setminus
B_{1}.\end{cases}$

Then, using that
$\big{|}\frac{x}{|x|^{2}}-\frac{y}{|y|^{2}}\big{|}=\frac{|x-y|}{|x||y|}$ and
that the Jacobain of the transformation $x\to\frac{x}{|x|^{2}}$ is
$\frac{1}{|x|^{n}}$, we obtain

$\displaystyle\int_{\mathbb{R}^{n}\setminus
B_{1}}\int_{\mathbb{R}^{n}\setminus
B_{1}}\frac{|Eu(x)-Eu(y)|^{p}}{|x-y|^{n+\alpha
p}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle=\int_{B_{1}}\int_{B_{1}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+\alpha
p}}|x|^{\alpha p}|y|^{\alpha p}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq\int_{B_{1}}\int_{B_{1}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+\alpha
p}}\mathop{}\\!dy\mathop{}\\!dx.$

Similarly, using that
$\big{|}x-\frac{y}{|y|^{2}}\big{|}^{2}=\frac{1}{|y|^{2}}(|x-y|^{2}+(1-|x|^{2})(1-|y|^{2})\geq\frac{|x-y|^{2}}{|y|^{2}}$
for all $x,y\in B_{1}$, we obtain

$\displaystyle\int_{B_{1}}\int_{\mathbb{R}^{n}\setminus
B_{1}}\frac{|Eu(x)-Eu(y)|^{p}}{|x-y|^{n+\alpha
p}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq\int_{B_{1}}\int_{B_{1}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+\alpha
p}}|y|^{\alpha p}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle\leq\int_{B_{1}}\int_{B_{1}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+\alpha
p}}\mathop{}\\!dy\mathop{}\\!dx.$

Hence, $[Eu]_{W^{\alpha,p}(\mathbb{R}^{n})}^{p}\leq
4[u]_{W^{\alpha,p}(B_{1})}^{p}$, so applying [BGV21, Corollary 2.5] to $Eu$
gives

$\displaystyle\|u-u_{B_{1}}\|_{L^{p}(B_{1})}$
$\displaystyle=\|Eu-(Eu)_{B_{1}}\|_{L^{p}(B_{1})}$ $\displaystyle\leq
C(n,p)\alpha(1-\alpha)[Eu]_{W^{\alpha,p}(\mathbb{R}^{n})}$ $\displaystyle\leq
C(n,p)\alpha(1-\alpha)[u]_{W^{\alpha,p}(B_{1})}.$

∎

Furthermore, we have the following proposition.

######  Proposition 4.4.

GeFXLo81 Let $E\subset\mathbb{R}^{n}$ be measurable with locally finite
perimeter. Then, for all $x\in\partial^{\ast}E$,

(4.5) $\displaystyle\lim_{R\to 0^{+}}\Phi_{E,x}(R)$
$\displaystyle=\Phi_{\mathbb{R}^{n}_{+},0}(1)$

where

$\displaystyle\Phi_{\mathbb{R}^{n}_{+},0}(1)=\frac{2\pi^{\frac{n}{2}-1}\Gamma\big{(}\frac{s+1}{2}\big{)}\Gamma\big{(}\frac{1-s}{2}\big{)}}{\Gamma\big{(}\frac{s}{2}\big{)}\Gamma\big{(}\frac{n-s}{2}+1\big{)}}>0.$

In particular,

$\displaystyle\lim_{R\to 0^{+}}\Phi_{E,x}(R)\geq\frac{Cs}{1-s}$

with $C$ depending only on $n$.

This proposition is the nonlocal analogue of [Mag12, Equation (15.9) in
Corollary 15.8] and is essentially proven in [CRS10, Theorem 9.2]; however,
the reader should be aware that the statement of [CRS10, Theorem 9.2] assumes
that $E$ is $s$-minimal, but this is assumption is not used in the proof of
GeFXLo81. Moreover, they do not explicitly say (4.5) (merely that $\lim_{R\to
0^{+}}\Phi_{E,x}(R)\geq C(n,s)$); however, if $E_{x,R}=\frac{E-x}{R}$ then,
from the scaling properties of $\Phi_{E,x}$,
$\Phi_{E,x}(R)=\Phi_{E_{x,R},0}(1)$, which, in conjunction with the
convergence properties of $E_{x,R}$ as $R\to 0^{+}$ for
$x\in\partial^{\ast}E$, see [Mag12, Theorem 15.5], and [CRS10, Proposition
9.1], immediately implies (4.5). The value of $\Phi_{\mathbb{R}^{n}_{+},0}(1)$
is given the proposition below.

######  Proposition 4.5.

We have that

$\displaystyle\Phi_{\mathbb{R}^{n}_{+},0}(1)=\frac{2\pi^{\frac{n}{2}-1}\Gamma\big{(}\frac{s+1}{2}\big{)}\Gamma\big{(}\frac{1-s}{2}\big{)}}{\Gamma\big{(}\frac{s}{2}\big{)}\Gamma\big{(}\frac{n-s}{2}+1\big{)}}.$

###### Proof.

First, we claim that

(4.6) $\displaystyle U_{\mathbb{R}^{n}_{+}}(X)$
$\displaystyle=-\frac{\Gamma\big{(}\frac{s+1}{2}\big{)}}{\pi^{\frac{1}{2}}\Gamma\big{(}\frac{s}{2}\big{)}}h\bigg{(}\frac{x_{n}}{x_{n+1}}\bigg{)}$

where

$\displaystyle h(\tau)$
$\displaystyle=\int_{0}^{\tau}\frac{\mathop{}\\!dt}{\big{(}t^{2}+1\big{)}^{\frac{1+s}{2}}}.$

Indeed, if $X_{\ast}:=(x^{\prime},-x_{n},x_{n+1})$ with
$x^{\prime}=(x_{1},\dots,x_{n-1})\in\mathbb{R}^{n-1}$ then the change of
variables $y\to(y^{\prime},-y_{n})$ in the second integral below gives

$\displaystyle U_{\mathbb{R}^{n}_{+}}(X)$
$\displaystyle=-a(n,s)x_{n+2}^{s}\bigg{(}\int_{\mathbb{R}^{n}_{+}}\frac{1}{\big{(}|x-y|^{2}+x_{n+1}^{2}\big{)}^{\frac{n+s}{2}}}\mathop{}\\!dy-\int_{\mathbb{R}^{n}_{-}}\frac{1}{\big{(}|x-y|^{2}+x_{n+1}^{2}\big{)}^{\frac{n+s}{2}}}\mathop{}\\!dy\bigg{)}$
(4.7) $\displaystyle=-a(n,s)x_{n+2}^{s}\big{(}f(X)-f(X_{\ast})\big{)}$

where

$\displaystyle
f(X):=\int_{\mathbb{R}^{n}_{+}}\frac{\mathop{}\\!dy}{\big{(}|x-y|^{2}+x_{n+1}^{2}\big{)}^{\frac{n+s}{2}}}.$

Observe that, via a translation in the first $n-1$ coordinates, we have

$\displaystyle f(X)$
$\displaystyle=\int_{0}^{+\infty}\int_{\mathbb{R}^{n-1}}\frac{\mathop{}\\!dy^{\prime}\mathop{}\\!dt}{\big{(}|y^{\prime}|^{2}+(x_{n}-t)^{2}+x_{n+1}^{2}\big{)}^{\frac{n+s}{2}}}.$

Next, the rescaling $y^{\prime}\to y^{\prime}\sqrt{(x_{n}-t)^{2}+x_{n+1}^{2}}$
in the inner integral gives

$\displaystyle f(X)$
$\displaystyle=\bigg{(}\int_{\mathbb{R}^{n-1}}\frac{\mathop{}\\!dy^{\prime}}{\big{(}|y^{\prime}|^{2}+1\big{)}^{\frac{n+s}{2}}}\bigg{)}\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(x_{n}-t)^{2}+x_{n+1}^{2}\big{)}^{\frac{1+s}{2}}}$

An application of polar coordinates gives

(4.8)
$\displaystyle\int_{\mathbb{R}^{n-1}}\frac{\mathop{}\\!dy^{\prime}}{\big{(}|y^{\prime}|^{2}+1\big{)}^{\frac{n+s}{2}}}=\frac{\pi^{\frac{n-1}{2}\Gamma\big{(}\frac{s+1}{2}\big{)}}}{\Gamma\big{(}\frac{n+s}{2}\big{)}},$

see [Gar19, Proposition 4.1] for the explicit computation. Furthermore, the
change of variables $t\to x_{n+1}t$ gives

$\displaystyle\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(x_{n}-t)^{2}+x_{n+1}^{2}\big{)}^{\frac{1+s}{2}}}$
$\displaystyle=x_{n+1}\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(x_{n}-x_{n+1}t)^{2}+x_{n+1}^{2}\big{)}^{\frac{1+s}{2}}}$
(4.9)
$\displaystyle=x_{n+1}^{-s}\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(x_{n}/x_{n+1}-t)^{2}+1\big{)}^{\frac{1+s}{2}}}.$

If we define

$\displaystyle
g(\tau)=\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(\tau-t)^{2}+1\big{)}^{\frac{1+s}{2}}}-\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(\tau+t)^{2}+1\big{)}^{\frac{1+s}{2}}}$

and

$\displaystyle\tilde{a}(n,s)=2a(n,s)\frac{\pi^{\frac{n-1}{2}\Gamma\big{(}\frac{s+1}{2}\big{)}}}{\Gamma\big{(}\frac{n+s}{2}\big{)}}=\frac{2\Gamma\big{(}\frac{s+1}{2}\big{)}}{\pi^{\frac{1}{2}}\Gamma\big{(}\frac{s}{2}\big{)}}$

then it follows from (4.7), (4.8), and (4.9) that

$\displaystyle U_{\mathbb{R}^{n}_{+}}(X)$
$\displaystyle=-\frac{1}{2}\tilde{a}(n,s)\bigg{[}\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(x_{n}/x_{n+1}-t)^{2}+1\big{)}^{\frac{1+s}{2}}}-\int_{0}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}(x_{n}/x_{n+1}+t)^{2}+1\big{)}^{\frac{1+s}{2}}}\bigg{]}$
$\displaystyle=-\frac{1}{2}\tilde{a}(n,s)g\bigg{(}\frac{x_{n}}{x_{n+1}}\bigg{)}.$

Now,

$\displaystyle g(\tau)$
$\displaystyle=\int_{-\tau}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}t^{2}+1\big{)}^{\frac{1+s}{2}}}-\int_{\tau}^{+\infty}\frac{\mathop{}\\!dt}{\big{(}t^{2}+1\big{)}^{\frac{1+s}{2}}}$
$\displaystyle=\int_{-\tau}^{\tau}\frac{\mathop{}\\!dt}{\big{(}t^{2}+1\big{)}^{\frac{1+s}{2}}}$
$\displaystyle=2h(\tau)$

which proves (4.6).

From (4.6), it follows that

$\displaystyle\nabla U_{\mathbb{R}^{n}_{+}}(X)$
$\displaystyle=-\tilde{a}(n,s)\bigg{(}0,\dots,0,\frac{1}{x_{n+1}}h^{\prime}\bigg{(}\frac{x_{n}}{x_{n+1}}\bigg{)},-\frac{x_{n}}{x_{n+1}}h^{\prime}\bigg{(}\frac{x_{n}}{x_{n+1}}\bigg{)}\bigg{)}$
$\displaystyle=-\tilde{a}(n,s)\bigg{(}0,\dots,0,\frac{x_{n+1}^{s}}{\big{(}x_{n}^{2}+x_{n+1}^{2}\big{)}^{\frac{1+s}{2}}},-\frac{x_{n}x_{n+1}^{s-1}}{\big{(}x_{n}^{2}+x_{n+1}^{2}\big{)}^{\frac{1+s}{2}}}\bigg{)},$

so

$\displaystyle x_{n+1}^{1-s}|\nabla U_{E}|^{2}$
$\displaystyle=\tilde{a}(n,s)^{2}\frac{x_{n+1}^{s-1}}{\big{(}x_{n}^{2}+x_{n+1}^{2}\big{)}^{s}}$

Using the notation, $B^{2,+}_{1}:=\\{(x_{n},x_{n+1})\in\mathbb{R}^{2}\text{
s.t. }x_{n}^{2}+x_{n+1}^{2}<1,x_{n+1}>0\\}$, we have

$\displaystyle\tilde{B}_{1}^{+}$
$\displaystyle=\big{\\{}X\in\mathbb{R}^{n+1}\text{ s.t. }(x_{n},x_{n+1})\in
B^{2,+}_{1},x^{\prime}\in B^{n-1}_{\sqrt{1-x_{n}^{2}-x_{n+1}^{2}}}\big{\\}},$

so we obtain

$\displaystyle\int_{\tilde{B}_{1}^{+}}x_{n+1}^{1-s}|\nabla
U_{\mathbb{R}^{n}_{+}}|^{2}\mathop{}\\!dX$
$\displaystyle=\tilde{a}(n,s)^{2}\int_{B^{2,+}_{1}}\int_{B^{n-1}_{\sqrt{1-x_{n}^{2}-x_{n+1}^{2}}}}\frac{x_{n+1}^{s-1}}{\big{(}x_{n}^{2}+x_{n+1}^{2}\big{)}^{s}}\mathop{}\\!dx^{\prime}\mathop{}\\!dx_{n}\mathop{}\\!dx_{n+1}$
$\displaystyle=\tilde{a}(n,s)^{2}\omega_{n-1}\int_{B^{2,+}_{1}}\frac{x_{n+1}^{s-1}\big{(}1-x_{n}^{2}-x_{n+1}^{2}\big{)}^{\frac{n-1}{2}}}{\big{(}x_{n}^{2}+x_{n+1}^{2}\big{)}^{s}}\mathop{}\\!dx_{n}\mathop{}\\!dx_{n+1}.$

Hence, polar coordinates $(x_{n},x_{n+1})=(r\cos\theta,r\sin\theta)$ gives

$\displaystyle\int_{\tilde{B}_{1}^{+}}x_{n+1}^{1-s}|\nabla
U_{\mathbb{R}^{n}_{+}}|^{2}\mathop{}\\!dX$
$\displaystyle=\tilde{a}(n,s)^{2}\omega_{n-1}\bigg{(}\int_{0}^{1}r^{-s}(1-r^{2})^{\frac{n-1}{2}}\mathop{}\\!dr\bigg{)}\bigg{(}\int_{0}^{\pi}(\sin\theta)^{s-1}\mathop{}\\!d\theta\bigg{)}.$

One can show that

$\displaystyle\int_{0}^{1}r^{-s}\big{(}1-r^{2}\big{)}^{\frac{n-1}{2}}\mathop{}\\!dr=\frac{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{1-s}{2}\right)}{2\Gamma\left(\frac{n-s}{2}+1\right)}\text{
and
}\int_{0}^{\pi}(\sin\theta)^{s-1}\mathop{}\\!d\theta=\frac{\pi^{\frac{1}{2}}\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{s+1}{2}\right)}$

Thus,

$\displaystyle\int_{\tilde{B}_{1}^{+}}x_{n+1}^{1-s}|\nabla
U_{\mathbb{R}^{n}_{+}}|^{2}\mathop{}\\!dX$
$\displaystyle=\tilde{a}(n,s)^{2}\omega_{n-1}\bigg{(}\frac{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{1-s}{2}\right)}{2\Gamma\left(\frac{n-s}{2}+1\right)}\bigg{)}\bigg{(}\frac{\pi^{\frac{1}{2}}\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{s+1}{2}\right)}\bigg{)}$
$\displaystyle=\frac{2\pi^{\frac{n}{2}-1}\Gamma\big{(}\frac{s+1}{2}\big{)}\Gamma\big{(}\frac{1-s}{2}\big{)}}{\Gamma\big{(}\frac{s}{2}\big{)}\Gamma\big{(}\frac{n-s}{2}+1\big{)}}$

as required. ∎

##  5\. Proof of main theorem

In this section, we give the proof of jG1dQXin.

###### Proof of jG1dQXin.

By translating, we may assume, without loss of generality, that
$x=0\in\partial^{\ast}E$. Let $U_{E}$ be given by (4.1). From uUBAsC54 and
GeFXLo81, we have that

$\displaystyle\int_{\tilde{B}_{R}^{+}}x_{n+1}^{1-s}|\nabla
U_{E}|^{2}\mathop{}\\!dX\geq R^{n-s}\lim_{r\to
0^{+}}\Phi_{E,0}(r)\geq\frac{CsR^{n-s}}{1-s}$

for all $R>0$. Moreover, since

$\displaystyle|\tilde{\chi}_{E}(x)-\tilde{\chi}_{E}(y)|^{2}=4|\chi_{E}(x)-\chi_{E}(y)|\qquad\text{for
all}x,y\in\mathbb{R}^{n}$

by Qu1H6WRf and 60sDucQ0, we obtain

$\displaystyle\frac{sR^{n-s}}{1-s}$ $\displaystyle\leq
C(n)\iint_{\mathcal{Q}(B_{2R})}\frac{|\tilde{\chi}_{E}(x)-\tilde{\chi}_{E}(y)|^{2}}{|x-y|^{n+s}}\mathop{}\\!dy\mathop{}\\!dx$
$\displaystyle=\frac{C(n)}{s}\operatorname{Per}_{s}(E;B_{2R})$
$\displaystyle\leq\frac{C(n)}{s}\bigg{(}\frac{\varepsilon^{-\frac{1-s}{s}}R^{1-s}}{1-s}\operatorname{Per}(E;B_{\delta
R})+\frac{\varepsilon R^{n-s}}{s}\big{)}$

with $\delta=1+\varepsilon^{-1/s}$ provided $\varepsilon<3^{-1/s}$. Choosing
$\varepsilon=\frac{1}{2}\min\big{\\{}\frac{s^{3}}{C(1-s)},3^{-\frac{1}{s}}\big{\\}}$,
we find

$\displaystyle
R^{n-1}\leq\frac{C(n)}{s^{2}}\varepsilon^{-\frac{1-s}{s}}\operatorname{Per}(E;B_{\delta
R}),$

which, after rescaling, further gives

$\displaystyle
R^{n-1}\leq\frac{C(n)}{s^{2}}\delta^{n-1}\varepsilon^{-\frac{1-s}{s}}\operatorname{Per}(E;B_{R})\qquad\text{for
all }R>0.$

Finally, note $s^{-2}\delta^{n-1}\varepsilon^{-\frac{1-s}{s}}\leq
C(n)s^{-2}3^{\frac{n-s}{s^{2}}}$ which remains bounded away from $0$ as $s\to
1^{-}$. ∎

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