# Characterizations of fractional Sobolev–Poincaré and (localized) Hardy
inequalities

Firoj Sk  Department of Mathematics, Indian Institute of Science Education and
Research, Dr. Homi Bhaba road, Pune–411008, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove capacitary versions of the fractional Sobolev–Poincaré
inequalities. We characterize localized variant of the boundary fractional
Sobolev–Poincaré inequalities through uniform fatness condition of the domain
in $\mathbb{R}^{n}$. Existence type results on the fractional Hardy inequality
are established in the supercritical case $sp>n$ for $s\in(0,1)$, $p>1$.
Characterization of the fractional Hardy inequality through weak supersolution
of the associate problem is also addressed.

###### Key words and phrases:

Fractional Sobolev–Poincaré inequality; fractional Hardy inequality;
variational Sobolev capacity, fractional–Sobolev spaces; fractional eigenvalue
problem.

###### 1991 Mathematics Subject Classification:

31B15; 35A23; 46E35; 49R05.

##  1\. Introduction and main results

The central aim of this paper is to study the Sobolev–Poincaré inequality,
pointwise Hardy inequality and the Hardy inequality under some assumptions on
the domain in fractional setting. Precise condition on the domain will be
clarified later. It is well known that the classical Sobolev–Poincaré
inequality states that for a bounded domain $\Omega\subset\mathbb{R}^{n}$ with
$C^{1}$ boundary and $1\leq p<n$, there exists a constant $C=C(n,p)>0$ such
that

(1.1) $\left(\int_{\Omega}|u(x)-u_{\Omega}|^{p^{*}}dx\right)^{1/p^{*}}\leq
C\left(\int_{\Omega}|\nabla u(x)|^{p}dx\right)^{1/p}\text{ for all }u\in
W^{1,p}(\Omega),$

where $p^{*}=\frac{np}{n-p}$ denotes the Sobolev critical exponent and the
space $W^{1,p}(\Omega)$ is the usual classical Sobolev space, see for example
[14, Chapter 4] in the case of ball. A capacitary variant of the
Sobolev–Poincaré inequality eq. 1.1 were considered in [20] and for weighted
case, see [27]. The well known classical (boundary) Hardy inequality states
that for a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{n}$, $1\leq
p<\infty$, there exists a constant $C=(n,p)>0$ such that for any $u\in
C_{c}^{\infty}(\Omega)$

(1.2) $\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{p}}dx\leq
C\int_{\Omega}|\nabla u(x)|^{p}dx,$

where $\delta(x):=\text{dist}(x,\Omega^{c})$. One can obtain the classical
Hardy inequality eq. 1.2 by applying appropriately the Hardy–Littlewood–Wiener
maximal function theorem on a pointwise Hardy inequality, see [17, 22] where
they have introduced pointwise Hardy inequality through a maximal operator.
Necessary and sufficient conditions are provided for pointwise Hardy
inequalities in [23] and see [25] for weighted case. In order to get a deeper
insight of the best constant in the Hardy inequality eq. 1.2 for a domain
$\Omega\subset\mathbb{R}^{n}$, one can consider the following minimization
problem which naturally related to Hardy’s inequality

(1.3) $\lambda_{p}(\Omega)=\inf_{\begin{subarray}{c}u\in
W^{1,p}_{0}(\Omega)\\\ u\neq
0\end{subarray}}\frac{\displaystyle\int_{\Omega}|\nabla
u(x)|^{p}dx}{\displaystyle\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{p}}dx},$

where the space $W^{1,p}_{0}(\Omega)$ denotes the closure of the space
$C_{c}^{\infty}(\Omega)$ with respect to the Sobolev norm
$\left(\int_{\Omega}|u|^{p}dx+\int_{\Omega}|\nabla u|^{p}dx\right)^{1/p}$. It
is immediate to observe that the constant $\lambda_{p}(\Omega)>0$ if and only
if the classical Hardy inequality eq. 1.2 holds true in $\Omega$. One more
important observation is that a minimizer of the problem eq. 1.3 gives the
existence of a weak solution of the following nonlinear eigenvalue problem

(1.4) $\text{div}\left(|\nabla u|^{p-2}\nabla
u\right)+\lambda_{p}(\Omega)\frac{|u|^{p-2}u}{\delta(x)^{p}}=0\text{ in
}\Omega,$

and vise versa. Interestingly, in [21], the authors addressed this issue and
established characterization of Hardy’s inequality eq. 1.2 through weak
supersolutions of the nonlinear eigenvalue problem eq. 1.4 for a domain
$\Omega$ in $\mathbb{R}^{n}$. This paper aims to extend further research on
this topic in fractional Sobolev spaces.

For this, let us introduce the fractional Sobolev space. Let
$\Omega\subseteq\mathbb{R}^{n}$ be any open set, $0<s<1$, $1\leq p<\infty$, we
define the fractional Sobolev space

$W^{s,p}(\Omega):=\left\\{u\in
L^{p}(\Omega):[u]_{s,p,\Omega}<\infty\right\\},$

endowed with the so-called fractional Sobolev norm, given by

$||u||_{s,p,\Omega}:=\left(\|u\|_{L^{p}(\Omega)}^{p}+[u]_{s,p,\Omega}^{p}\right)^{\frac{1}{p}},$

where

$[u]_{s,p,\Omega}^{p}:=\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy,$

is the Gagliardo seminorm. For the study of fractional Sobolev spaces in a
systematic way we refer to [8, 10, 12, 16] and references therein. At this
stage, we consider two more Banach spaces $W^{s,p}_{0}(\Omega)$ and
$W^{s,p}_{\Omega}(\mathbb{R}^{n})$ defined as the closure of the space
$C_{c}^{\infty}(\Omega)$ with the norms $||\cdot||_{s,p,\Omega}$ and
$||\cdot||_{s,p,\mathbb{R}^{n}}$ respectively. These two spaces arise
naturally in studying weak solutions of the Dirichlet problems involving
regional fractional $p$-Laplacian and fractional $p$-Laplacian operators
respectively, see [6, 9, 11, 15, 28] and references therein. If $\Omega$ is a
bounded Lipschitz domain and $1<p<\infty$, then one has

$W^{s,p}_{\Omega}(\mathbb{R}^{n})=\\{u\in W^{s,p}(\mathbb{R}^{n}):u=0\text{ in
}\Omega^{c}\\},$

see [9, Proposition B.1]. Moreover,
$W^{s,p}_{\Omega}(\mathbb{R}^{n})=W^{s,p}_{0}(\Omega)$ provided $sp\neq 1$,
see for instance [6, Proposition B.1].

In the spirit of local case, we introduce what we call variational Sobolev
capacity in fractional Sobolev spaces.

######  Definition 1.1.

Let $0<s<1$, $p\in[1,\infty)$ and $\Omega\subseteq\mathbb{R}^{n}$ be an open
set. For a compact set $K\subset\Omega$, variational $(s,p)$-Sobolev capacity
is defined by

(1.5) $\text{Cap}_{s,p}(K,\Omega):=\inf\left\\{[u]_{s,p,\Omega}^{p}:u\in
C_{c}^{\infty}(\Omega),\;u\geq 1\text{ on }K\right\\}.$

For an open set $A\subset\Omega$, variational $(s,p)$-Sobolev capacity is
defined by

$\text{Cap}_{s,p}(A,\Omega)=\sup\left\\{\text{Cap}_{s,p}(K,\Omega):K\subset
A,\;K\text{ is compact }\right\\},$

and for an arbitrary set $E\subset\Omega$, variational $(s,p)$-Sobolev
capacity is defined by

$\text{Cap}_{s,p}(E,\Omega)=\inf\left\\{\text{Cap}_{s,p}(A,\Omega):E\subset
A,\;A\text{ is open }\right\\}.$

It is to be noted that by standard approximation argument, we can replace
$C_{c}^{\infty}(\Omega)$ by $W^{s,p}_{0}(\Omega)$ in the definition of
capacity eq. 1.5.

######  Remark 1.2.

It is worth mentioning that in the definition of capacity (1.5) one can
restricts the function $u\in C_{c}^{\infty}(\Omega)$ such that $u=1$ in a
neighbourhood $\mathcal{N}(K)\subset\Omega$ of $K$ and $0\leq u\leq 1$ in
$\Omega$, see for instance [29, Theorem 2.1].

######  Definition 1.3.

Let $0<s<1$, $p\in[1,\infty)$. We say that a property holds $(s,p)$-quasi
everywhere (in short $(s,p)$-q.e.) if it holds except for a set of capacity
zero.
We say a function $u:\Omega\to\mathbb{R}$ is $(s,p)$-quasi continuous (in
short $(s,p)$-q.c.) in $\Omega$ if for every $\epsilon>0$ there exists an open
set $E\subset\Omega$ such that $\text{ Cap}_{s,p}(E,\Omega)<\epsilon$ and
$u|_{\Omega\setminus E}$ is continuous.

######  Remark 1.4.

It is important to note that from [30, Theorem 2.2] we have the following
characterization for $\text{Cap}_{s,p}(K,\Omega)$ via $(s,p)$-q.e. property

$\text{Cap}_{s,p}(K,\Omega)=\inf\left\\{[u]_{s,p,\Omega}^{p}:u\in
W^{s,p}_{0}(\Omega),\;u\geq 1\;(s,p)\text{-q.e. on }K\right\\}.$

In recent years, many researchers have shown their interest in studying
variational Sobolev capacities, see [1, 2, 3, 18] for the case of classical
Sobolev spaces and [10, 29, 30, 31, 32] for the case of fractional Sobolev
spaces.

Before outlining the main results in the present paper in a precise manner, we
need to introduce some terminologies and definitions. Let $0\leq\alpha<1$ and
$R>0$. For a locally integrable function $u$, the fractional maximal function
is defined by

$M_{\alpha,R}(u)(x):=\sup\limits_{0<r<R}r^{\alpha}\fint_{B_{r}(x)}|u(y)|dy.$

If $R=\infty$, then we shall simply write $M_{\alpha,R}$ by $M_{\alpha}$ and
for $\alpha=0,\;R=\infty$, we have the usual maximal function. Let
$0<\beta<\infty$, the fractional sharp maximal function of a locally
integrable function $u$ is defined by

$u^{\\#}_{\beta,R}(x):=\sup\limits_{0<r<R}r^{-\beta}\fint_{B_{r}(x)}|u(y)-u_{B_{r}(x)}|dy.$

If $R=\infty$, then we shall simply write $u^{\\#}_{\beta,R}$ by
$u^{\\#}_{\beta}.$

######  Definition 1.5 (Pointwise fractional $p$-Hardy inequality).

Let $0<s<1$, $0\leq\alpha<1$ and $p\in[1,\infty)$. We say that an open set
$\Omega\subsetneq\mathbb{R}^{n}$ with non-empty boundary admits pointwise
fractional $p$-Hardy’s inequality if there exist constants $C>0$ and
$\sigma\geq 1$ such that

(1.6) $|u(x)|\leq
C\delta(x)^{s-\frac{\alpha}{p}}\left(M_{\alpha,\;\sigma\delta(x)}(|D^{s}_{p}u|)^{p}(x)\right)^{1/p}\text{
for all }u\in W^{s,p}_{0}(\Omega),$

and for almost every all $x\in\Omega$, where
$|D^{s}_{p}u|(x):=\left(\displaystyle\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dy\right)^{\frac{1}{p}}.$

######  Definition 1.6 (Uniformly $(s,p)$-fat set).

Let $0<s<1$ and $1\leq p<\infty.$ We say that a closed set
$E\subset\mathbb{R}^{n}$ is uniformly $(s,p)$-fat set if there exists a
constant $\gamma>0$ such that

$\text{Cap}_{s,p}(E\cap\overline{B_{r}(x)},2B_{r}(x))\geq\gamma\;\text{Cap}_{s,p}(\overline{B_{r}(x)},2B_{r}(x)),$

for all $x\in E$ and $r>0.$

Next we consider the following fractional $p$-Laplacian eigenvalue problem on
$\Omega\subset\mathbb{R}^{n}$ open set, $1<p<\infty$:

(1.7)
$\begin{cases}(-\Delta_{p})^{s}u=\lambda_{s,p}(\Omega)\frac{|u|^{p-2}u}{\delta(x)^{sp}}\text{
in }\Omega,\\\ u\in W^{s,p}_{\Omega}(\mathbb{R}^{n}),\;u\neq 0,\end{cases}$

where the fractional $p$-Laplacian operator $(-\Delta_{p})^{s}$, defined by

$(-\Delta_{p})^{s}u(x):=\lim_{\epsilon\to 0}\int_{\mathbb{R}^{n}\setminus
B_{\epsilon}(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}dy,\;x\in\mathbb{R}^{n}.$

The corresponding minimization problem of eq. 1.7 is

$\lambda_{s,p}(\Omega):=\inf_{\begin{subarray}{c}u\in
W^{s,p}_{\Omega}(\mathbb{R}^{n})\\\ u\neq
0\end{subarray}}\frac{[u]_{s,p,\mathbb{R}^{n}}^{p}}{\displaystyle\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{sp}}dx}.$

The tail space $L_{sp}^{p-1}(\mathbb{R}^{n})$ (see [24]), defined by

$L_{sp}^{p-1}(\mathbb{R}^{n})=\left\\{u\in
L^{p-1}_{\text{loc}}(\mathbb{R}^{n}):\int_{\mathbb{R}^{n}}\frac{|u(x)|^{p}}{(1+|x|)^{n+sp}}dx<\infty\right\\}.$

######  Definition 1.7 (see [24]).

We say that a function $u\in W^{s,p}_{\text{loc}}(\Omega)$ such that $u_{-}\in
L^{p-1}_{sp}(\mathbb{R}^{n})$ is a fractional weak $p$-supersolution of eq.
1.7 if

(1.8)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{n+sp}}dxdy\\\
\geq\lambda_{s,p}(\Omega)\int_{\Omega}\frac{|u(x)|^{p-2}u(x)\phi(x)}{\delta(x)^{sp}}dx,$

for all $\phi\in C_{c}^{\infty}(\Omega)$ with $\phi\geq 0.$

######  Definition 1.8.

Let $0<s<1$ and $1<p<\infty.$ We say that an open subset $\Omega$ of
$\mathbb{R}^{n}$ admits fractional $p$-Hardy’s inequality if there exists an
uniform constant $C_{H}>0$ such that

(1.9) $\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{sp}}dx\leq
C_{H}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\;\text{
for all }u\in W^{s,p}_{\Omega}(\mathbb{R}^{n}).$

We remark that very recently, in [13] the authors studied capacitary versions
of fractional Poincaré, pointwise, and localized fractional Hardy inequalities
in a metric measure space. However, their results involve the Assouad
codimension of the domain, and the function is continuous.

Let us now describe our results in this paper before formulating these.
Theorem 1.9 is a capacitary version of the fractional Sobolev–Poincaré
inequality, which is motivated by the result of [20], and whereas, theorem
1.10 gives a characterization of uniformly $(s,p)$-fat set through a boundary
fractional Sobolev–Poincaré type inequality. As an application of theorem
1.10, at the end of section 3, we provide various classes of domains that are
uniformly $(s,p)$-fat set. The existence issue regarding fractional
$p$-Hardy’s inequality eq. 1.9 in the supercritical case $sp>n$ for any proper
open set is addressed in theorem 1.11. This result can be obtained by proving
an appropriate pointwise fractional Hardy type inequality and applying the
maximal function theorem. Inspired by [21], Theorem 1.12 gives a
characterization of fractional $p$-Hardy’s inequality eq. 1.9 in terms of
fractional weak $p$-supersolution of the problem eq. 1.7.

Our main results are stated below.

######  Theorem 1.9.

Suppose $u\in W^{s,p}(B)$ be a $(s,p)$-quasi continuous function, where
$B=B_{r}(x_{0})\subset\mathbb{R}^{n}(n\geq 2)$ is an open ball of radius
$r>0.$ Let $Z(u)=\left\\{x:u(x)=0\right\\}$ be the zero set of the function
$u$ and $0<\lambda<1$. Then there exists a constant $C=C(n,s,p,\lambda)>0$
such that

$\left(\fint_{B}|u(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}\leq
C\left(\frac{1}{\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}},$

where $p^{*}_{s}=\frac{np}{n-sp}$ is the fractional Sobolev critical exponent,
$1\leq p<\frac{n}{s}$.

######  Theorem 1.10.

Let $0<s<1$, $p\in[1,\infty)$ such that $sp<n$, and let $\Omega$ be any proper
open set in $\mathbb{R}^{n}(n\geq 2)$. Then $\mathbb{R}^{n}\setminus\Omega$ is
uniformly $(s,p)$-fat set if and only if for any
$z\in\mathbb{R}^{n}\setminus\Omega$, $r>0$

(1.10)
$\left(\int_{B_{r}(z)}|u(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}\leq
C\left(\int_{B_{r}(z)}\int_{B_{r}(z)}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}\;\text{
for all }u\in C_{c}^{\infty}(\Omega),$

where $C>0$ is a constant independent of $r$ and the function $u$.

######  Theorem 1.11.

Let $\Omega$ be any open set in $\mathbb{R}^{n}$ with
$\Omega\neq\mathbb{R}^{n}$. Let $0<s<1$ and $p>1$ such that $sp>n$. Then we
have the following

$\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{sp}}dx\leq
C\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\;\text{
for all }u\in C_{c}^{\infty}(\Omega),$

where $C$ is a positive constant independent of the function. Furthermore,

$\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{sp}}dx\leq
C\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\;\text{
for all }u\in C_{c}^{\infty}(\Omega).$

######  Theorem 1.12.

Let $0<s<1$ and $1<p<\infty.$ Then fractional $p$-Hardy’s inequality eq. 1.9
holds true in $\Omega$ if and only if there exists an eigenvalue
$\lambda_{s,p}(\Omega)>0$ and a positive fractional weak $p$-supersolution
$u\in W^{s,p}_{\Omega}(\mathbb{R}^{n})$ of eq. 1.7 in $\Omega.$

This paper organized in the following way: In section 2 we collect some known
results and discussed some necessary preliminaries. Proofs of theorems 1.9,
1.10 and 1.11 along with some further results are given in section 3. In
section 4 we prove theorem 1.12.

##  2\. Preliminaries and Known results

Throughout the paper we shall assume the following notations, unless mentioned
otherwise explicitly:

  * •

$\Omega$ is an open connected set in $\mathbb{R}^{n}$, $0<s<1$, $1\leq
p<\infty,\;n\in\mathbb{N}.$

  * •

$\overline{\Omega}$ is the closure of $\Omega.$

  * •

$|\Omega|$ is the Lebesgue measure of $\Omega.$

  * •

$u_{\Omega}=\fint_{\Omega}udx=\frac{1}{|\Omega|}\int_{\Omega}udx$ is the
average of the function $u$ in $\Omega.$

  * •

$X^{c}$ is the complement of the set $X$ in the appropriate universal set.

  * •

$B_{r}(x)$ is an open ball centered at $x$ of radius $r>0$.

  * •

$\mathbb{S}^{k-1}$ is the unit sphere in $\mathbb{R}^{k}.$

  * •

$c,\;C,\;C(*,*,\cdots,*)>0$ denote generic constants that will appear in the
estimate and need not be the same as in the preceding steps; the value depends
on the quantities indicted by $*$’s.

We start with some known results and some technical lemmas that will be
required to proof our results.

######  Lemma 2.1 (Discrete Picone inequality, [7]).

Let $p\in(1,\infty)$ and let $f,g:\mathbb{R}^{n}\to\mathbb{R}$ be two
measurable functions with $f\geq 0,\;g>0$, then $L(f,g)\geq 0\text{ in
}\mathbb{R}^{n}\times\mathbb{R}^{n},$ where

$L(f,g)(x,y)=|f(x)-f(y)|^{p}-|g(x)-g(y)|^{p-2}(g(x)-g(y))\left(\frac{f(x)^{p}}{g(x)^{p-1}}-\frac{f(y)^{p}}{g(y)^{p-1}}\right).$

The equality holds if and only if $f=\alpha g$ a.e. in $\mathbb{R}^{n}$ for
some constant $\alpha.$

######  Lemma 2.2 (see [4, Lemma 2.2]).

Let $1<q<\infty$ and $a,\;b\in\mathbb{R}$. Then there exists a constant
$C(q)>0$ such that

(2.1) $\left||a|^{q-2}a-|b|^{q-2}b\right|\leq
C(q)\left(|a|+|b|\right)^{q-2}|a-b|.$

######  Theorem 2.3 (see [21, 27]).

Let $X$ be a reflexive Banach space and $X^{*}$ denotes the dual space of $X$.
If the operator $T:X\to X^{*}$ satisfies the following conditions:

  1. (1)

(Boundedness:) T is bounded that is there exists $C>0$ such that
$||Tf||_{X^{*}}\leq C||f||_{X}$, for all $f\in X.$

  2. (2)

(Demicontinuity:) For all $g\in X$, we have $\langle Tf_{k},g\rangle\to\langle
Tf,g\rangle$, whenever $f_{k}\to f$ in $X$, as $k\to\infty,$ where
$\langle,\rangle$ denotes the duality bracket.

  3. (3)

(Coercivity:) if $\\{f_{k}\\}$ is a sequence in $X$ with
$||f_{k}||_{X}\to\infty$, as $k\to\infty$ then

$\frac{\langle
Tf_{k},f_{k}\rangle}{||f_{k}||_{X}}\xrightarrow{k\to\infty}\infty.$

Then for all $g\in X^{*}$ there exists $u\in X$ such that $Tu=g.$

######  Lemma 2.4 (see [5, Lemma 3.1]).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set and $s\in(0,1)$,
$1\leq p<\infty$. Then

(2.2)
$\int_{\Omega}|u-u_{\Omega}|^{p}dx\leq\frac{\text{diam}(\Omega)^{n+sp}}{|\Omega|}\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\;\text{
for all }u\in W^{s,p}(\Omega).$

######  Lemma 2.5.

Let $s\in(0,1)$, $1\leq p<\infty.$ Then the following properties of capacity
hold:

  1. $a)$

(Ball estimate:)
$\text{Cap}_{s,p}\left(\overline{B_{r}(x)},2B_{r}(x)\right)=C\;r^{n-sp}$, for
some constant $C>0.$

  2. $b)$

(Monotonicity:) If $K_{1}\subseteq K_{2}\subset\Omega$, where $K_{i}$’s are
compact sets, one has

$\text{Cap}_{s,p}(K_{1},\Omega)\leq\text{Cap}_{s,p}(K_{2},\Omega).$

###### Proof.

$a)$ It follows from [29, Theorem 2.2] by choosing the radius of the ball
appropriately.
$b)$ It is an immediate consequence of the definition of
$\text{Cap}_{s,p}(\cdot,\Omega)$. ∎

We believe that the following two results are well known to the experts, but
we include their proof for completeness.

######  Lemma 2.6.

Let $\Omega\subsetneq\mathbb{R}^{n}$ be an open set, $s\in(0,1)$ and
$p\in(0,\infty)$ such that $sp>1.$ Then

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\leq
C\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\;\text{
for all }u\in C^{\infty}_{c}(\Omega),$

where $C=C(n,s,p)$ is a positive constant and does not depend on the domain
$\Omega$.

###### Proof.

Let $u\in C_{c}^{\infty}(\Omega)$. Then we have

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
=\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+2\int_{\Omega}|u(x)|^{p}\left(\int_{\Omega^{c}}\frac{dy}{|x-y|^{n+sp}}\right)dx.$

Note that, for $x\in\Omega$, we have

$\int_{\Omega^{c}}\frac{dy}{|x-y|^{n+sp}}\leq\int_{\mathbb{S}^{n-1}}dw\int_{d_{w,\Omega}(x)}^{\infty}\frac{dr}{r^{sp+1}}=\frac{1}{sp}\int_{\mathbb{S}^{n-1}}\frac{dw}{d_{w,\Omega}(x)^{sp}}=\frac{C(n,s,p)}{m_{sp}(x)^{sp}},$

where $d_{w,\Omega}(x)=\inf\\{|t|:x+tw\in\Omega^{c}\\}$ (see, [26]) and

$m_{sp}(x)^{sp}=\left.\frac{2\pi^{\frac{n-1}{2}}\Gamma\left(\frac{1+sp}{2}\right)}{\Gamma\left(\frac{n+sp}{2}\right)}\middle/\int_{\mathbb{S}^{n-1}}\frac{dw}{d_{w,\Omega}(x)}\right..$

Thus, we obtain

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\leq\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+2C(n,s,p)\int_{\Omega}\frac{|u(x)|^{p}}{m_{sp}(x)^{sp}}dx\\\
\leq
C(n,s,p)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy,$

where in the last step we have used the fractional Hardy inequality of
Loss–Sloane [26, Theorem 1.2]. Hence the lemma follows. ∎

######  Lemma 2.7.

Suppose $u\in L^{1}_{\text{loc}}(\mathbb{R}^{n})$ and let $0<\beta<\infty$.
Then there is a constant $C=C(n,\beta)$ and a set $A\subset\mathbb{R}^{n}$
with $|A|=0$ such that

$|u(x)-u(y)|\leq
C|x-y|^{\beta}\left(u^{\\#}_{\beta,4|x-y|}(x)+u^{\\#}_{\beta,4|x-y|}(y)\right),\;\text{
for all }x,y\in\mathbb{R}^{n}\setminus A.$

###### Proof.

Let $S_{u}$ be the set of all Lebesgue points of the function $u$ and set
$A=S_{u}^{c}$. Then, by Lebesgue differentiation theorem we have $|A|=0$. Now
fix $x\in S_{f}$, $0<r<\infty$ and we denote $B_{k}=B_{\frac{r}{2^{k}}}(x)$,
$k=0,1,2,\cdots$

$|u(x)-u_{B_{r}(x)}|=\left|\lim\limits_{m\to\infty}\sum_{k=0}^{m}\left(u_{B_{k+1}}-u_{B_{k}}\right)\right|\leq\sum_{k=0}^{\infty}\left|u_{B_{k+1}}-u_{B_{k}}\right|\\\
=\sum_{k=0}^{\infty}\left|\frac{1}{|B_{k+1}|}\int_{B_{k+1}}u(z)dz-
u_{B_{k}}\right|\leq\sum_{k=0}^{\infty}\frac{1}{|B_{k+1}|}\int_{B_{k+1}}\left|u(z)-u_{B_{k}}\right|dz\\\
\leq\sum_{k=0}^{\infty}\frac{|B_{k}|}{|B_{k+1}|}\fint_{B_{k}}\left|u(z)-u_{B_{k}}\right|dz=C(n)\sum_{k=0}^{\infty}\left(\frac{r}{2^{k}}\right)^{\beta}\left(\frac{r}{2^{k}}\right)^{-\beta}\fint_{B_{k}}\left|u(z)-u_{B_{k}}\right|dz\\\
\leq C(n,\beta)r^{\beta}u^{\\#}_{\beta,r}(x).$

Let $y\in B_{r}(x)\setminus A$ and we have $B_{r}(x)\subset B_{2r}(y)$. Now

$|u(y)-u_{B_{r}(x)}|\leq|u(y)-u_{B_{2r}(y)}|+|u_{B_{2r}(y)}-u_{B_{r}(x)}|\\\
\leq
C_{1}\;r^{\beta}u^{\\#}_{\beta,2r}(y)+\fint_{B_{r}(x)}|u(z)-u_{B_{2r}(y)}|dz\\\
\leq
C_{1}r^{\beta}u^{\\#}_{\beta,2r}(y)+C_{2}(n)\fint_{B_{2r}(y)}|u(z)-u_{B_{2r}(y)}|dz\leq
C(n,\beta)r^{\beta}u^{\\#}_{\beta,2r}(y).$

Let $x,y\in\mathbb{R}^{n}\setminus A$ with $x\neq y$ and let $r=2|x-y|$. Then
$x,y\in B_{r}(x)$ and hence we have

$|u(x)-u(y)|\leq|u(x)-u_{B_{r}(x)}|+|u(y)-u_{B_{r}(x)}|\leq
C(n,\beta)r^{\beta}\left(u^{\\#}_{\beta,2r}(x)+u^{\\#}_{\beta,2r}(y)\right).$

This completes the proof of the lemma. ∎

######  Lemma 2.8.

Let $0\leq\alpha<s<1,\;R>0$. Suppose $u\in W^{s,p}(\mathbb{R}^{n})$. Then for
any $x\in\mathbb{R}^{n}$ there is a constant $C$ such that

$u^{\\#}_{s-\alpha,R}(x)\leq CM_{\alpha,R}\left(|D^{s}_{p}u|\right)(x),$

where
$|D^{s}_{p}u|(x)=\left(\displaystyle\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dy\right)^{1/p}$.

###### Proof.

Using Hölder inequality we have

$|u((y)-u_{B_{r}(x)}|=\left|u(y)-\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}u(z)dz\right|\leq\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}|u(y)-u(z)|dz\\\
\leq\frac{1}{|B_{r}(x)|^{1/p}}\left(\int_{B_{r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}|y-z|^{n+sp}dz\right)^{1/p}.$

Integrating over the ball $B_{r}(x)$ we obtain

$\fint_{B_{r}(x)}|u((y)-u_{B_{r}(x)}|dy\leq\frac{Cr^{\frac{n+sp}{p}}}{|B_{r}(x)|^{1/p}}\fint_{B_{r}(x)}\left(\int_{B_{r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}dz\right)^{1/p}dy\\\
\leq\frac{Cr^{\frac{n+sp}{p}}}{|B_{r}(x)|^{1/p}}\fint_{B_{r}(x)}|D^{s}_{p}u|(y)dy\leq
Cr^{s-\alpha}r^{\alpha}\fint_{B_{r}(x)}|D^{s}_{p}u|(y)dy.$

Thus we get

$r^{\alpha-s}\fint_{B_{r}(x)}|u((y)-u_{B_{r}(x)}|dy\leq
Cr^{\alpha}\fint_{B_{r}(x)}|D^{s}_{p}u|(y)dy,$

and consequently we have

$u^{\\#}_{s-\alpha,R}(x)\leq CM_{\alpha,R}\left(|D^{s}_{p}u|\right)(x),\text{
for any }x\in\mathbb{R}^{n}\;\text{ and }R>0.$

∎

By the above two lemmas we obtain the following result.

######  Corollary 2.9.

Let $u\in W^{s,p}_{\text{loc}}(\mathbb{R}^{n})$ and $0\leq\alpha<s<1$. Then,
there is a set $A\subset\mathbb{R}^{n}$ with $|A|=0$ such that for all
$x,y\in\mathbb{R}^{n}\setminus A$ we have

$|u(x)-u(y)|\leq
C|x-y|^{s-\alpha}\left(M_{\alpha,4|x-y|}(|D^{s}_{p}u|)(x)+M_{\alpha,4|x-y|}(|D^{s}_{p}u|)(y)\right),$

where $C$ is a positive constant independent of the function $u.$

##  3\. On Sobolev-Poincaré and localized Hardy inequalities

This section is devoted to proofs of theorems 1.9, 1.10 and 1.11. We also
address some further results in this context.

###### Proof of theorem 1.9..

If $u_{B}=\fint_{B}u=0$. Then, applying improved fractional Sobolev–Poincaré
inequality (see, [19, 30]) for $B_{r}(x_{0})$ we have

(3.1) $\left(\fint_{B}|u(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}\leq
C\left(r^{sp-n}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}.$

Using the capacity estimate lemma 2.5 we get

$\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)\leq\text{Cap}_{s,p}(\lambda\overline{B},B)=C(n,p,\lambda)r^{n-sp}.$

By exploiting this estimate in eq. 3.1 to get the desired result in this case.

If $u_{B}\neq 0$, without loss of generality we can take $u_{B}=1.$ Choose
$\phi\in C_{c}^{\infty}(B)$ such that $\phi=1\text{ in }\lambda\overline{B}$,
$|\nabla\phi|\leq c/r$ and $0\leq\phi\leq 1.$ We define $\psi=\phi(u_{B}-u)$.
Clearly, $\psi\in W^{s,p}_{0}(B)$ be a $(s,p)$-quasi continuous function and
$\psi=1\text{ in }Z(u)\cap\lambda\overline{B}.$ Therefore we obtain

(3.2)
$\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)\leq\int_{B}\int_{B}\frac{|\psi(x)-\psi(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
\leq
2^{p-1}\left(\int_{B}\int_{B}\frac{|\phi(x)|^{p}|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+\int_{B}\int_{B}\frac{|u_{B}-u(y)|^{p}|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)\\\
\leq
2^{p-1}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+2^{p-1}\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|\leq\eta\\}}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
+2^{p-1}\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|>\eta\\}}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
=:2^{p-1}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+2^{p-1}(I_{1}+I_{2}),$

where $\eta>0$ will be chosen later,

$I_{1}:=\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|\leq\eta\\}}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy,$

and

$I_{2}:=\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|>\eta\\}}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy.$

We estimate the above integrals $I_{1},\;I_{2}$ in the following:

Estimate for $I_{1}$ : Using the properties of the function $\phi$, we have

$I_{1}=\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|\leq\eta\\}}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
\leq\frac{c^{p}}{r^{p}}\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|\leq\eta\\}}\frac{dx}{|x-y|^{n+sp-p}}dy\\\
\leq
c^{p}r^{-p}\int_{B}|u(y)-u_{B}|^{p}\int_{|x-y|\leq\eta}\frac{dx}{|x-y|^{n+sp-p}}dy\leq\frac{C\eta^{p-sp}}{r^{p}}\int_{B}|u(y)-u_{B}|^{p}dy.$

Estimate for $I_{2}$ :

$I_{2}=\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|>\eta\\}}\frac{|\phi(x)-\phi(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
\leq
2^{p}\int_{B}|u(y)-u_{B}|^{p}\int_{B\cap\\{x:|x-y|>\eta\\}}\frac{dx}{|x-y|^{n+sp}}dy\\\
\leq
2^{p}\int_{B}|u(y)-u_{B}|^{p}\int_{|x-y|>\eta}\frac{dx}{|x-y|^{n+sp}}dy\leq
C\eta^{-sp}\int_{B}|u(y)-u_{B}|^{p}dy.$

Plugging the above two estimates in eq. 3.2, we obtain

$\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)\\\ \leq
2^{p-1}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+2^{p-1}C\left(\frac{\eta^{p-sp}}{r^{p}}+\eta^{-sp}\right)\int_{B}|u(y)-u_{B}|^{p}dy.$

Now, choose $\eta=r$ the radius of the ball $B$ and by fractional
Poincaré–Wirtinger inequality eq. 2.2 we have

$\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)\\\ \leq
C\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy.$

Therefore, by above estimate we get

(3.3) $u_{B}=1\leq
C\left(\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)^{-1}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}.$

Now using the estimate eq. 3.3 together with improved fractional
Sobolev–Poincaré inequality (see, [19, 30]), we have

$\left(\fint_{B}|u(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}\leq
2^{1-\frac{1}{p^{*}_{s}}}\left[\left(\fint_{B}|u(x)-u_{B}|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}+u_{B}\right]\\\
\leq
C\;r^{s-\frac{n}{p}}\left(\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}\\\
+C\left(\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)^{-1}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}\\\
\leq
C\left(\frac{1}{\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}.$

In the last estimate we used lemma 2.5. This completes the proof of the
theorem. ∎

######  Remark 3.1.

Theorem 1.9 is essentially the best possible in the following way: Let $1\leq
p<s/n$ and $F\subset\mathbb{R}^{n}(n\geq 2)$ closed set. Take $x_{0}\in F$ be
any and consider the ball $B_{r}(x_{0})$ with $r>0$ small enough. Suppose
there exists $C_{F}>0$ such that

$\left(\int_{B_{r}(x_{0})}|u|^{p^{*}_{s}}\right)^{1/p^{*}_{s}}\leq
C_{F}\left(\int_{B_{r}(x_{0})}\int_{B_{r}(x_{0})}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{1/p},$

where $u\in W^{s,p}(B_{r}(x_{0}))$ is a $(s,p)$-quasi continuous function with
$u=0$ in $F\cap\frac{1}{2}\overline{B_{r}(x_{0})}$. Let $v\in
W^{s,p}_{0}(\frac{3}{4}B_{r}(x_{0}))$ be a $(s,p)$-quasi continuous function
with $v=1$ in $F\cap\frac{1}{2}\overline{B_{r}(x_{0})}$. Define $w:=1-v\in
W^{s,p}(B_{r}(x_{0}))$. Then $w$ is a $(s,p)$-quasi continuous function and
$w=0$ in $F\cap\frac{1}{2}\overline{B_{r}(x_{0})}$. Hence,

$C_{F}\left(\int_{B_{r}(x_{0})}\int_{B_{r}(x_{0})}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}\\\
=C_{F}\left(\int_{B_{r}(x_{0})}\int_{B_{r}(x_{0})}\frac{|w(x)-w(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}\\\
\geq\left(\int_{B_{r}(x_{0})}|w|^{p^{*}_{s}}\right)^{\frac{1}{p^{*}_{s}}}\geq\left|B_{r}(x_{0})\setminus\frac{3}{4}B_{r}(x_{0})\right|^{\frac{1}{p^{*}_{s}}}\geq
c|B_{r}(x_{0})|^{\frac{1}{p^{*}_{s}}}.$

Therefore, taking infimum over all such $v$ we infer that

$C_{F}\;\text{Cap}_{s,p}\left(F\cap\frac{1}{2}\overline{B_{r}(x_{0})},B_{r}(x_{0})\right)^{1/p}\geq
c|B_{r}(x_{0})|^{\frac{1}{p^{*}_{s}}}=cr^{\frac{n-sp}{p}}.$

###### Proof of theorem 1.10.

It is easy to see that $\mathbb{R}^{n}\setminus\Omega$ is uniformly
$(s,p)$-fat set implies the desired result by theorem 1.9 and lemma 2.5.

To prove other implication. Let $0<\lambda<2^{\frac{1-p^{*}_{s}}{p^{*}_{s}}}.$
Suppose $u\in C_{c}^{\infty}(B_{r}(z))$ such that $u(x)=1$ for
$x\in\left(\mathbb{R}^{n}\setminus\Omega\right)\cap\lambda\overline{B_{r}(z)}.$
We divide the proof in the following two steps:

Step 1: If

$\displaystyle\fint_{\lambda
B_{r}(z)}|u(x)|^{p^{*}_{s}}\geq\lambda^{p^{*}_{s}}.$

Then by hypothesis eq. 1.10 we have

$\lambda|\lambda B_{r}(z)|^{\frac{1}{p^{*}_{s}}}\leq\left(\int_{\lambda
B_{r}(z)}|u(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}\leq
Cr^{sp}\int_{B_{r}(z)}|u(x)|^{p}dx\\\ \leq C\left(\int_{\lambda
B_{r}(z)}\int_{\lambda
B_{r}(z)}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}},$

and consequently we obtain

$\int_{B_{r}(z)}\int_{B_{r}(z)}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\geq
C(n,p,s,\lambda)r^{n-sp}.$

Step 2: If

$\displaystyle\fint_{\lambda B_{r}(z)}|u(x)|^{p^{*}_{s}}<\lambda^{p^{*}_{s}}.$

Note that

$1=\fint_{\lambda B_{r}(z)}dx=\fint_{\lambda
B_{r}(z)}|1-u(x)+u(x)|^{p^{*}_{s}}dx\\\ \leq
2^{p^{*}_{s}-1}\left(\fint_{\lambda
B_{r}(z)}|1-u(x)|^{p^{*}_{s}}dx+\fint_{\lambda
B_{r}(z)}|u(x)|^{p^{*}_{s}}dx\right),$

and this yields that

$\fint_{\lambda B_{r}(z)}|1-u(x)|^{p^{*}_{s}}dx\geq
2^{1-p^{*}_{s}}-\fint_{\lambda B_{r}(z)}|u(x)|^{p^{*}_{s}}dx\geq
2^{1-p^{*}_{s}}-\lambda^{p^{*}_{s}}=C(n,s,p,\lambda)>0.$

Hence

(3.4) $\int_{\lambda B_{r}(z)}|1-u(x)|^{p^{*}_{s}}dx\geq
C(n,s,p,\lambda)r^{n}.$

Now we choose a cut-off function $\phi\in C_{c}^{\infty}(B_{r}(z))$ such that
$\phi(x)=1$ for every $x\in\lambda B_{r}(z)$ and define $f:=\phi(1-u).$ Then,
clearly Supp$(f)\subset\Omega$ and $f\in C_{c}^{\infty}(\Omega)$. Thus, by eq.
3.4 and then applying the hypothesis eq. 1.10 for $f$ we obtain

$C(n,s,p,\lambda)\;r^{\frac{n}{p^{*}_{s}}}\leq\left(\int_{\lambda
B_{r}(z)}|1-u(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}=\left(\int_{\lambda
B_{r}(z)}|f(x)|^{p^{*}_{s}}dx\right)^{\frac{1}{p^{*}_{s}}}\\\ \leq
C_{1}\left(\int_{\lambda B_{r}(z)}\int_{\lambda
B_{r}(z)}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}}=C_{1}\left(\int_{\lambda
B_{r}(z)}\int_{\lambda
B_{r}(z)}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{\frac{1}{p}},$

and consequently we have

$\int_{B_{r}(z)}\int_{B_{r}(z)}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\geq
C(n,s,p,\lambda)r^{n-sp}.$

Thus, taking infimum over all such functions $u$ to get the required result in
this case. This completes the proof of the theorem. ∎

###### Proof of theorem 1.11.

Let us choose $q\geq 1$ such that $n<sq<sp$. Now fix $x\in\Omega$ and
$x_{0}\in\partial\Omega$ such that
$|x-x_{0}|=\text{dist}(x,\partial\Omega)=\delta(x)=R$. We denote
$\chi=\chi_{B_{2R}(x_{0})}$ the characteristic function of $B_{2R}(x_{0})$.
Let $u\in C_{c}^{\infty}(\Omega)$, then by corollary 2.9 we have

(3.5) $|u(x)|=|u(x)-u(x_{0})|\leq
C|x-x_{0}|^{s-\frac{n}{q}}\left(M_{\frac{n}{q}}(|D^{s}_{p}u\chi|)(x)+M_{\frac{n}{q}}(|D^{s}_{p}u\chi|)(x_{0})\right).$

Using Hölder inequality we have

$\frac{r^{\frac{n}{q}}}{|B_{r}(x)|}\int_{B_{r}(x)}\left(|D^{s}_{p}u|\chi\right)(y)dy\leq\frac{r^{\frac{n}{q}}}{|B_{r}(x)|^{1/q}}\left(\int_{B_{r}(x)}\left(\left(|D^{s}_{p}u|\chi\right)(y)\right)^{q}dy\right)^{1/q}\\\
\leq
C(n)\left(\int_{\mathbb{R}^{n}}\left(\left(|D^{s}_{p}u|\chi\right)(y)\right)^{q}dy\right)^{1/q},$

and consequently we get

$M_{\frac{n}{q}}(|D^{s}_{p}u\chi|)(x)\leq
C(n)\;||(|D^{s}_{p}u|\chi)||_{L^{q}(\mathbb{R}^{n})}.$

Similar computation yields that

$M_{\frac{n}{q}}(|D^{s}_{p}u\chi|)(x_{0})\leq
C(n)\;||(|D^{s}_{p}u|\chi)||_{L^{q}(\mathbb{R}^{n})}.$

Combining the above two estimates in eq. 3.5 we obtain

$|u(x)|\leq
C|x-x_{0}|^{s-\frac{n}{q}}\left(\int_{B_{2R}(x_{0})}(|D^{s}_{p}u|(z))^{q}dz\right)^{1/q}\\\
\leq
CR^{s-\frac{\alpha}{q}}\left(R^{\alpha-n}\int_{B_{3R}(x)}(|D^{s}_{p}u|(z))^{q}dz\right)^{1/q}\\\
\leq C\left(\text{
dist}(x,\partial\Omega)\right)^{s-\frac{\alpha}{q}}\left(M_{\alpha,3R}(|D^{s}_{p}u|)^{q}(x)\right)^{1/q}.$

Integrating with respect to the variable $x$ over $\Omega$ with $\alpha=0$, we
infer that

$\int_{\Omega}\frac{|u(x)|^{p}}{\text{dist}(x,\partial\Omega)^{sp}}dx\leq
C\int_{\Omega}\left(M(|D^{s}_{p}u|^{q}(x))\right)^{\frac{p}{q}}dx\leq
C\int_{\mathbb{R}^{n}}|D^{s}_{p}u|^{p}(x)dx\\\
=C\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy.$

In the above estimate we used the Hardy-Littlewood-Wiener maximal function
theorem. This proved the first part of theorem 1.11. The second part follows
from the first part together with lemma 2.6. This completes the proof of the
theorem. ∎

######  Corollary 3.2.

Suppose $u\in W^{s,p}(B)$ be a $(s,p)$-quasi continuous function, where
$B=B_{r}(x_{0}).$ Let $Z(u)=\left\\{x:u(x)=0\right\\}$ be the zero set of the
function $u$ and $0<\lambda<1$. Then there exists a constant $C$ such that

$\fint_{B}|u(x)|^{p}dx\leq\frac{C}{\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)}\int_{B}\int_{B}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy.$

###### Proof.

We proceed exactly same as it is done in the proof of theorem 1.9 with
necessary changes. ∎

######  Corollary 3.3.

Let $0<s<1$ and $1\leq p<\infty.$ Let $\Omega$ be any proper open set in
$\mathbb{R}^{n}$. Then $\mathbb{R}^{n}\setminus\Omega$ is uniformly
$(s,p)$-fat set if and only if there exists a constant $C>0$ such that

$\int_{B_{r}(z)}|u(x)|^{p}dx\leq
Cr^{sp}\int_{B_{r}(z)}\int_{B_{r}(z)}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy,$

for any $z\in\mathbb{R}^{n}\setminus\Omega$, $r>0$ and $u\in
C_{c}^{\infty}(\Omega).$

###### Proof.

We proceed exactly same as it is done in the proof of theorem 1.10 with
necessary changes. ∎

######  Theorem 3.4.

Let $0<s<1$, $1\leq p<\infty$ such that $sp\leq n$, $0\leq\alpha<p$, and let
$\Omega$ be an open set in $\mathbb{R}^{n}$ such that it’s complement that is
$\mathbb{R}^{n}\setminus\Omega$ is uniformly $(s,p)$\- fat. Assume that $u\in
C^{\infty}_{c}(\Omega)$. Then pointwise fractional $p$-Hardy inequality eq.
1.6 holds true for $\Omega$ that is there exist constants $C=C(n,p,\alpha)>0$
and $\sigma>1$ such that for all $x\in\Omega$

$|u(x)|\leq
C\delta(x)^{s-\frac{\alpha}{p}}\left(M_{\alpha,\;\sigma\delta(x)}|D^{s}_{p}u|^{p}(x)\right)^{\frac{1}{p}}.$

###### Proof.

Let $x\in\Omega.$ Let us choose $x_{0}\in\partial\Omega$ for which
$|x-x_{0}|=\delta(x)=R.$ Then, by using the standard telescoping argument as
in the proof of lemma 2.7 we obtain

$|u(x)-u_{B_{R}(x_{0})}|\leq
CR^{s-\frac{\alpha}{p}}\left(M_{\alpha,\;2R}|D^{s}_{p}u|^{p}(x)\right)^{\frac{1}{p}}.$

Thus, by above estimate we have

(3.6) $|u(x)|\leq|u(x)-u_{B_{R}(x_{0})}|+|u_{B_{R}(x_{0})}|\leq
CR^{s-\frac{\alpha}{p}}\left(M_{\alpha,\;2R}|D^{s}_{p}u|^{p}(x)\right)^{\frac{1}{p}}+|u_{B_{R}(x_{0})}|.$

Now consider the set $Z(u)=\\{x\in\mathbb{R}^{n}:u(x)=0\\}$. Then by using
corollary 3.2 and the monotonicity property of the capacity along with
hypothesis, we obtain

(3.7)
$|u_{B_{R}(x_{0})}|\leq\left(C\;\text{Cap}_{s,p}(Z(u)\cap\lambda\overline{B},B)^{-1}\int_{B_{R}(x_{0})}|D^{s}_{p}u|^{p}(x)dx\right)^{\frac{1}{p}}\\\
\leq\left(C\;\text{Cap}_{s,p}(\Omega^{c}\cap\lambda\overline{B},B)^{-1}\int_{B_{R}(x_{0})}|D^{s}_{p}u|^{p}(x)dx\right)^{\frac{1}{p}}\\\
\leq
C\left(R^{sp-n}\int_{B_{R}(x_{0})}|D^{s}_{p}u|^{p}(x)dx\right)^{\frac{1}{p}}\leq
CR^{s-\frac{\alpha}{p}}\left(M_{\alpha,R}|D^{s}_{p}u|^{p}(x)\right)^{\frac{1}{p}}.$

Plugging the estimate eq. 3.7 into eq. 3.6 we obtain

$|u(x)|\leq CR^{s-\frac{\alpha}{p}}\left(M_{\alpha,\;\sigma
R}|D^{s}_{p}u|^{p}(x)\right)^{\frac{1}{p}}.$

Since we have chosen $x\in\Omega$ arbitrarily and hence this completes the
proof of the theorem. ∎

As an application of theorem 1.10 and theorem 3.4 we discuss some examples of
some class of domains that are uniformly $(s,p)$-fat set.

######  Example 3.5.

For $s\in(0,1),\;1<p<\infty$. Then all nonempty closed sets in
$\mathbb{R}^{n}$ are uniformly $(s,p)$-fat set provided $sp>n.$

###### Proof.

Suppose $E\subset\mathbb{R}^{n}$ be an nonempty closed set. Let $x\in E$ and
$r>0$. Consider $u\in C^{\infty}_{c}(2B_{r}(x))$ such that $u\geq 1$ in
$E\cap\overline{B_{r}(x)}$. Choose a cutoff function $\rho\in
C^{\infty}_{c}(3B_{r}(x))$ such that $0\leq\rho\leq 1$, $\rho=1$ in
$\overline{B_{r}(x)}$ and $|\nabla\rho|\leq c/r$. Then, by fractional Morrey’s
inequality (see, [8, Corollary 2.7]) we have

(3.8) $|\left(\rho u\right)(y)-\left(\rho u\right)(z)|\leq
C|y-z|^{s-\frac{n}{p}}[\rho u]_{s,p,\mathbb{R}^{n}}.$

Now, let $y\in E\cap\overline{B_{r}(x)}$ and $z\in B_{3r}(x)\setminus
B_{2r}(x)$. Then, we have $(\rho u)(y)\geq 1$ and $(\rho u)(z)=0.$ Thus by eq.
3.8, we obtain

(3.9) $[\rho u]_{s,p,\mathbb{R}^{n}}^{p}\geq C|y-z|^{n-sp}|\left(\rho
u\right)(y)-\left(\rho u\right)(z)|\geq Cr^{n-sp}.$

Therefore, using lemma 2.6 and eq. 3.9 we have

(3.10) $[\rho u]_{s,p,B_{2r}(x)}^{p}\geq Cr^{n-sp}.$

Also, note that

(3.11) $[\rho
u]_{s,p,B_{2r}(x)}^{p}=\int_{B_{2r}(x)}\int_{B_{2r}(x)}\frac{|\left(\rho
u\right)(y)-\left(\rho u\right)(z)|^{p}}{|y-z|^{n+sp}}dydz\\\
=\int_{B_{2r}(x)}\int_{B_{2r}(x)}\frac{|\rho(y)\left(u(y)-u(z)\right)+u(z)\left(\rho(y)-\rho(z)\right)|^{p}}{|y-z|^{n+sp}}dydz\\\
\leq
2^{p-1}\left(\int_{B_{2r}(x)}\int_{B_{2r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}dydz+\frac{C}{r^{p}}\int_{B_{2r}(x)}\int_{B_{4r}(z)}\frac{|u(z)|^{p}}{|y-z|^{n+sp-p}}dydz\right)\\\
\leq
2^{p-1}\left(\int_{B_{2r}(x)}\int_{B_{2r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}dydz+\frac{C}{r^{sp}}\int_{B_{2r}(x)}|u(z)|^{p}dz\right)\\\
\leq
C\int_{B_{2r}(x)}\int_{B_{2r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}dydz.$

In the last estimate we have used the fractional Poincaré inequality (see,
[6]) and then lemma 2.6. Now, combining the estimates eq. 3.10, eq. 3.11 we
obtain

$\int_{B_{2r}(x)}\int_{B_{2r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}dydz\geq
Cr^{n-sp}.$

Since $u$ is arbitrary and thus taking infimum over all such functions to get
the desired result. ∎

######  Example 3.6.

Let $0<s<1$ and $1\leq p<\infty$. Suppose $E\subset\mathbb{R}^{n}$ be a closed
set such it satisfies the measure density condition

(3.12) $|E\cap B_{r}(x)|\geq c\;|B_{r}(x)|,$

for all $x\in E$ and radii $r>0$, and for some constant $c>0$. Then $E$ is
uniformly $(s,p)$-fat set.

###### Proof.

Let $x\in E$. Set $K=E\cap\overline{B_{r}(x)}$. Let $u\in
C_{c}^{\infty}(B_{2r}(x))$ such that $0\leq u\leq 1$, $u=1$ in $K$. Then, we
have

$0\leq u_{4B_{r}(x)}=\fint_{4B_{r}(x)}u<1.$

Then, by above we obtain $|u(y)-u_{4B_{r}(x)}|=1-u_{4B_{r}(x)}=C>0$ for $y\in
K.$ Now using the measure density condition eq. 3.12 and the fractional
Poincaré–Wirtinger inequality eq. 2.2 we obtain

$c|B_{r}(x)|\leq|K|\leq C(p)\int_{K}|u(y)-u_{4B_{r}(x)}|^{p}dy\leq
C(p)\int_{4B_{r}(x)}|u(y)-u_{4B_{r}(x)}|^{p}dy\\\ \leq
Cr^{sp}\int_{4B_{r}(x)}\int_{4B_{r}(x)}\frac{|u(y)-u(z)|^{p}}{|y-z|^{n+sp}}dydz.$

Now, in the right hand side of the above we make change the variables to get
the required result. ∎

##  4\. Nonlocal eigenvalue problem

In this section we present the proof of theorem 1.12.

###### Proof of theorem 1.12.

Suppose we have a fractional weak $p$-supersolution $u\in
W^{s,p}_{\Omega}(\mathbb{R}^{n})$ of eq. 1.7 which is positive in $\Omega.$
Let $v\in C_{c}^{\infty}(\Omega)$. Without loss of generality we may assume
that $v\geq 0$ in $\Omega$. Define $\phi=\frac{v^{p}}{u^{p-1}}$. We claim that
the function $\phi$ is an admissible test function. In order to prove this,
let $\\{\rho_{k}\\}$ be the standard mollifiers in $\mathbb{R}^{n}$,
$k\in\mathbb{N}$. Define $\phi_{k}=\frac{v^{p}}{u_{k}^{p-1}}$, where
$u_{k}=u*\rho_{k}$. Then $\phi_{k}\in C_{c}^{\infty}(\Omega)$. Moreover, as
$u>0$ in $\Omega$, we have $u_{k}>0$ in $\Omega$ for any $k$. Also, note that
there exists $\alpha>0$ such that $u,\;u_{k}>\alpha$ in Supp$(v)$. Therefore,
for all $x,\;y\in\text{Supp}(v)$,

$|\phi_{k}(x)-\phi_{k}(y)|=\left|\frac{v(x)^{p}}{u_{k}(x)^{p-1}}-\frac{v(y)^{p}}{u_{k}(y)^{p-1}}\right|\\\
=\left|\frac{v(x)^{p}-v(y)^{p}}{u_{k}(x)^{p-1}}+\frac{v(y)^{p}\left(u_{k}(y)^{p-1}-u_{k}(x)^{p-1}\right)}{u_{k}(x)^{p-1}u_{k}(y)^{p-1}}\right|\\\
\leq\alpha^{p-1}\left|v(x)^{p}-v(y)^{p}\right|+||v||_{\infty}^{p}\frac{\left|u_{k}(y)^{p-1}-u_{k}(x)^{p-1}\right|}{u_{k}(x)^{p-1}u_{k}(y)^{p-1}}\\\
\leq p\alpha^{p-1}\left(v(x)^{p-1}+v(y)^{p-1}\right)|v(x)-v(y)|\\\
+(p-1)||v||_{\infty}^{p}\frac{\left(u_{k}(x)^{p-2}+u_{k}(y)^{p-2}\right)}{u_{k}(x)^{p-1}u_{k}(y)^{p-1}}|u_{k}(x)-u_{k}(y)|\\\
\leq C(p,\alpha,||v||_{\infty})\left(|v(x)-v(y)|+|u_{k}(x)-u_{k}(y)|\right).$

Thus, we obtain

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|\phi_{k}(x)-\phi_{k}(y)|^{p}}{|x-y|^{n+sp}}dxdy\\\
\leq
C(p,\alpha,||v||_{\infty})\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}dxdy+\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u_{k}(x)-u_{k}(y)|^{p}}{|x-y|^{n+sp}}dxdy\right).$

Since, $u_{k}\to u$ in $W^{s,p}_{\Omega}(\mathbb{R}^{n})$ (see, [16, Lemma
11]) and $u_{k}(x)\to u(x)$ pointwise, as $k\to\infty.$ Then, by generalized
dominated convergence theorem to ensure that $\phi\in
W^{s,p}_{\Omega}(\mathbb{R}^{n}).$ Hence the claim follows. Now by eq. 1.8
with $\phi$ as a test function and using discrete Picone inequality lemma 2.1
we obtain

$\lambda_{s,p}(\Omega)\int_{\Omega}\frac{u(x)^{p-1}v(x)^{p}}{\delta(x)^{sp}\;u(x)^{p-1}}dx=\lambda_{s,p}(\Omega)\int_{\Omega}\frac{u(x)^{p-1}\phi(x)}{\delta(x)^{sp}}dx\\\
\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{n+sp}}dxdy\\\
\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}dxdy,$

and this yields that

$\lambda_{s,p}(\Omega)\int_{\Omega}\frac{v(x)^{p}}{\delta(x)^{sp}}dx\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}dxdy.$

Thus, we conclude that the fractional $p$-Hardy inequality eq. 1.9 holds true
for $\Omega$ with $C_{H}=\frac{1}{\lambda_{s,p}(\Omega)}$.

To prove other implication, suppose that the domain $\Omega$ admits the
fractional $p$-Hardy’s inequality eq. 1.9 and let
$\lambda_{s,p}(\Omega)<\frac{1}{C_{H}}$. Consider the following space

$X=\left\\{u\in W^{s,p}_{\text{loc}}(\Omega):\frac{u}{\delta(x)^{s}}\in
L^{p}(\Omega),[u]_{s,p,\mathbb{R}^{n}}^{p}<\infty\right\\},$

endowed with the norm

$||u||_{X}=||u\;\delta(x)^{-s}||_{L^{p}(\Omega)}+[u]_{s,p,\mathbb{R}^{n}}.$

Since the domain $\Omega$ admits the fractional $p$-Hardy’s inequality eq.
1.9, we then have $W^{s,p}_{\Omega}(\mathbb{R}^{n})=X$ in the sense of
equivalent norms. The operator $T:X\to X^{*}$ is defined by

$\langle
Tu,\phi\rangle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{n+sp}}dxdy\\\
-\lambda_{s,p}(\Omega)\int_{\Omega}\frac{|u(x)|^{p-2}u(x)\phi(x)}{\delta(x)^{sp}}dx,\text{
for }u,\phi\in X.$

(1) Boundedness: By Hölder inequality we obtain

$||Tu||_{X^{*}}=\sup_{||v||_{X}\leq 1}|\langle Tu,v\rangle|\\\
\leq\sup_{||v||_{X}\leq
1}\left\\{\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-1}|v(x)-v(y)|}{|x-y|^{n+sp}}dxdy+\lambda_{s,p}(\Omega)\int_{\Omega}\frac{|u(x)|^{p-1}|v(x)|}{\delta(x)^{sp}}dx\right\\}\\\
\leq\sup_{||v||_{X}\leq
1}\left\\{[u]_{s,p,\mathbb{R}^{n}}^{p-1}\;[v]_{s,p,\mathbb{R}^{n}}+\lambda_{s,p}(\Omega)||u\;\delta(x)^{-s}||_{L^{p}(\Omega)}^{p-1}\;||v\;\delta(x)^{-s}||_{L^{p}(\Omega)}\right\\}\\\
\leq(1+\lambda_{s,p}(\Omega))||u||_{X}^{p-1}.$

Therefore, $T$ is bounded.

(2) Demicontinuity: Let $\\{u_{k}\\}$ be a sequence of functions in $X$ which
converges to $u$ in $X$. For $g\in X$, by Hölder inequality we obtain

(4.1) $\left|\langle Tu_{k},g\rangle-\langle Tu,g\rangle\right|=\left|\langle
Tu_{k}-Tu,g\rangle\right|\\\
=\left|\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\left(U_{k}(x,y)-U(x,y)\right)(g(x)-g(y))}{|x-y|^{n+sp}}dxdy\right.\\\
\left.-\lambda_{s,p}(\Omega)\int_{\Omega}\frac{\left(|u_{k}(x)|^{p-2}u_{k}(x)-|u(x)|^{p-2}u(x)\right)g(x)}{\delta(x)^{sp}}dx\right|\\\
\leq\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\left|U_{k}(x,y)-U(x,y)\right|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}dxdy\right)^{\frac{p-1}{p}}\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|g(x)-g(y)|^{p}}{|x-y|^{n+sp}}dxdy\right)^{1/p}\\\
+\lambda_{s,p}(\Omega)\left(\int_{\Omega}\frac{\left|u_{k}(x)|^{p-2}u_{k}(x)-|u(x)|^{p-2}u(x)\right|^{\frac{p}{p-1}}}{\delta(x)^{sp}}dx\right)^{\frac{p-1}{p}}\left(\int_{\Omega}\frac{|g(x)|^{p}}{\delta(x)^{sp}}dx\right)^{1/p}\\\
\leq||g||_{X}\;\left[\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\left|U_{k}(x,y)-U(x,y)\right|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}dxdy\right)^{\frac{p-1}{p}}\right.\\\
\left.+\lambda_{s,p}(\Omega)\left(\int_{\Omega}\frac{\left|u_{k}(x)|^{p-2}u_{k}(x)-|u(x)|^{p-2}u(x)\right|^{\frac{p}{p-1}}}{\delta(x)^{sp}}dx\right)^{\frac{p-1}{p}}\right]\\\
=:||g||_{X}(I_{1}+\lambda_{s,p}(\Omega)I_{2}),$

where $U_{k}(x,y)=|u_{k}(x)-u_{k}(y)|^{p-2}(u_{k}(x)-u_{k}(y))$,
$U(x,y)=|u(x)-u(y)|^{p-2}(u(x)-u(y))$ and

$I_{1}:=\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\left|U_{k}(x,y)-U(x,y)\right|^{\frac{p}{p-1}}}{|x-y|^{n+sp}}dxdy\right)^{\frac{p-1}{p}},$

$I_{2}:=\left(\int_{\Omega}\frac{\left|u_{k}(x)|^{p-2}u_{k}(x)-|u(x)|^{p-2}u(x)\right|^{\frac{p}{p-1}}}{\delta(x)^{sp}}dx\right)^{\frac{p-1}{p}}.$

We shall estimate the integrals $I_{1},\;I_{2}$ in the following:

Estimate for $I_{1}$: Using eq. 2.1 we obtain

$I_{1}^{\frac{p}{p-1}}\leq
C\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\Sigma_{k}(x,y)}{|x-y|^{n+sp}}dxdy,$

where

$\Sigma_{k}(x,y):=|(u_{k}(x)-u_{k}(y))-(u(x)-u(y))|^{\frac{p}{p-1}}(|u_{k}(x)-u_{k}(y)|+|u(x)-u(y)|)^{\frac{p(p-2)}{p-1}}.$

Using Hölder inequality with the exponents $p-1$ and $\frac{p-1}{p-2}$, and
then by convexity we obtain

$I_{1}^{\frac{p}{p-1}}\leq
C[u_{k}-u]_{s,p,\mathbb{R}^{n}}^{\frac{p}{p-1}}([u_{k}]_{s,p,\mathbb{R}^{n}}^{p}+[u]^{p}_{s,p,\mathbb{R}^{n}})^{\frac{p-2}{p-1}}\leq
C||u_{k}-u||_{X}^{\frac{p}{p-1}}(||u_{k}||_{X}^{p}+||u||_{X}^{p})^{\frac{p-2}{p-1}}.$

Estimate for $I_{2}$: Similarly as above we get

$I_{2}^{\frac{p}{p-1}}\leq
C\int_{\Omega}\frac{|u_{k}(x)-u(x)|^{\frac{p}{p-1}}(|u_{k}(x)|+|u(x)|)^{\frac{p(p-2)}{p-1}}}{\delta(x)^{sp}}dx\\\
\leq
C\left(\int_{\Omega}\frac{|u_{k}(x)-u(x)|^{p}}{\delta(x)^{sp}}dx\right)^{\frac{1}{p-1}}\left(\int_{\Omega}\frac{|u_{k}(x)|^{p}}{\delta(x)^{sp}}dx+\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{sp}}dx\right)^{\frac{p-2}{p-1}}\\\
\leq
C||u_{k}-u||_{X}^{\frac{p}{p-1}}(||u_{k}||_{X}^{p}+||u||_{X}^{p})^{\frac{p-2}{p-1}}.$

Now, plugging the above two estimates into eq. 4.1 and since $u_{k}\to u$ in
$X$ we get

$|\langle Tu_{k},g\rangle-\langle Tu,g\rangle\leq
C||g||_{X}(1+\lambda_{s,p}(\Omega))||u_{k}-u||_{X}(||u_{k}||_{X}^{p}+||u||_{X}^{p})^{\frac{p-2}{p}}\xrightarrow{k\to\infty}0.$

This proves that $T$ is demicontinuous.

(3) Coercivity: Let $\\{u_{k}\\}$ be a sequence of functions in $X$ such that
$||u_{k}||_{X}\to\infty$, as $k\to\infty.$ Then, by fractional $p$-Hardy’s
inequality eq. 1.9 we obtain

$\langle
Tu_{k},u_{k}\rangle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy-\lambda_{s,p}(\Omega)\int_{\Omega}\frac{|u(x)|^{p}}{\delta(x)^{sp}}dx\\\
\geq(1-C_{H}\;\lambda_{s,p}(\Omega))\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy\geq\frac{1-C_{H}\;\lambda_{s,p}(\Omega)}{\left(1+\tilde{C}_{H}\right)^{p}}||u_{k}||_{X}^{p},$

and consequently this yields that

$\frac{\langle
Tu_{k},u_{k}\rangle}{||u_{k}||_{X}}\geq\frac{1-C_{H}\;\lambda_{s,p}(\Omega)}{\left(1+\tilde{C}_{H}\right)^{p}}||u_{k}||_{X}^{p-1}\xrightarrow{k\to\infty}\infty.$

Hence, $T$ is coercive. Therefore, the operator $T$ satisfied all the
requirements of theorem 2.3. This implies that for a $g\in X^{*}$ with $g\geq
0$, $g\not\equiv 0$, there exists a solution $u\in
X=W^{s,p}_{\Omega}(\mathbb{R}^{n})$ of the equation $Tu=g.$ In other words,
$u$ is a fractional weak $p$-supersolution of eq. 1.7. Furthermore, the
function $u$ is positive in $\Omega$. Indeed, as $u\in X$ then $u_{-}$ is also
in $X$ (see, [32, Lemma 2.6]), where $u_{-}:=\max\\{0,-u\\}$ is the negative
part of the function $u.$ Note that again by using fractional $p$-Hardy’s
inequality eq. 1.9 we obtain

$0\leq\langle
Tu,u_{-}\rangle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(u_{-}(x)-u_{-}(y))}{|x-y|^{n+sp}}dxdy\\\
-\lambda_{s,p}(\Omega)\int_{\Omega}\frac{|u(x)|^{p-2}u(x)u_{-}(x)}{\delta(x)^{sp}}dx\\\
=-\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u_{-}(x)-u_{-}(y)|^{p}}{|x-y|^{n+sp}}dxdy+\lambda_{s,p}(\Omega)\int_{\Omega}\frac{|u_{-}(x)|^{p}}{\delta(x)^{sp}}dx\\\
\leq\left(\lambda_{s,p}(\Omega)-\frac{1}{C_{H}}\right)\int_{\Omega}\frac{|u_{-}(x)|^{p}}{\delta(x)^{sp}}dx\leq
0,$

and consequently this implies that $u_{-}(x)=0$ a.e. in $\Omega$. Hence $u$ is
non negative and the positiveness of $u$ follows from the minimum principle
(see, [7, Theorem A.1]). This completes the proof of the theorem. ∎

Acknowledgement: The research of the author is supported by the SERB WEA grant
no. WEA/2020/000005.

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