# Stability of complement value problems for $p$-Lévy operators

Guy Foghem  Technische Universtät Dresden, Fakultät für Mathematik, Institut
für Wissenschaftliches Rechnen, Zellescher Weg 23-25 01217, Dresden, Germany.
guy.foghem[at]tu-dresden.de

###### Abstract.

We set-up a general framework tailor made to solve complement value problems
governed by symmetric nonlinear integrodifferential $p$-Lévy operators. A
prototypical example of integrodifferential $p$-Lévy operators is the well-
known fractional $p$-Laplace operator. Our main focus is on nonlinear IDEs in
presence of Dirichlet, Neumann and Robin conditions and we show well posedness
results. Several results are new even for the fractional $p$-Laplace operator
but we develop the approach for general translation-invariant nonlocal
operators. We also bridge a gap from nonlocal to local, by showing that
solutions to the local Dirichlet and Neumann boundary value problems
associated with $p$-Laplacian are strong limits of the nonlocal ones.

###### Key words and phrases:

Nonlinear nonlocal operators, $p$-Lévy operators, IntegroDifferential
Equations(IDEs), Nonlocal Sobolev spaces, Stability of weak solutions

###### 2020 Mathematics Subject Classification:

35D30,35J20, 34B15,35J60,35J66,35B35, 46E35, 49J40, 49J45

###### Contents

  1. 1 Introduction
  2. 2 Basic concepts and notations
  3. 3 Nonlocal function spaces
  4. 4 Characterization of the $p$-Lévy integrability
  5. 5 Nonlocal Trace Theorem
  6. 6 Compact embeddings
  7. 7 Nonlocal Poincaré type inequalities
  8. 8 Existence of weak solutions
  9. 9 Transition from nonlocal to local
  10. 10 Robust Poincaré inequality
  11. 11 Convergence of weak solutions
  12. A
  13. B

##  1\. Introduction

In this article we study certain class of nonlinear IntegroDifferential
Equations (IDEs) associated with symmetric nonlinear nonlocal operators of
$p$-Lévy operators, $1<p<\infty$ which are integrodifferential operators of
the form

$\displaystyle\begin{split}Lu(x)&=2\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p-2}(u(x)-u(y))\nu(x\\!-\\!y)\mathrm{d}y,\qquad\quad(x\in\operatorname{\mathbb{R}}^{d})\\\
&=2\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))\nu(x\\!-\\!y)\mathrm{d}y.\end{split}$
(1.1)

Throughout, we assume that $d\geq 1$,
$\psi(t)=|t|^{p-2}t,\,t\in\operatorname{\mathbb{R}}$ and the function
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is the
symmetric density of a symmetric $p$-Lévy measure, i.e., $\nu$ satisfies

$\displaystyle\begin{split}\nu(h)=\nu(-h)\quad&\text{ $h\neq 0$ and\,\,
}\qquad\int_{\operatorname{\mathbb{R}}^{d}}(1\land|h|^{p})\nu(h)\mathrm{d}h<\infty.\end{split}$
(L)

Moreover, if $\nu$ is radial we use same notation for its radial profile,
i.e., $\nu(h)=\nu(|h|)$. The notation $\operatorname{\operatorname{p.\\!v.}}$
stands for the principal value, whereas $a\land b$ denotes $\min(a,b)$,
$a,b\in\operatorname{\mathbb{R}}$. More succinctly, given an open set
$\Omega\subset\operatorname{\mathbb{R}}^{d}$ we are interested in the problems
of the forms

$\displaystyle Lu=f\,\,\text{ in }\,\,\Omega\quad\text{ and
}\quad\tau\mathcal{N}u+(1-\tau)\beta|u|^{p-2}u=g\,\,\text{ on
}\,\,\Omega^{c},$ ($P_{\nu,\tau}$)

where $\tau\in\\{0,\frac{1}{2},1\\}$ and the data
$f:\Omega\to\operatorname{\mathbb{R}}$,
$g,\beta:\Omega^{c}\to\operatorname{\mathbb{R}}$ are given. The operator
$u\mapsto\operatorname{\mathcal{N}}u$ is the nonlocal $p$-normal derivative
across the boundary of $\Omega$,

$\displaystyle\operatorname{\mathcal{N}}u(y)=2\int_{\Omega}\psi(u(y)-u(x))\nu(x\\!-\\!y)\mathrm{d}x\qquad\quad(y\in\Omega^{c}).$
(1.2)

The nonlinear operator $u\mapsto\operatorname{\mathcal{N}}u$ appears naturally
while deriving the nonlocal Gauss-Green formula (see Appendix B.2) and was
introduce in the linear setting in [DROV17]. We point out that another type of
such an operator appeared earlier in literature see for instance [DGLZ12]. In
view of problem ($P_{\nu,\tau}$), it is necessary and sufficient to prescribe
the complement condition on $\Omega_{e}:=\Omega_{\nu}\setminus\Omega$ with
$\Omega_{\nu}=\Omega+\operatorname{supp}\nu$ which we name as _the nonlocal
boundary of the exterior boundary_ of $\Omega$ with respect to $\nu$. As well,
we also call $\Omega_{\nu}=\Omega+\operatorname{supp}\nu$ to be _the nonlocal
hull_ of $\Omega$ with respect to $\nu$; see in Section 8 for more details.
Genuinely speaking, PDEs oriented folks may view the problem ($P_{\nu,\tau}$)
as the nonlocal counterpart of the nonlinear local problem of the form

$\displaystyle-\Delta_{p}u=f\text{ in }\Omega\quad\text{ and
}\quad\tau\partial_{n,p}u+(1-\tau)\beta|u|^{p-2}u=g\text{ on }\partial\Omega$
($P_{\tau}$)

where $\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ denotes the
$p$-Laplace operator and when $\Omega$ is smooth, $\partial_{n,p}u=|\nabla
u|^{p-2}\nabla u\cdot n$ is the $p$-normal derivative on the boundary
$\partial\Omega$ of $\Omega$, viz., at a point $x\in\partial\Omega$, $n(x)$ is
the outward normal derivative on the boundary $\partial\Omega$. Of course, one
immediately recognizes that the complement (resp. boundary) condition the
problem ($P_{\nu,\tau}$) (resp. ($P_{\tau}$)) is the Neumann condition
$\operatorname{\mathcal{N}}u=g$ (resp. $\partial_{n,p}u=g$) for $\tau=1$, the
Robin condition $\operatorname{\mathcal{N}}u+\beta|u|^{p-2}u=2g$ (resp.
$\partial_{n,p}u+\beta|u|^{p-2}u=2g$) for $\tau=\frac{1}{2}$ and the Dirichlet
condition $|u|^{p-2}u=g$ or equivalently $u=|g|^{p^{\prime}-2}g$ for $\tau=0$
and $\beta=1$ where we have $pp^{\prime}=p+p^{\prime}$. Actually, our second
goal is to show that weak solutions to the problem ($P_{\tau}$) $\tau=0,1$ are
strong limit of those of problems of the forms ($P_{\nu,\tau}$). Before we
explain our strategy, let us discuss the assumption in (L) which is utterly
decisive and give a prototypical example.

It remarkably appears that the assumption in (L) is unavoidable. Indeed, on
the one hand, the symmetry condition is natural in the sense that one has

$\displaystyle\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)=\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y\mathrm{d}x=\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p}\nu^{\operatorname{sym}}(x-y)\mathrm{d}y\mathrm{d}x.$

where $\nu^{\operatorname{sym}}(h)=\frac{1}{2}(\nu(h)+\nu(-h))$ is the
symmetric part of $\nu$. On the other hand, it appears that
$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty$ for
all $u\in C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$ if and only if
$\nu\in L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$; cf. Section 4 for
the details. Literally, it turns out that the symmetry and the $p$-Lévy
condition in (L) can be self-generated through the energy form
$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u).$
Furthermore, a heuristic computation reveals that the first variation of the
operator $u\mapsto
Q(u)=\frac{1}{p}\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)$
on the Banach space $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})=\\{u\in
L^{p}(\operatorname{\mathbb{R}}^{d})\,:\,\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty\\}$
with the norm
$\|u\|_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}=\big{(}\|u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}+\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)\big{)}^{1/p}$
gives rise to the $p$-Lévy operator $L$. Moreover, we have
$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)=\langle
Lu,u\rangle=Q^{\prime}(u)(u)$, $u\in
C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$. The density $\nu$ is somewhat
the “order” of the operator $L$, which becomes apparent in the case of
fractional kernel $\nu(h)=|h|^{-d-sp}$, $s\in(0,1)$ is fixed. For case $p=2$,
one immediately recognizes the so call Lévy condition in (1.7) while the
operator $L$ is translation invariant and generates a symmetric Lévy process.
In general, the pointwise evaluation of $Lu(x)$ and $-\Delta_{p}u(x)$ for
$u\in C^{2}_{b}(\operatorname{\mathbb{R}}^{d})$ is warranted in the degenerate
case (also often called the superquadratic case), i.e. $p\geq 2$. However, in
the situation singular case (also often called the subquadratic case), i.e.,
$1<p<2$, the pointwise definition of $Lu(x)$ and $-\Delta_{p}u(x)$ might not
exist even for _a bona fide_ function $u\in
C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$. In many situations, the
operators $L$ and $-\Delta_{p}$ do not always act on functions in a reasonable
pointwise sense. We refer interested reader in Appendix B.1 for some sketchy
examples. In both cases, i.e., $1<p<\infty$, a reasonable alternative is
rather to evaluate $Lu$ and $\Delta_{p}u$ in the generalized sense, i.e., in
the weak sense or via their respective associated energies forms. For instance
by duality (see Section 3.1), one finds that the nonlinear operators
$L:W^{p}_{\nu}(\operatorname{\mathbb{R}}^{d})\to(W^{p}_{\nu}(\operatorname{\mathbb{R}}^{d}))^{\prime}$
and
$\Delta_{p}:W^{1,p}(\operatorname{\mathbb{R}}^{d})\to(W^{1,p}(\operatorname{\mathbb{R}}^{d}))^{\prime}$
are well-defined. In particular, $Lu$ and $\Delta_{p}u$ are distributions.
Morally, the nonlocal operator $L$ may be as good or as bad the local operator
$-\Delta_{p}$. In fact, the operator $L$ can be seen as a prototype of a
nonlinear nonlocal operator of divergence just as the $-\Delta_{p}$ is a
prototype of a nonlinear local operator of divergence.

For prototypical example, consider $s\in\operatorname{\mathbb{R}}$ then an
effortless computation reveals that $\nu(h)=\frac{C_{d,p,s}}{2}|h|^{-d-sp}$
belongs to $L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$, i.e., $\nu$
satisfies (L) if and only if $s\in(0,1)$. In this case, the resulting Banach
space
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})=W^{s,p}(\operatorname{\mathbb{R}}^{d})$
is the usual fractional Sobolev-Slobodeckij space, whereas the associated
integrodifferential operator $L$ is the well-known fractional $p-$Laplace
operator $(-\Delta)^{s}_{p}$,

$\displaystyle(-\Delta)^{s}_{p}u(x)$
$\displaystyle=C_{d,p,s}\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}}\mathrm{d}y,$

where $C_{d,p,s}$ is a normalizing constant of the fractional $p-$Laplacian
which we define by

$\displaystyle
C_{d,p,s}=\frac{s(1-2s)\Gamma(\frac{d+sp}{2})}{\pi^{\frac{d-1}{2}}\Gamma(\frac{sp+1}{2})\Gamma(p(1-s))\cos(s\pi)}.$

Our choice of the constant $C_{d,p,s}$, see Section 9.4, guaranties the
following properties:

  * •

For $p=2$, by [Fog20, Proposition 2.21], $C_{d,2,s}$ is the unique normalizing
constant of the fractional Laplacian such that
$\widehat{(-\Delta)^{s}u}(\xi)=|\xi|^{2s}\widehat{u}(\xi)$,
$\xi\in\operatorname{\mathbb{R}}^{d}$ for all $u\in
C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$, where
$\widehat{u}(\xi)=\int_{\operatorname{\mathbb{R}}^{d}}e^{-ix\cdot\xi}u(x)\mathrm{d}x$
denotes the in Fourier transform of $u$. Namely, we have

$\displaystyle
C_{d,2,s}=\frac{s(1-2s)\Gamma(\frac{d+2s}{2})}{\pi^{\frac{d-1}{2}}\Gamma(\frac{2s+1}{2})\Gamma(2(1-s))\cos(s\pi)}=\frac{s2^{2s}\Gamma(\frac{d+2s}{2})}{\pi^{d/2}\Gamma(1-s)}.$

  * •

For any $u\in L^{\infty}(\operatorname{\mathbb{R}}^{d})\cap C^{2}(B_{1}(x))$,
$\nabla u(x)\neq 0$ we have $(-\Delta)^{s}_{p}u(x)\xrightarrow{s\to
1}-\Delta_{p}u(x)$.

  * •

For all $u\in W^{1,p}(\operatorname{\mathbb{R}}^{d})$ we have
$\|u\|_{W^{s,p}(\operatorname{\mathbb{R}}^{d})}\xrightarrow{s\to
1}\|u\|_{W^{1,p}(\operatorname{\mathbb{R}}^{d})}$; see Section 9.5.

  * •

Moreover we have the following asymptotic behaviors

$\displaystyle\lim_{s\to 0}\frac{C_{d,p,s}}{s(1-s)}$
$\displaystyle=\frac{2}{|\mathbb{S}^{d-1}|\Gamma(p)}\qquad\text{and}\qquad\lim_{s\to
1}\frac{C_{d,p,s}}{s(1-s)}=\frac{2p}{|\mathbb{S}^{d-1}|K_{d,p}}.$ (1.3)

Here, we emphasize that the constant $K_{d,p}$, see [Fog23, Fog20] for the
computation, plays a crucial role in our asymptotic analysis and is given by

$\displaystyle K_{d,p}$
$\displaystyle=\fint_{\mathbb{S}^{d-1}}|w_{d}|^{p}\mathrm{d}\sigma_{d-1}(w)=\frac{\Gamma\big{(}\frac{d}{2}\big{)}\Gamma\big{(}\frac{p+1}{2}\big{)}}{\Gamma\big{(}\frac{d+p}{2}\big{)}\Gamma\big{(}\frac{1}{2}\big{)}}.$
(1.4)

The asymptotic $s\to 1$, highlighting the factor $K_{d,p}$ is already
anticipated in [Fog20, Eq: 2.38] in the case $p=2$ since
$K_{d,2}=\frac{1}{d}$. The above asymptotic in (1.3) perfectly lines up with
the case $p=2$ as obtained in [DNPV12]. Despite the amusing fact of this
asymptotic, it is important for the reader to keep in mind that $C_{d,p,s}$ is
purely artificial and that only the case $p=2$ naturally appears as the unique
constant for which $\widehat{(-\Delta)^{s}u}(\xi)=|\xi|^{2s}\widehat{u}(\xi)$.
Another different normalizing constant for the fractional $p$-Laplacian is
proposed in [DGV21] wherein, one also finds other representations of the
fractional $p$-Laplace operator.

The main purpose of this article is twofold. The primary objective is to study
the well-posedness of the nonlocal problem ($P_{\nu,\tau}$) under additional
mild assumptions. For instance, as an avant-goût, let us illuminate what we do
with the particular case of the Neumann problem, i.e., $\tau=1$ and for the
particular fractional kernel $\nu(h)=\frac{C_{d,p,s}}{2}|h|^{-d-sp}$. We refer
the reader to Section 8 for the general setting and more details. We point
that $u$ is a weak solution to Neumann problem $(P_{\nu,1})$ if $u\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and satisfies

$\displaystyle\operatorname{\mathcal{E}}(u,v)=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\Omega^{c}}g(y)v(y)\mathrm{d}y\qquad\text{for
all $u\in W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$}.$ (1.5)

Here we consider
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})=\big{\\{}u:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}\,\text{meas.}:\,u|_{\Omega}\in
L^{p}(\Omega),\,\,\operatorname{\mathcal{E}}(u,u)<\infty\big{\\}}$ with energy
form $\operatorname{\mathcal{E}}(\cdot,\cdot)$,

$\displaystyle\operatorname{\mathcal{E}}(u,v)=\frac{C_{d,p,s}}{2}\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\frac{\psi(u(x)-u(y))(v(x)-v(y))}{|x-y|^{d+sp}}\mathrm{d}y\mathrm{d}x.$

In Theorem 8.12, we establish the well-posedness up to additive constants of
the variational problem (1.5) and hence of the Neumann problem $(P_{\nu,1})$
on the space $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ whenever
$f\in L^{p^{\prime}}(\Omega)$ and $g\in
L^{p^{\prime}}(\Omega^{c},\omega^{1-p^{\prime}})$ with
$\omega(h)=(1+|h|)^{-d-sp}$ are compatible, i.e.,

$\displaystyle\int_{\Omega}f(x)\mathrm{d}x+\int_{\Omega^{c}}g(y)\mathrm{d}y=0.$

Let us highlight two important observations regarding the weak formulation
(1.5) at this stage. First of all, one observes in this particular that the
weight $\omega^{1-p^{\prime}}$ is rapidly increasing, hence the Neumann data
$g\in L^{p^{\prime}}(\Omega^{c},\omega^{1-p^{\prime}})$ is required to decay
rapidly at infinity. Although this may seems restrictive, it is however
counterintuitive since the space
$L^{p^{\prime}}(\Omega^{c},\omega^{1-p^{\prime}})$ is in fact the largest
admissible function space for the Neumann data $g$. Indeed, we establish a
non-existence result in Theorem 8.14 where we exhibit some examples of Neumann
data $g$ not belonging to $L^{p^{\prime}}(\Omega^{c},\omega^{1-p^{\prime}})$
and compatible with $f=0$ the variational Neumann problem (1.5) has no
solution in $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. As the second
observation, one notices that [DROV17, Definition 3.6] and subsequent
definitions like [MPL19, Definition 2.7] look very similar to (1.5) at first
glance; for the general case, we refer the reader to Definition 8.6. However,
the test space defined in [DROV17, Eq. (3.1)], [MPL19, Section 2] depends on
the Neumann data $g$, which is somewhat unaccustomed. We emphasize that our
test space $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in the weak
formulation (1.5) does not depend on the Neumann data $g$. Next, we would like
to mention that our strategy of studying the problem ($P_{\nu,\tau}$) in the
general setting also includes a Dirichlet and Robin complement problems,
requires to bring into play certain crucial tools. These include nonlocal
functions spaces, nonlocal Poincaré types inequalities and nonlocal trace
spaces. On the one hand, the on Poincaré inequalities we establish in Section
7 encoding the coercivity of the nonlocal energies, include the Poincaré type
inequality when $\Omega$ is not necessarily connected but has finitely many
connected components and the Poincaré-Friedrichs type inequality when $\Omega$
is bounded only in one direction. To the best of our knowledge, such
generalization of the Poincaré inequalities do not exist in the literature so
far. On the other hand, the nonlocal trace spaces truly embody the Dirichlet,
Neumann and/or Robin complement data $g$ of the problem ($P_{\nu,\tau}$).
There are several new findings on nonlocal trace spaces, we refer interested
reader for instance to [GH22, DTWY22, BGPR20, Rut20, DK19]. It is worth
mentioning that the result [GH22] provides a robust nonlocal fractional trace
space, viz., one is able to recover the classical local trace from the
nonlocal one.

The secondary objective is to prove the $L^{p}$-convergence of nonlocal to
local weak solutions; cf Section 11. Let us explain for which family of
problems we are able to prove the convergence. Given a family
$(\nu_{\varepsilon})_{\varepsilon}$, $\nu_{\varepsilon}\geq 0$, of radial
$p$-Lévy kernel, we prove in Theorem 9.6 that, there holds the formula

$\displaystyle\lim_{\varepsilon\to
0}\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p}\nu_{\varepsilon}(x-y)\mathrm{d}y\mathrm{d}x=K_{d,p}\int_{\operatorname{\mathbb{R}}^{d}}|\nabla
u(x)|^{p}\mathrm{d}x$ (1.6)

for all $u\in W^{1,p}(\operatorname{\mathbb{R}}^{d})$ if and only if the
family $(\nu_{\varepsilon})_{\varepsilon}$ satisfies

$\displaystyle\lim_{\varepsilon\to
0}\int_{\operatorname{\mathbb{R}}^{d}}(1\land|h|^{p})\nu_{\varepsilon}(h)\mathrm{d}h=1\quad\text{and
for all $\delta>0,$}\quad\lim_{\varepsilon\to
0}\int_{|h|>\delta}(1\land|h|^{p})\nu_{\varepsilon}(h)\mathrm{d}h=0.$ (1.7)

In other words, condition (1.7) is the sharpest for which in the formula (1.6)
(a.k.a in literature the BBM-formula) holds true. Note in passing that a
typical example of family $(\nu_{\varepsilon})_{\varepsilon}$ satisfying
(1.7), is to consider the normalized fractional kernels
$\nu_{\varepsilon}(h)=a_{\varepsilon,d,p}|h|^{-d-(1-\varepsilon)p}$ with
$\varepsilon=1-s$ and
$a_{\varepsilon,d,p}=\frac{p\varepsilon(1-\varepsilon)}{|\mathbb{S}^{d-1}|}$;
see Example 9.4. Another fascinating example of
$(\nu_{\varepsilon})_{\varepsilon}$ satisfying (1.7) is obtained from any
radial function
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ that is
$p$-Lévy normalized, i.e.,
$\int_{\operatorname{\mathbb{R}}^{d}}(1\land|h|^{p})\,\nu(h)\mathrm{d}h=1$, by
considering $\nu_{\varepsilon}$ defined as the rescaled version of $\nu$ with

$\displaystyle\begin{split}\nu_{\varepsilon}(h)=\begin{cases}\varepsilon^{-d-p}\nu\big{(}h/\varepsilon\big{)}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ |h|\leq\varepsilon,\\\
\varepsilon^{-d}|h|^{-p}\nu\big{(}h/\varepsilon\big{)}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \varepsilon<|h|\leq 1,\\\
\varepsilon^{-d}\nu\big{(}h/\varepsilon\big{)}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ |h|>1.\end{cases}\end{split}$

Let $L_{\varepsilon}$ and $\operatorname{\mathcal{N}}_{\varepsilon}$ be the
nonlocal operators associated with $\nu_{\varepsilon}$ and put
$\mu=K_{d,p}^{-1}$. Consider the problem

$\displaystyle\mu L_{\varepsilon}u=f_{\varepsilon}\,\,\text{ in
}\,\,\Omega\quad\text{ and
}\quad\tau\mu\mathcal{N}_{\varepsilon}u+(1-\tau)|u|^{p-2}u=g_{\varepsilon}\,\,\text{
on }\,\,\Omega^{c},$ ($P_{\nu_{\varepsilon},\tau}$)

Surprisingly, under the condition in(1.7) and some mild conditions on
$\Omega$, $f_{\varepsilon}$ and $g_{\varepsilon}$ we prove, cf. Section 11,
that weak solutions to the problem $\eqref{eq:main-problem-esp}$ with
$\tau=\\{0,1\\}$ strongly converge in $L^{p}(\Omega)$ as $\varepsilon\to 0$,
to the weak solutions of the corresponding local problem $\eqref{eq:main-
problem-local}$. We also prove the convergence of weak associated with
regional type operators. Another appealing effect of the approximation family
$(\nu_{\varepsilon})_{\varepsilon}$ is that for $u\in
L^{\infty}(\operatorname{\mathbb{R}}^{d})\cap C^{2}(B_{1}(x))$ with $\nabla
u(x)\neq 0$ there holds,

$\displaystyle\lim_{\varepsilon\to 0}L_{\varepsilon}u(x)=\lim_{\varepsilon\to
0}2\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p-2}(u(x)-u(y))\nu(x-y)\mathrm{d}y=-K_{d,p}\Delta_{p}u(x).$

In particular, we find that $(-\Delta)^{s}_{p}u(x)\to-\Delta_{p}u(x)$ as $s\to
1$. A similar pointwise asymptotic from the fractional $p$-Laplacian to the
$p$-Laplacian $-\Delta_{p}$ can be found in [BS22, DL21, IN10]. We refer to
Section 9 for more details. In our strategy of proving the convergence of
solutions we need, cf. Section 10, to establish the robust Poincaré type
inequalities à la A. Ponce [Pon04a] including in the situation where $\Omega$
is only bounded in one direction.

Let us comment on related works in the literature. For an outstanding
reference on basics related to the $p$-Laplace operator we refer the reader to
the classical lecture notes [Lin19]. The study of nonlocal operators driven by
Lévy kernel are becoming popular. For recent studies of IntegroDifferential
Equations(IDEs) involving the Lé vy operator of type $L$ (for the case p=2)
see [FK22, DFK22]. See also [Rut18, Rut20] and [ROV16] for the studies of
Dirichlet problems associated Lé vy operator driven by singular Lévy measure.
For problems related to general nonlocal elliptic type operators of Lévy see
[Fog20, Voi17]. We also point out [BGPR23, Rut20] where the nonlinear Douglas
identity for $p$-Lévy type operators are investigated. The fractional
$p$-Laplacian is one of the most studied $p$-Lévy operator with the framework
of IDEs. For instance, the study of Dirichlet problems can be found in [FP14,
DKP16, ILPS16, Pal18, QX16, BP16, FI22] and for the study of Neumann problems
see [MPL19, MPL21]. Note that [MPL19] is somewhat the extension of the idea
and the set up from [DROV17] to the nonlinear setting. We also refer to
[MRT16, AMRT10] for the studies of problems related to the regional type
operators. For regularity of solutions associated to the fractional
$p$-Laplacian and its alike see [IMS16a, IMS16b, BLS18, CK22, CKW22]. It is
noteworthy to emphasize that our setting is sufficiently general and includes
kernel with bounded support. Nonlocal problems driven by integrodifferential
operators associated with kernels of bounded support appear as core models to
several problems in peridynamic. In Section 8, we explain how to deal with
such type of kernels. For recent studies of nonlocal problem aiming at the
application to peridynamics see [FK22, FVV22, DTZ22], several additional
references can be found therein.

Let us comment about convergence from nonlocal to local. In fact under the
assumption in (1.7), the convergence in (1.6) remains true with
$\operatorname{\mathbb{R}}^{d}$ replaced by any extension domain
$\Omega\subset\operatorname{\mathbb{R}}^{d}$; see for instance [Fog23].
Interestingly, the latter convergence also holds in the sense of the gamma
convergence or Mosco convergence; see for instance [Fog20, FKV20, Pon04b].
This type of limit was originally studied for Lipschitz bounded domain in
[BBM01] and several generalizations have recently emerged, see for instance
[BMR20, DB22, DD22, Fog23, PS17] along with the references contained there. To
the best of our knowledge, rigorous proofs of convergence of weak solutions to
nonlocal problems with complement Neumann condition to the local ones appear
in [Fog20, FK22, GH22]. A heuristic proof of the convergence for fractional
Laplacian $(-\Delta)^{s}$ was investigated in [DROV17]. Note however that the
convergence of nonlocal Neumann problems associated with regional type
operators can be found in [AMRT08, AMRT10, DPS15]. In contrast to the Neumann
problems, there is a substantial amount of works treating the convergence from
nonlocal to local of weak solutions of complement boundary Dirichlet problems
associated with the fractional $p$-Laplacian $(-\Delta)^{s}_{p}$; we refer
interested reader for instance to [BPS16, FBS20, BO21a, BO21b, SV22]. The
convergence of weak solutions of Dirichlet problems for fractional
$g$-Laplacian where $g$ is an Orlicz function see [BS19]. For convergence of
solutions to elliptic problems, see [Fog20, Voi17]. Last, a uniform
convergence of viscosity solutions to the constrained fractional $p$-Laplace
operator is established in [IN10]. For more on viscosity solutions associated
with the $p$-Laplacian and the fractional $p$-Laplacian see for instance the
recent results [Lin16, Lin19, DL21] as well as the references therein.

The rest of this article is organized as follows. We introduce some basics
concepts regarding the support of $\nu$ in Section 2. In Section 3 we study
nonlocal Sobolev spaces, whereas Section 5 is dedicated to the study of the
corresponding nonlocal trace spaces. Meanwhile we give an analytic
characterization of $p$-Lévy integrability in Section 4. In Section 6 we give
some compact embeddings. Nonlocal Poincaré and nonlocal Poincaré-Friedrichs
inequalities are established in Section 7, while their robust versions are
provided in Section 10. Section 9 and Section 11 focuses on convergences of
from nonlocal to local objects; this includes convergence of forms, weak
solutions, nonlinear operators. We also characterize the family
$(\nu_{\varepsilon})_{\varepsilon}$ satisfying (1.7). In Appendix A and
Appendix B we provide some basics results such as nonlocal Gauss-Green
formula, pointwise evaluations, elementary inequalities that are useful for
the study and the understanding of nonlinear operators under considerations
namely, $L$ and $\operatorname{\mathcal{N}}$.

_Acknowledgment:_ Several results included in this work were obtained in the
author PhD thesis during his studies at Bielefeld University, in the framework
of the International Research Training Group 2235 “Searching for the regular
in the irregular: Analysis of random and singular systems”. The author thanks
the host institution and the DFG for the financial support.

##  2\. Basic concepts and notations

###  2.1. Notations

Throughout this article unless otherwise stated, we assume that
$\Omega\subset\operatorname{\mathbb{R}}^{d}$ is an open set,
$1<p^{\prime},p<\infty$ with $p+p^{\prime}=pp^{\prime}$ and $\nu$ is a
symmetric kernel satisfying (L). We frequently use the convex inequality
$(a+b)^{p}\leq 2^{p-1}(a^{p}+b^{p})$ for $a>0,b>0$. If there is no specific
mention, all functions and sets are assumed to be at least Borel measurable
and are understood in the almost everywhere sense. Given two quantities $F$
and $G$ the relation $F\asymp G$ means that there are positive constants
$C_{1}$ and $C_{2}$ such that $C_{1}\leq F/G\leq C_{2}$. In general, $C>0$
denotes a generic constant depending on the local and $\varepsilon>0$ is a
small quantity tending to $0$. The Euclidean scalar product of
$x=(x_{1},x_{2},\cdots,x_{d})\in\operatorname{\mathbb{R}}^{d}$ and
$y=(y_{1},y_{2},\cdots,y_{d})\in\operatorname{\mathbb{R}}^{d}$ is $x\cdot
y=x_{1}x_{1}+x_{2}y_{2}+\cdots+x_{d}y_{d}$ and denote the norm of $x$ by
$|x|=\sqrt{x\cdot x}$.

###  2.2. Support and nonlocal boundary

We introduce some basics definitions in connection with the measure theoretic
support of a measurable function.

######  Definition 2.1.

The support of a measurable function
$\omega:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}$ is defined
as

$\displaystyle\begin{split}\operatorname{supp}\omega&=\operatorname{\mathbb{R}}^{d}\setminus\mathcal{O}\quad\text{with}\quad\mathcal{O}=\bigcup\big{\\{}O\,:\,\text{$O$
is open and $\omega=0$ a.e. on $O$}\big{\\}}\\\
&=\big{\\{}x\in\operatorname{\mathbb{R}}^{d}\,:\,\text{$\omega>0$ a.e. on any
open set $O$ s.t. $x\in O$}\big{\\}}.\end{split}$

Note that $\operatorname{supp}\omega$ is a closed set and $\omega=0$ a.e.
$\operatorname{\mathbb{R}}^{d}\setminus\operatorname{supp}\omega$. In
particular, $\omega=0$ a.e. if and only if
$\operatorname{supp}\omega=\emptyset.$ If $\omega$ is a continuous function
then
$\operatorname{supp}\omega=\overline{\big{\\{}h\in\operatorname{\mathbb{R}}^{d}\,:\,\omega(h)\neq
0\big{\\}}}.$ The latter is not true in general. For instance, on the real
line we put $\omega=\mathds{1}_{\mathbb{Q}}$, where $\mathbb{Q}$ is the set of
rational numbers, then
$\overline{\big{\\{}h\in\operatorname{\mathbb{R}}\,:\,\omega(h)\neq
0\big{\\}}}=\operatorname{\mathbb{R}}$ but
$\operatorname{supp}\omega=\emptyset$.

######  Definition 2.2.

We say that $\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$
has full support if $\operatorname{supp}\nu=\operatorname{\mathbb{R}}^{d}$,
equivalently $\nu(h)>0$ for almost every $h\in\operatorname{\mathbb{R}}^{d}$
that is $|\\{\nu=0\\}|=0$.

######  Remark 2.3.

It is very tempting to think that the set of zeros of $\nu$ in
$\operatorname{supp}\nu$ is a null set, i.e.,
$|\operatorname{supp}\nu\cap\\{\nu=0\\}|=0$. In other words is true that
$\nu>0$ a.e. on $\operatorname{supp}\nu$? This is not always true, indeed
consider $O\subset\operatorname{\mathbb{R}}^{d}$ a non-empty open set whose
boundary has positive measure, i.e., $|\partial O|>0$. The function
$g(x)=\operatorname{dist}(x,\operatorname{\mathbb{R}}^{d}\setminus O)$ is
continuous. Note that $\\{g\neq 0\\}=O$ and thus
$\operatorname{supp}g=\overline{O}$. However, we have
$|\operatorname{supp}g\cap\\{g=0\\}|=|\partial O|>0$.

######  Definition 2.4 (Nonlocal hull and nonlocal boundary).

Given a measurable set $S\subset\operatorname{\mathbb{R}}^{d}$:

  1. $(i)$

We define the nonlocal hull with respect to $\nu$ (or simply the
$\nu$-nonlocal hull) of $S$ to be the set $S_{\nu}=S+\operatorname{supp}\nu$.
Note that if $0\in\operatorname{supp}\nu$ then $S\subset S_{\nu}$.

  2. $(ii)$

We define the exterior boundary (or the nonlocal boundary) of $S$ with respect
to $\nu$ denoted $S_{e}$ or $\partial_{\nu}S$, to be the set
$S_{e}=S_{\nu}\setminus S=(S+\operatorname{supp}\nu)\setminus S$ (or
$\partial_{\nu}S=S_{\nu}\setminus S$).

The terminologies are justified by the following facts. From an analysis point
of view, the set $S_{\nu}=S+\operatorname{supp}\nu$ is smallest set needed to
evaluate $Lu$ on $S$ that is we have

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))\nu(x-y)\mathrm{d}y=\int_{S_{\nu}}\psi(u(x)-u(y))\nu(x-y)\mathrm{d}y\quad\text{for
all $x\in S$}.$

In other words, the values $Lu(x)$ for $x\in S$ solely depend of the values of
$u$ in $S_{\nu}$. Therefore, to solve a nonlocal equation of the form $Lu=f$
in $S$, it is sufficient to prescribe exterior boundary condition on
$S_{\nu}\setminus S$. For instance if we consider $S=\Omega$, then the Neumann
condition $\operatorname{\mathcal{N}}u=g$, only makes if $g=0$ on
$\operatorname{\mathbb{R}}^{d}\setminus\Omega_{\nu}$; see Section 8 for
additional details. Indeed, for $x\in\Omega$ and
$y\in\operatorname{\mathbb{R}}^{d}\setminus\Omega_{\nu}$ we have
$x-y\not\in\operatorname{supp}\nu$ so that $\nu(x-y)=0$, and hence

$\displaystyle\operatorname{\mathcal{N}}u(y):=\int_{\Omega}\psi(u(y)-u(x))\nu(x-y)\mathrm{d}x=0\qquad\text{for
all $y\in\operatorname{\mathbb{R}}^{d}\setminus\Omega_{\nu}$.}$

From a probabilistic point of the view, nonlocal hull $S_{\nu}$ can be seen as
the reachable area of a jump process starting from a point in $S$. In some
sense, any jump starting in $S$ cannot reach farther beyond the set $S_{\nu}$.
Therefore, if a jump process is censored (restricted) at the $S$ then the
whole universe the process is $S_{\nu}$. In other words any censored process
on $\Omega$ is never aware of anything happening on
$\operatorname{\mathbb{R}}^{d}\setminus\Omega_{\nu}$.

######  Example 2.5.

Let us mention two classes of examples that are well-known in the literature.

  1. $(i)$

If $\nu$ has full support, i.e.
$\operatorname{supp}\nu=\operatorname{\mathbb{R}}^{d}$ then
$S_{\nu}=S+\operatorname{\mathbb{R}}^{d}=\operatorname{\mathbb{R}}^{d}$ and
$S_{e}=\operatorname{\mathbb{R}}^{d}\setminus S$. That is the exterior
boundary of $S$ with respect to $\nu$ coincides with its whole complement. A
typical example include $\nu(h)=|h|^{-d-sp}$. IntegroDifferential Equation
(IDEs) associated with nonlocal operators driven kernels $\nu$ with
$\operatorname{supp}\nu=\operatorname{\mathbb{R}}^{d}$, are often called
complement values problems in the modern literature.

  2. $(ii)$

If $\operatorname{supp}\nu=\overline{B_{\delta}(0)}$, $\delta>0$ then
$S_{\nu}=S+\overline{B_{\delta}(0)}=\\{x\in\operatorname{\mathbb{R}}^{d}:\operatorname{dist}(x,S)\leq\delta\\}$,
i.e., $S_{\nu}$ is the $\delta$-tubular thickening neighborhood of $S$. On
other hand we have $S_{e}=S_{\nu}\setminus
S=\\{x\in\operatorname{\mathbb{R}}^{d}\setminus
S:\operatorname{dist}(x,S)\leq\delta\\}$.

For instance, if $S=B_{r}(a)$, $r>0$ and $a\in\operatorname{\mathbb{R}}^{d}$
then $S_{\nu}=B_{r+\delta}(a)$ and
$S_{e}=B_{r+\delta}(a)\setminus\overline{B_{r}(a)}$. For a concrete examples
we have $\nu(h)=\mathds{1}_{B_{\delta}(0)}(h)$ or
$\nu(h)=|h|^{-d-sp}\mathds{1}_{B_{\delta}(0)}(h)$. These types of kernels
often appear in the area of peridynamics, wherein the exterior boundary
$S_{e}=S_{\nu}\setminus S$ is also known as the volume constraint. Thus,
IntegroDifferential Equation (IDEs) associated with nonlocal operators driven
kernels $\nu$ with $\operatorname{supp}\nu=\overline{B_{\delta}(0)}$, are
called in the area of peridynamic as volume constrained problems. See for
instance the recent works [FK22, FVV22, DTZ22] and several additional
references therein.

######  Remark 2.6.

Let us highlight some remarks concerning $\operatorname{supp}\nu$ and
$S_{\nu}$.

  1. $(i)$

If $0\in\operatorname{supp}\nu$ then $S\subset S_{\nu}$. This property is
sometimes important to makes of many IDEs.

  2. $(ii)$

If the set $S$ is open then $S_{\nu}$ is also open. This stems from the fact
that each $S+h$, $h\in\operatorname{supp}\nu$ is open and
$S_{\nu}=S+\operatorname{supp}\nu=\bigcup_{h\in\operatorname{supp}\nu}S+h$.

  3. $(iii)$

If the set $S$ is compact or $\operatorname{supp}\nu$ is compact then
$S_{\nu}$ is a closed set.

##  3\. Nonlocal function spaces

In this section we introduce generalized Sobolev-Slobodeckij-like function
spaces with respect to a Lévy measure $\nu$ and an open subset
$\Omega\subset\operatorname{\mathbb{R}}^{d}$, in particular the space
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and its nonlocal trace
space $T^{p}_{\nu}(\Omega^{c})$. The function spaces are tailor-made for
IntegroDifferential Equations (IDEs) with Neumann and Dirichlet complement
conditions. Recall that
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is the
density of a symmetric $p$-Lévy measure, that is,

$\displaystyle\begin{split}\nu(h)=\nu(-h)\quad&\text{ $h\neq 0\,\,$ and
}\qquad\int_{\operatorname{\mathbb{R}}^{d}}(1\land|h|^{p})\nu(h)\mathrm{d}h<\infty.\end{split}$
(L)

In fact $\nu$ is said to be $p$-Lévy integrable, i.e. $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$.

###  3.1. Nonlocal energy forms

Recall $\psi(t)=|t|^{p-2}t$. Given a symmetric kernel
$k:(\operatorname{\mathbb{R}}^{d}\times\operatorname{\mathbb{R}}^{d})\setminus\operatorname{diag}\to[0,\infty)$
we consider the energy form $\operatorname{\mathcal{E}}^{k}(\cdot,\cdot)$ by

$\displaystyle\operatorname{\mathcal{E}}^{k}(u,v)=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))(v(x)-v(y))\,k(x,y)\mathrm{d}y\mathrm{d}x.$

######  Proposition 3.1.

  1. $(i)$

If $\operatorname{\mathcal{E}}^{k}(u,u)$ and
$\operatorname{\mathcal{E}}^{k}(v,v)$ are finite so is
$\operatorname{\mathcal{E}}^{k}(u,v)$ and we have

$|\operatorname{\mathcal{E}}^{k}(u,v)|\leq\operatorname{\mathcal{E}}^{k}(u,u)^{1/p^{\prime}}\operatorname{\mathcal{E}}^{k}(v,v)^{1/p}.$

  2. $(ii)$

Consider the Banach space $W^{p}_{k}(\operatorname{\mathbb{R}}^{d})=\\{u\in
L^{p}(\operatorname{\mathbb{R}}^{d})\,:\,\operatorname{\mathcal{E}}^{k}(u,u)<\infty\\}$
with
$\|u\|^{p}_{W^{p}_{k}(\operatorname{\mathbb{R}}^{d})}=\|u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}+\operatorname{\mathcal{E}}^{k}(u,u)$
. Then the operator
$L:W^{p}_{k}(\operatorname{\mathbb{R}}^{d})\to(W^{p}_{k}(\operatorname{\mathbb{R}}^{d}))^{\prime}$
with $Lu:=\operatorname{\mathcal{E}}^{k}(u,\cdot)$ is nonlinear and bounded
with

$\displaystyle|\langle
Lu,v\rangle|=|\operatorname{\mathcal{E}}^{k}(u,v)|\leq\operatorname{\mathcal{E}}^{k}(u,u)^{1/p^{\prime}}\operatorname{\mathcal{E}}^{k}(v,v)^{1/p},$
$\displaystyle\|Lu\|_{(W^{p}_{k}(\operatorname{\mathbb{R}}^{d}))^{\prime}}\leq\operatorname{\mathcal{E}}^{k}(u,u)^{1/p^{\prime}}\leq\|u\|^{p-1}_{W^{p}_{k}(\operatorname{\mathbb{R}}^{d})}.$

###### Proof.

$(ii)$ follows from $(i)$ whereas, $(i)$ is an immediate consequence of the
Hölder inequality. ∎

######  Remark 3.2.

Formally, the operators $L$ is denoted as

$\displaystyle
Lu(x):=2\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))k(x,y)\mathrm{d}y,$

whenever this expression makes sense. For instance if $k(x,y)=\nu(x-y)$ with
$\nu\in L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$ then under
additional mild conditions $Lu(x)$ can be defined for smooth functions (see
Appendix B.1). Beside this, a curious reader easily verifies that, a similar
analysis can be achieved with the spaces
$W_{k}^{p}(\operatorname{\mathbb{R}}^{d})$ and the nonlocal operator $L$
replaced with the space $W^{1,p}(\operatorname{\mathbb{R}}^{d})$ and the local
$p$-Laplace operator $-\Delta_{p}$.

From now we assume that $\operatorname{\mathcal{E}}^{k}(u,u)<\infty$ and
$\operatorname{\mathcal{E}}^{k}(v,v)<\infty$. We will consider special cases
of the $k(x,y)$ that are of paramount interest. First we consider the form
$\operatorname{\mathcal{E}}_{\Omega}(\cdot,\cdot)$ given by

$\displaystyle\operatorname{\mathcal{E}}_{\Omega}(u,v)$
$\displaystyle=\iint\limits_{\Omega\Omega}\psi(u(x)-u(y))(v(x)-v(y))\,\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))(v(x)-v(y))\,\min(\mathds{1}_{\Omega}(x),\mathds{1}_{\Omega}(y))\nu(x-y)\mathrm{d}y\mathrm{d}x.$

That is, $k(x,y)=\min(\mathds{1}_{\Omega}(x),\mathds{1}_{\Omega}(y))\nu(x-y)$.
Next, the Gauss-Green formula (see Appendix B.2) motivates us to consider the
special energy form $\operatorname{\mathcal{E}}(\cdot,\cdot)$, define by

$\displaystyle\operatorname{\mathcal{E}}(u,v)$
$\displaystyle=\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\psi(u(x)-u(y))(v(x)-v(y))\,\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))(v(x)-v(y))\,\max(\mathds{1}_{\Omega}(x),\mathds{1}_{\Omega}(y))\nu(x-y)\mathrm{d}y\mathrm{d}x.$

That is, in this case we have
$k(x,y)=\max(\mathds{1}_{\Omega}(x),\mathds{1}_{\Omega}(y))\nu(x-y)$. Taking
$k(x,y)=\frac{1}{2}(\mathds{1}_{\Omega}(x),\mathds{1}_{\Omega}(y))\nu(x-y)$ we
obtain another related form $\operatorname{\mathcal{E}}_{+}(\cdot,\cdot)$,
that is

$\displaystyle\operatorname{\mathcal{E}}_{+}(u,v)$
$\displaystyle=\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))(v(x)-v(y))\,\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle=\frac{1}{2}\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\psi(u(x)-u(y))(v(x)-v(y))\,(\mathds{1}_{\Omega}(x)+\mathds{1}_{\Omega}(y))\nu(x-y)\mathrm{d}y\mathrm{d}x.$

As shown in the result, the forms $\operatorname{\mathcal{E}}(\cdot,\cdot)$
and $\operatorname{\mathcal{E}}_{+}(\cdot,\cdot)$ are equivalents.

######  Proposition 3.3.

We have $\operatorname{\mathcal{E}}(u,u)\leq
2\operatorname{\mathcal{E}}_{+}(u,u)\leq 2\operatorname{\mathcal{E}}(u,u)$ and
$\operatorname{\mathcal{E}}_{\Omega}(u,u)\leq\operatorname{\mathcal{E}}(u,u)$.

###### Proof.

The proof is immediate since $\min(a,b)\leq\max(a,b)\leq a+b\leq 2\max(a,b)$.
∎

######  Remark 3.4.

For brevity let us denote $\chi(x,y)=|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))$.
Recall that $\Omega_{e}=\Omega_{\nu}\setminus\Omega$ and
$\Omega_{\nu}=\Omega+\operatorname{supp}\nu$. Assume
$0\in\operatorname{supp}\nu$ so that $\Omega\subset\Omega_{\nu}$, then

$\displaystyle\operatorname{\mathcal{E}}(u,v)$
$\displaystyle=\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\chi(x,y)\nu(x-y)\mathrm{d}y\,\mathrm{d}x=\Big{(}\iint\limits_{\Omega\Omega}+2\iint\limits_{\Omega^{c}\Omega}\Big{)}\chi(x,y)\nu(x-y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=\Big{(}\iint\limits_{\Omega\Omega}+2\iint\limits_{\Omega_{e}\Omega}\Big{)}\chi(x,y)\nu(x-y)\mathrm{d}y\,\mathrm{d}x=\iint\limits_{(\Omega_{\nu}\times\Omega_{\nu})\setminus(\Omega_{e}\times\Omega_{e})}\chi(x,y)\nu(x-y)\mathrm{d}y\,\mathrm{d}x.$

Analogously we have

$\displaystyle\operatorname{\mathcal{E}}_{+}(u,v)$
$\displaystyle=\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}\chi(x,y)\nu(x-y)\mathrm{d}y\,\mathrm{d}x=\iint\limits_{\Omega\Omega_{\nu}}\chi(x,y)\nu(x-y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}\chi(x,x+h)\nu(h)\mathrm{d}h\,\mathrm{d}x=\iint\limits_{\Omega\,\operatorname{supp}\nu}\chi(x,x+h)\nu(h)\mathrm{d}h\,\mathrm{d}x.$

###  3.2. Nonlocal Sobolev-Slobodeckij-like spaces

We define the space $W^{p}_{\nu}(\Omega)$ by

$\displaystyle W^{p}_{\nu}(\Omega)=\Big{\\{}u\in
L^{p}(\Omega)\,:\,|u|^{p}_{W^{p}_{\nu}(\Omega)}:=\operatorname{\mathcal{E}}_{\Omega}(u,u)<\infty\Big{\\}}\,,$
(3.1)

equipped with the norm

$\displaystyle\|u\|_{W^{p}_{\nu}(\Omega)}=\big{(}\|u\|^{p}_{L^{p}(\Omega)}+|u|^{p}_{W^{p}_{\nu}(\Omega)}\big{)}^{1/p}=\big{(}\|u\|^{p}_{L^{p}(\Omega)}+\operatorname{\mathcal{E}}_{\Omega}(u,u)\big{)}^{1/p}.$

For $\nu(h)=|h|^{-d-sp},s\in(0,1)$, then $W^{p}_{\nu}(\Omega)=W^{s,p}(\Omega)$
is the classical fractional Sobolev-Slobodeckij space. However if $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d})$ then
$W^{p}_{\nu}(\Omega)=L^{p}(\Omega)$. Indeed,

$\displaystyle\iint\limits_{\Omega\Omega}$
$\displaystyle\big{|}u(x)-u(y)\big{|}^{p}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y\leq
2^{p}\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}|u(x)|^{p}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y=2^{p}\|\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\|u\|^{p}_{L^{p}(\Omega)}.$

For various choices of $\nu$ it appears (see Section 7) that,
$\operatorname{\mathcal{E}}(u,u)<\infty$ implies $u\in L^{p}(\Omega)$.
Motivated by this remark and following [SV13, FKV15, Fog20] we introduce the
space $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, which is crucial,

$\displaystyle\begin{split}W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})&=\Big{\\{}u:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}\text{
meas.}\,:\,u|_{\Omega}\in L^{p}(\Omega)\,\\!\text{ and
}\,\\!|u|^{p}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}:=\operatorname{\mathcal{E}}_{+}(u,u)<\infty\Big{\\}}\,\\\
&=\Big{\\{}u:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}\text{
meas.}\,:\,u|_{\Omega}\in L^{p}(\Omega)\,\\!\text{ and
}\,\\!\operatorname{\mathcal{E}}(u,u)<\infty\Big{\\}}.\end{split}$ (3.2)

The equality in (3.2) follows from Proposition 3.3. The space
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a seminormed spaces
equipped with the seminorm defined by

$\displaystyle\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=\big{(}\|u\|^{p}_{L^{p}(\Omega)}+|u|^{p}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\big{)}^{1/p}\asymp\big{(}\|u\|^{p}_{L^{p}(\Omega)}+\operatorname{\mathcal{E}}(u,u)\big{)}^{1/p}.$

Last, we define $W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ as
the space of functions that vanish on the complement of $\Omega$:

$\displaystyle\begin{split}W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})&=\\{u\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
:\leavevmode\nobreak\ u=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{a.e.
on }\operatorname{\mathbb{R}}^{d}\setminus\Omega\\}\,\\\ &=\\{u\in
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
:\leavevmode\nobreak\ u=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{a.e.
on }\operatorname{\mathbb{R}}^{d}\setminus\Omega\\}\,.\end{split}$ (3.3)

Naturally, $W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ is
endowed with the
$\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ which is
equivalent therein with the norm
$\|\cdot\|_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}$, viz. for $u\in
W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ we have

$\displaystyle\|u\|_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}=\big{(}\|u\|^{p}_{L^{p}(\Omega)}+\operatorname{\mathcal{E}}(u,u)\big{)}^{1/p}\asymp\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}.$

It is not difficult to check that
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a closed
subspace of $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ and
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. Another closely related
closed subspace of $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is
defined as

$\displaystyle
W_{\nu,0}^{p}(\Omega|\operatorname{\mathbb{R}}^{d})=\overline{C_{c}^{\infty}(\Omega)}^{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=\\{\text{the
closure of $C_{c}^{\infty}(\Omega)$ in
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$}\\}.$

It is worth emphasizing that if $\Omega$ has a continuous boundary then
$W_{\nu,0}^{p}(\Omega|\operatorname{\mathbb{R}}^{d})=W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$
(see [Fog20, Theorem 3.76] and also Theorem 3.11). However this equality does
not hold true in general. For $\nu(h)=|h|^{-d-sp},s\in(0,1)$ we denote
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})=W^{s,p}(\Omega|\operatorname{\mathbb{R}}^{d})$,
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})=W^{s,p}_{\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$
and
$W_{\nu,0}^{p}(\Omega|\operatorname{\mathbb{R}}^{d})=W^{s,p}_{0}(\Omega|\operatorname{\mathbb{R}}^{d})$.

######  Remark 3.5.

As a direct consequence of the Proposition 3.1 we have the following.

  1. $(i)$

If $u,v\in W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ then
$|\operatorname{\mathcal{E}}(u,v)|\leq\operatorname{\mathcal{E}}(u,u)^{1/p^{\prime}}\operatorname{\mathcal{E}}(v,v)^{1/p}$
and
$\operatorname{\mathcal{E}}(u,\cdot)\in(W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}))^{\prime}$.

  2. $(ii)$

If $u,v\in W^{p}_{\nu}(\Omega)$ then
$|\operatorname{\mathcal{E}}_{\Omega}(u,v)|\leq\operatorname{\mathcal{E}}_{\Omega}(u,u)^{1/p^{\prime}}\operatorname{\mathcal{E}}_{\Omega}(v,v)^{1/p}$
and
$\operatorname{\mathcal{E}}_{\Omega}(u,\cdot)\in(W^{p}_{\nu}(\Omega))^{\prime}$.

A simple proof of the following Theorem can be found in [Fog20, Theorem 3.46].

######  Theorem 3.6.

The spaces $W^{p}_{\nu}(\Omega)$ and
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ are separable
reflexive Banach spaces.

######  Remark 3.7.

It is worthwhile noticing that
$\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ is always a
norm on $W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$, but not in
general a norm on $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ if $\nu$
is not fully supported. A simple counterexample is given by
$\nu(h)=\mathds{1}_{B_{1}(0)}(h)$ and $\Omega=B_{1}(0)$. For the function
$u(x)=\mathds{1}_{B^{c}_{2}(0)}(x)$ we have
$\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=0$ whereas $u\neq
0$. More generally, assume that $\Omega$ is bounded and $\nu$ has a compact
support. Let
$S=\operatorname{\mathbb{R}}^{d}\setminus\big{(}\Omega\cup\operatorname{supp}\nu+\Omega\big{)}$
and consider the function $u(x)=\mathds{1}_{S}(x)$. A routine verification
shows that $\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=0$ but
$u\neq 0$. This means that
$(W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}),\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})})$
cannot be a normed space.

However
$(W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}),\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})})$
is a Banach space when if $\nu$ is of full support; cf. [Fog20, FK22] for the
proof. Let us say some few words for the general case where $\nu$ is not
necessarily fully supported. In view of the Remark 3.4 we introduce another
version of the space $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$.
Namely, recall $\Omega_{\nu}=\operatorname{supp}\nu+\Omega$, we consider the
space

$\displaystyle W^{p}_{\nu}(\Omega|\Omega_{\nu})=$
$\displaystyle=\big{\\{}u:\Omega\cup\Omega_{\nu}\to\operatorname{\mathbb{R}}\,\text{meas.}:u|_{\Omega}\in
L^{p}(\Omega),\,\iint\limits_{\Omega\Omega_{\nu}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\mathrm{d}x<\infty\big{\\}}$

equipped with the seminorm

$\displaystyle\|u\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}=\Big{(}\|u\|^{p}_{L^{p}(\Omega)}+\iint\limits_{\Omega\Omega_{\nu}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\mathrm{d}x\Big{)}^{1/p}.$

It worth noticing that if $0\in\operatorname{supp}\nu$ then
$\Omega\subset\Omega_{\nu}$ so that $\Omega\cup\Omega_{\nu}=\Omega_{\nu}$.

######  Remark 3.8.

We have $W^{p}_{\nu}(\Omega|\Omega_{\nu})\equiv
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in the sense that both
spaces are isometric isomorphic. Clearly if $u\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ then
$u_{\nu}=u|_{\Omega_{\nu}}\in W^{p}_{\nu}(\Omega|\Omega_{\nu})$ and we have
$\|u_{\nu}\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}=\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$.
Conversely, if $u\in W^{p}_{\nu}(\Omega|\Omega_{\nu})$ then $\widetilde{u}\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ where $\widetilde{u}$ is
the zero extension of $u$ off $\Omega_{\nu}$ and
$\|u\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}=\|\widetilde{u}\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$.
Note that $u\in W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is the
function null function if and only if $u=0$ a.e. on $\Omega_{\nu}$. The
notation $W^{p}(\Omega|\Omega_{\nu})$ instead of
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ should be also
appropriate. However we deliberately keep the latter notation for simplicity.

In view of the Remark 2.3, it is legitimate to assume that
$|\operatorname{supp}\nu\cap\\{\nu=0\\}|=0$. The latter seems not enough to
show that $\|\cdot\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}$ is norm. More
precisely:

Open Question: Assume that $\nu(h)>0$ for all $h\in\operatorname{supp}\nu$.
Prove or disprove that is $\|\cdot\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}$
defines a norm on $W^{p}_{\nu}(\Omega|\Omega_{\nu})$.

Next we provide a condition under which we are able to show that
$W^{p}_{\nu}(\Omega|\Omega_{\nu})$ is a Banach space.

######  Theorem 3.9.

The space $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\equiv
W^{p}_{\nu}(\Omega|\Omega_{\nu})$ is a separable and reflexive Banach space,
whenever $\Omega$ and $\nu$ satisfy the condition

$\displaystyle\omega(x):=\int_{\Omega}1\land\nu(x-y)\mathrm{d}y>0\quad\text{for
almost all $x\in\Omega_{\nu}\setminus\Omega$}.$ ($S_{1}$)

In particular, if $\nu$ is of full support.

###### Proof.

Observe that if $x\not\in\Omega_{\nu}$, for all $y\in\Omega$ we have
$x-y\not\in\operatorname{supp}\nu$. It follows that $\omega(x)=0$ on
$\operatorname{\mathbb{R}}^{d}\setminus\Omega_{\nu}$. By Theorem 5.8 we have
$W^{p}_{\nu}(\Omega|\Omega_{\nu})\equiv
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\operatorname{\mathbb{R}}^{d},\omega)\equiv L^{p}(\Omega_{\nu},\omega).$
Whence, there is $C>0$ such that

$\displaystyle\|u\|_{L^{p}(\Omega_{\nu},\omega)}\leq
C\|u\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}\quad\text{for all $u\in
W^{p}_{\nu}(\Omega|\Omega_{\nu})$}.$ (3.4)

Assume that $\eqref{eq:positive-Oe}$ holds, if
$\|u\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}=0$ then $u=0$ a.e. on $\Omega$ and
by (3.4), $u\omega=0$ a.e. on $\Omega_{\nu}\setminus\Omega$. There holds that
$u=0$ a.e. $\Omega_{\nu}$, since $\omega>0$ a.e. on
$\Omega_{\nu}\setminus\Omega$. Whence the seminorm
$\|\cdot\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}$ is a norm on
$W^{p}_{\nu}(\Omega|\Omega_{\nu})$. Now, consider $(u_{n})_{n}$ be a Cauchy
sequence in $W^{p}_{\nu}(\Omega|\Omega_{\nu})$, then taking into account the
estimate (3.4), the sequences $(u_{n})_{n}$ and $(w_{n})_{n}$ with
$w_{n}=u_{n}\omega$ are Cauchy sequence in $L^{p}(\Omega)$ and
$L^{p}(\Omega_{\nu})$ respectively. Thus, passing to subsequence if necessary,
we can assume there exist $u\in L^{p}(\Omega)$ and $w\in L^{p}(\Omega_{\nu})$
such that

  1. -

$(u_{n})$ converges in $L^{p}(\Omega)$ and pointwise almost everywhere on
$\Omega$ to some $u\in L^{p}(\Omega)$,

  2. -

$(w_{n})$ converges in $L^{p}(\Omega_{\nu})$ and pointwise almost everywhere
in $\Omega_{\nu}$ to some $w\in L^{p}(\Omega_{\nu})$.

We extend $u$ for $x\in\Omega_{\nu}\setminus\Omega$ by $u(x)=w(x)/\omega(x)$
which is well defined since $\omega>0$ a.e. on $\Omega_{\nu}\setminus\Omega$.
It follows that $u:\Omega\cup\Omega_{\nu}\to\operatorname{\mathbb{R}}$ is
measurable since $u_{n}\to u$ a.e. on $\Omega$ and for almost all
$x\in\Omega_{\nu}\setminus\Omega$ we have
$u_{n}(x)=w_{n}(x)/\omega(x)\xrightarrow{n\to\infty}w(x)/\omega(x)=u(x).$ It
turns out that $(u_{n})_{n}$ is a Cauchy sequence in
$W^{p}_{\nu}(\Omega|\Omega_{\nu})$ converging to $u$ on $L^{p}(\Omega)$ and
a.e. on $\Omega_{\nu}$. This together with Fatou’s lemma implies that

$\displaystyle|u_{n}-u|^{p}_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}\leq\liminf_{m\to\infty}\iint\limits_{\Omega\Omega_{\nu}}|[u_{n}-u_{m}](x)-[u_{n}-u_{m}](y)|^{p}\nu(x-y)\mathrm{d}y\mathrm{d}x\xrightarrow{n\to\infty}0.$

We can therefore easily conclude that
$\|u_{n}-u\|_{W^{p}_{\nu}(\Omega|\Omega_{\nu})}\xrightarrow{n\to\infty}0$ and
$u\in W^{p}_{\nu}(\Omega|\Omega_{\nu})$ which show that completeness of
$W^{p}_{\nu}(\Omega|\Omega_{\nu})$. The completeness is proved. Now, consider
the isometry $\mathcal{I}:W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\to
L^{p}(\Omega)\times L^{p}(\Omega\times\operatorname{\mathbb{R}}^{d})$ with
$\mathcal{I}(u)=(u(x),(u(x)-u(y))\nu^{1/p}(x-y))$. From its Banach structure,
the space
$\big{(}W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}),\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\big{)}$,
identified with
$\mathcal{I}\big{(}W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\big{)}$,
is separable and reflexive as a closed subspace of the separable and reflexive
space $L^{p}(\Omega)\times L^{p}(\Omega\times\operatorname{\mathbb{R}}^{d})$.
∎

######  Example 3.10.

For a typical situation where $\nu>0$ a.e. on $\operatorname{supp}\nu$ with
$\operatorname{supp}\nu=\overline{B_{1}(0)}$ and $\Omega=B_{1}(0)$, for
instance $\nu(h)=\mathds{1}_{B_{1}(0)}$ or
$\nu(h)=\mathds{1}_{B_{1}(0)}|h|^{-d-sp}$, $s\in(0,1)$ we have
$\Omega_{\nu}=B_{2}(0)$ and
$W^{p}_{\nu}(\Omega|\Omega_{\nu})=W^{p}_{\nu}(B_{1}(0)|B_{2}(0))$ is a Banach
space.

###  3.3. Density of smooth functions

For many results it is crucial that smooth functions with compact support are
dense in the function space under consideration. Let us summarize some
important results in this direction.

######  Theorem 3.11.

Let $\nu$ satisfies (L) with full support and let
$\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open.

  1. $(i)$

$C^{\infty}(\Omega)\cap W^{p}_{\nu}(\Omega)$ is dense in
$W^{p}_{\nu}(\Omega)$.

  2. $(ii)$

If $\Omega$ has a continuous boundary $\partial\Omega$, then
$C_{c}^{\infty}(\overline{\Omega})$ is dense in $W^{p}_{\nu}(\Omega)$.

  3. $(iii)$

If $\Omega$ has a continuous boundary $\partial\Omega$, then
$C_{c}^{\infty}(\Omega)$ is dense in
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$.

  4. $(iv)$

If $\Omega$ has a Lipschitz boundary $\partial\Omega$, then
$C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$ is dense in
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$.

  5. $(v)$

$C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$ is dense in
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$.

The proofs of the first and second statement can be found in [Fog20] and
[DK21]. The first statement reminisces a sort of Meyer-Serrin density type
result (see [Fog20, Theorem 3.67]). Note that
$C_{c}^{\infty}(\overline{\Omega})$ is defined as
$\\{v|_{\overline{\Omega}}:v\in
C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})\\}$. The proof of the third
statement is given in [FSV15] for a special choice of $\nu$ and in [Fog20],
[BGPR20] for the general case. Note however that the third may fail if
$\partial\Omega$ is not continuous see for instance [FSV15, Remark 7]. The
proof of the fourth and the fifth assertions are given in [FKV20, Fog20].

###  3.4. Connection with classical Sobolev spaces

There are connections between the nonlocal and the local Sobolev spaces.
Recall that $W^{1,p}(\Omega)$ denotes the classical Sobolev space endowed with
the norm

$\|u\|^{p}_{W^{1,p}(\Omega)}=\|u\|^{p}_{L^{p}(\Omega)}+\|\nabla
u\|^{p}_{L^{p}(\Omega)}\,.$

While, $W^{1,p}_{0}(\Omega)$ is the closure of $C_{c}^{\infty}(\Omega)$ with
respect to the $W^{1,p}(\Omega)$. We emphasize that $W^{1,p}_{0}(\Omega)$ also
coincides with the closure of $C_{c}^{\infty}(\Omega)$ in
$W^{1,p}(\operatorname{\mathbb{R}}^{d})$. Let $W^{p}_{\nu,0}(\Omega)$ be the
closure of $C_{c}^{\infty}(\Omega)$ in $W^{p}_{\nu}(\Omega)$. Note that, the
zero extension to $\operatorname{\mathbb{R}}^{d}$ of any function in
$W^{1,p}_{0}(\Omega)$ belongs to $W^{1,p}(\operatorname{\mathbb{R}}^{d})$.
Recall that $\Omega$ is an $W^{1,p}$-extension domain if there is an operator
$E:W^{1,p}(\Omega)\to W^{1,p}(\operatorname{\mathbb{R}}^{d})$ and $C>0$ such
that $Eu|_{\Omega}=u$ and $\|Eu\|_{W^{1,p}(\operatorname{\mathbb{R}}^{d})}\leq
C\|u\|_{W^{1,p}(\Omega)}$ for all $u\in W^{1,p}(\Omega)$.

######  Proposition 3.12.

Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open. The following
embeddings hold true.

  1. $(i)$

$W^{1,p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
W^{p}_{\nu}(\operatorname{\mathbb{R}}^{d}).$

  2. $(ii)$

If $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is an $W^{1,p}$-extension
domain then $W^{1,p}(\Omega)\hookrightarrow W^{p}_{\nu}(\Omega)$.

  3. $(iii)$

$W^{p}_{\nu}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
W^{p}_{\nu}(\Omega)\hookrightarrow L^{p}(\Omega)\,.$

  4. $(iv)$

$W^{1,p}_{0}(\Omega)\hookrightarrow
W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
W^{p}_{\nu}(\Omega).$

  5. $(v)$

If $\partial\Omega$ continuous then
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
W^{p}_{\nu,0}(\Omega)\hookrightarrow L^{p}(\Omega)$.

###### Proof.

Note that $(ii)$ is implied by $(i)$. Whereas $(v)$ follows from the fact that
$C_{c}^{\infty}(\Omega)$ is dense in
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$; see Theorem 3.11.
We only prove $(i)$, i.e.,
$W^{1,p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
W^{p}_{\nu}(\operatorname{\mathbb{R}}^{d})$ since the remaining ones are
trivial. A routine argument yields that, for $u\in
W^{1,p}(\operatorname{\mathbb{R}}^{d})$ and
$h\in\operatorname{\mathbb{R}}^{d}$

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}|u(x+h)-u(x)|^{p}\mathrm{d}x\leq|h|^{p}\|\nabla
u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})},$

whereas $\|u(\cdot+h)-u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}\leq
2^{p}\|u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}$. Therefore we get

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}|u(x+h)-u(x)|^{p}\mathrm{d}x\leq
2^{p}(1\land|h|^{p})\|u\|^{p}_{W^{1,p}(\operatorname{\mathbb{R}}^{d})}\,.$

Integrating both side over $\operatorname{\mathbb{R}}^{d}$ with respect to the
measure $\nu(h)\mathrm{d}h$ yields

$\displaystyle\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x\leq
2^{p}\|\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p}\mathrm{d}h)}\|u\|^{p}_{W^{1,p}(\operatorname{\mathbb{R}}^{d})}.$
(3.5)

The desired embedding readily follows. ∎

######  Remark 3.13.

The embedding $W^{1,p}(\Omega)\hookrightarrow W^{p}_{\nu}(\Omega)$ may fail if
$\Omega$ is not an extension domain (see [Fog23, Counterexample 3.10] or
[DNPV12, Example 9.1]). More importantly, [Fog23, Fog20] $W^{1,p}(\Omega)$ can
be viewed as limiting space of the nonlocal space of type
$W^{p}_{\nu}(\Omega)$ and $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$;
see for instance [Fog23].

##  4\. Characterization of the $p$-Lévy integrability

We will now see that the $p$-Lévy integrability condition (L) is optimal and
can be self-generated from the associated energy form. In fact the $p$-Lévy
integrability condition (L) renders the space
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ more consistent, in a sense that
it warrants the space $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ to contain
smooth functions.

######  Theorem 4.1.

Assume $\nu\geq 0$ is symmetric. The following assertions are equivalent.

  1. $(i)$

The $p$-Lévy condition (L) holds, i.e. $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$.

  2. $(ii)$

The embedding $W^{1,p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ is continuous.

  3. $(iii)$

$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty$ for
all $u\in W^{1,p}(\operatorname{\mathbb{R}}^{d})$.

  4. $(iv)$

$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty$ for
all $u\in C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$.

Moreover, this also remains true when $p=1$ with
$BV(\operatorname{\mathbb{R}}^{d})$ in place of
$W^{1,1}(\operatorname{\mathbb{R}}^{d})$.

###### Proof.

$(i)\implies(ii)$ The estimate (3.5) readily yields the continuity of the
embedding $W^{1,p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$. The implications
$(ii)\implies(iii)$ and $(iii)\implies(iv)$ are straightforward. It remains to
prove that $(iv)\implies(i)$. To this end, assume that
$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty$ for
all $u\in C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$. Let $\zeta\in
C_{c}^{\infty}(\operatorname{\mathbb{R}})$ with $\zeta\not\equiv 0$ and
$\operatorname{supp}\zeta\subset(0,1)$ and $\vartheta\in
C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d-1})$ such that $\vartheta=1$ on
$Q^{\prime}$ with $Q^{\prime}=[0,1]^{d-1}$. Consider
$u_{i}(x)=\zeta\otimes\vartheta(x_{i},x^{\prime}_{i})=\zeta(x_{i})\vartheta(x^{\prime}_{i})$
where $x^{\prime}_{i}=(x_{1},\cdots x_{i-1},x_{i+1}\cdots,x_{d})$. Clearly we
have $u_{i}\in C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$ and $\nabla
u_{i}(x)=\zeta^{\prime}(x_{i})e_{i}$ for $x\in Q=[0,1]^{d}$. By the continuity
of the shift, for each $\varepsilon>0,$ there is $\delta>0$ such that

$\|\nabla u(\cdot+h)-\nabla u\|^{p}_{L^{p}(Q)}<\varepsilon$ for all
$|h|\leq\delta.$

Using this and the inequality $b^{p}\geq 2^{1-p}(a+b)^{p}-a^{p}$, $a,b\geq 0$
we find that

$\displaystyle\begin{split}\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u_{i},u_{i})&\geq\int_{Q}\int_{B_{\delta}(0)}\Big{|}\int_{0}^{1}\nabla
u_{i}(x+th)\cdot h\mathrm{d}t\Big{|}^{p}\nu(h)\mathrm{d}h\,\mathrm{d}x\\\
&\geq 2^{1-p}\int_{Q}\int_{B_{\delta}(0)}|\nabla u_{i}(x)\cdot
h|^{p}\nu(h)\mathrm{d}h\,\mathrm{d}x-\varepsilon\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h\\\
&=2^{1-p}\int_{Q}|\zeta^{\prime}(x_{i})|^{p}\mathrm{d}x\int_{B_{\delta}(0)}|h_{i}|^{p}\nu(h)\mathrm{d}h-\varepsilon\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h.\end{split}$

Note that
$\int_{Q}|\zeta^{\prime}(x_{i})|^{p}\mathrm{d}x=\|\zeta^{\prime}\|^{p}_{L^{p}(0,1)}$.
We have $|h_{1}|^{p}+\cdots+|h_{d}|^{p}\geq c_{d,p}|h|^{p}$ for some
$c_{d,p}>0$, by equivalence of Euclidean norms. Taking
$\varepsilon=\frac{C}{2d}$ with
$C=2^{1-p}c_{d,p}\|\zeta^{\prime}\|^{p}_{L^{p}(0,1)}$ we get

$\displaystyle\sum_{i=1}^{d}\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u_{i},u_{i})$
$\displaystyle\geq(C-d\varepsilon)\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h=\frac{C}{2}\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h\,,$
(4.1)

Now, consider $u\in C_{c}^{\infty}(B_{\delta/2}(0)),$ $u\not\equiv 0$. Since
$B_{\delta/2}(0)\subset B_{\delta}(x)$ for all $x\in B_{\delta/2}(0)$ we get

$\displaystyle\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)$
$\displaystyle\geq
2\int_{B_{\delta/2}(0)}|u(x)|^{p}\int_{B^{c}_{\delta/2}(0)}\nu(x-y)\mathrm{d}y\,\mathrm{d}x\geq
2\|u\|^{p}_{L^{p}(B_{\delta/2}(0))}\int_{B^{c}_{\delta}(0)}\nu(h)\mathrm{d}h.$
(4.2)

Combing (4.1) and (4.2) yields $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$ that is $\nu$ satisfies
(L). The case $p=1$ follows analogously. ∎

######  Lemma 4.2.

Let $\phi\in L^{1}(\operatorname{\mathbb{R}}^{d})$ such that $\phi\geq 0$ and
$\int_{\operatorname{\mathbb{R}}^{d}}\phi(z)\mathrm{d}z=1$. For all $u\in
L^{1}_{\operatorname{loc}}(\operatorname{\mathbb{R}}^{d})$,

$\displaystyle\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u*\phi,u*\phi)\leq\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u).$

###### Proof.

From Jensen’s inequality and simple change of variables we find that

$\displaystyle\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{d}}\phi(z)\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x-z)-u(y-z)|^{p}\nu(x-y)\mathrm{d}y\mathrm{d}x\,\mathrm{d}z$
$\displaystyle\geq\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\Big{|}\int_{\operatorname{\mathbb{R}}^{d}}\phi(z)(u(x-z)-u(y-z))\mathrm{d}z\Big{|}^{p}\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u*\phi(x)-u*\phi(y)|^{p}\nu(x-y)\mathrm{d}y\mathrm{d}x=\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u*\phi,u*\phi).$

∎

######  Theorem 4.3.

Assume $\nu$ is radial, then $\nu$ satisfies (L) if and only if
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\neq\\{0\\}$.

###### Proof.

In view of Theorem 4.1 we know
$C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})\subset
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ if $\nu$ satisfies (L).
Conversely, assume there is $v\in W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$
such that $v\not\equiv 0$ then also $|v|\in
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ with
$\operatorname{\mathcal{E}}(|v|,|v|)\leq\operatorname{\mathcal{E}}(v,v)$,
since $||v(x)|-|v(y)||\leq|v(x)-v(y)|$. Let $\phi(x)=\kappa e^{-|x|^{2}}$
satisfies $\|\phi\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}=1$ for some
$\kappa>0$. By Young’s inequality and Lemma 4.2 we get $u=|v|*\phi\in
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$. Moreover,

$\displaystyle\operatorname{\mathcal{E}}(u,u)\leq\operatorname{\mathcal{E}}(|v|,|v|)\leq\operatorname{\mathcal{E}}(v,v)<\infty.$

Clearly $u\in C^{\infty}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ and $u(x)>0$ for each
$x\in\operatorname{\mathbb{R}}^{d}$, i.e.,
$\operatorname{supp}u=\operatorname{\mathbb{R}}^{d}$. On the other hand,
Young’s inequality implies $u\in W^{1,p}(\operatorname{\mathbb{R}}^{d})$. In
particular we have $\|\nabla u\|_{L^{p}(\operatorname{\mathbb{R}}^{d})}\neq
0$.

For each $\varepsilon>0$, there is $\delta=\delta(\varepsilon)>0$ such that
$\|\nabla u(\cdot+h)-\nabla
u\|_{L^{p}(\operatorname{\mathbb{R}}^{d})}<\varepsilon$ for all
$|h|\leq\delta$. Consider
$C_{\delta}=\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h$, then Minkowski’s
inequality implies

$\displaystyle\Big{(}\iint\limits_{\operatorname{\mathbb{R}}^{d}B_{\delta}(0)}|\nabla
u(x)\cdot h|^{p}\nu(h)\mathrm{d}h\mathrm{d}x\Big{)}^{1/p}$
$\displaystyle\leq\Big{(}\iint\limits_{\operatorname{\mathbb{R}}^{d}B_{\delta}(0)}\Big{|}\int_{0}^{1}\nabla
u(x+th)\cdot
h\mathrm{d}t\Big{|}^{p}\nu(h)\mathrm{d}h\mathrm{d}x\Big{)}^{1/p}+\varepsilon
C_{\delta}^{1/p}.$

The fundamental theorem of calculus and a passage to polar coordinates yield

$\displaystyle\begin{split}\operatorname{\mathcal{E}}(u,u)^{1/p}&\geq\Big{(}\iint\limits_{\operatorname{\mathbb{R}}^{d}\,B_{\delta}(0)}\Big{|}\int_{0}^{1}\nabla
u(x+th)\cdot
h\mathrm{d}t\Big{|}^{p}\nu(h)\mathrm{d}h\,\mathrm{d}x\Big{)}^{1/p}\\\
&\geq\Big{(}\int_{\operatorname{\mathbb{R}}^{d}}\int_{\mathbb{S}^{d-1}}|\nabla
u(x)\cdot
w|^{p}\mathrm{d}\sigma_{d-1}(w)\int_{0}^{\delta}r^{p+d-1}\nu(r)\mathrm{d}r\Big{)}^{1/p}-\varepsilon\Big{(}\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h\Big{)}^{1/p}\\\
&=\Big{(}K^{1/p}_{d,p}\|\nabla
u\|_{L^{p}(\operatorname{\mathbb{R}}^{d})}-\varepsilon\Big{)}\Big{(}\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h\Big{)}^{1/p}.\end{split}$

Necessarily, we get $\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h<\infty$
since, taking $\varepsilon=\frac{K^{1/p}_{d,p}}{2}\|\nabla
u\|_{L^{p}(\operatorname{\mathbb{R}}^{d})}$ we get

$\displaystyle\begin{split}\operatorname{\mathcal{E}}(u,u)\geq\frac{K_{d,p}}{2}\|\nabla
u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h.\end{split}$

Next, it is not difficult to show that $u(x)\to 0$ as $|x|\to\infty$. Thus for
$r>0$ there is $R>\delta>0$ such that $0<u(x)\leq r$ whenever
$|x|>\frac{R}{2}$. Now consider the 1-Lipschitz function

$\displaystyle\zeta_{r}(s)=\max(-r,\min(r,s))=\begin{cases}s&|s|\leq r\\\
r\operatorname{sgn}(s)&|s|\geq r.\end{cases}$

It is readily seen that $|\zeta_{r}(u(x))-\zeta_{r}(u(y))|\leq|u(x)-u(y)|$ and
$0\leq\zeta_{r}(u(x))\leq u(x)$. In particular,
$\operatorname{\mathcal{E}}(u,u)\geq\operatorname{\mathcal{E}}(\zeta_{r}(u),\zeta_{r}(u))$.
If we set $w_{r}(x)=u(x)-\zeta_{r}(u(x))$ then $w_{r}(x)=0$ whenever
$|x|>R/2$. Since $B_{R/2}(x)\subset B_{R}(0)$ for all $x\in B_{R/2}(0)$, we
find that

$\displaystyle
2^{p}\operatorname{\mathcal{E}}(u,u)\geq\operatorname{\mathcal{E}}(w_{r},w_{r})$
$\displaystyle\geq\int_{B_{R/2}(0)}|w_{r}(x)|^{p}\int_{\operatorname{\mathbb{R}}^{d}\setminus
B_{R/2}(0)}\nu(x-y)\mathrm{d}y\mathrm{d}x\geq\|w_{r}\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}\int_{|h|\geq
R}\nu(h)\mathrm{d}h.$

Given that $u$ is not constant we get
$\|w_{r}\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}\neq 0$. Therefore we
deduce that

$\displaystyle\int_{|h|\geq R}\nu(h)\mathrm{d}h<\infty.$

Last, note that for $\delta\leq|h|\leq R$ and $x\in B_{\frac{\delta}{4}}(0)$
we have $|x|<\frac{\delta}{4}<\frac{\delta}{2}\leq|x+h|\leq
R+\frac{\delta}{2}$. Since $u$ is smooth with full support we put

$\displaystyle M=\min_{\frac{\delta}{2}\leq z\leq
R+\frac{\delta}{2}}\int_{B_{\frac{\delta}{4}}(0)}|u(x)-u(z)|^{p}\mathrm{d}x>0.$

From this we find that

$\displaystyle\operatorname{\mathcal{E}}(u,u)$
$\displaystyle\geq\int_{B_{R}(0)\setminus
B_{\delta}(0)}\int_{B_{\frac{\delta}{4}}(0)}|u(x)-u(x+h)|^{p}\mathrm{d}x\,\nu(h)\mathrm{d}h\geq
M\int_{B_{R}(0)\setminus B_{\delta}(0)}\nu(h)\mathrm{d}h.$

Hence, it follows that

$\displaystyle\int_{B_{R}(0)\setminus B_{\delta}(0)}\nu(h)\mathrm{d}h<\infty.$

Finally, altogether we find that $\nu$ is $p$-Lévy integrable, i.e., $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$. ∎

The following result is a straightforwards consequence of Theorem 4.1 and
Theorem 4.3.

######  Theorem 4.4.

Assume $\nu\geq 0$ is radial. The following assertions are equivalent.

  1. $(i)$

The $p$-Lévy condition (L) holds, i.e. $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$.

  2. $(ii)$

The embedding $W^{1,p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ is continuous.

  3. $(iii)$

$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty$ for
all $u\in W^{1,p}(\operatorname{\mathbb{R}}^{d})$.

  4. $(iv)$

$\operatorname{\mathcal{E}}_{\operatorname{\mathbb{R}}^{d}}(u,u)<\infty$ for
all $u\in C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$.

  5. $(v)$

The space $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ is nontrivial, i.e.,
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\neq\\{0\\}$.

Moreover, this also remains true when $p=1$ with
$BV(\operatorname{\mathbb{R}}^{d})$ in place of
$W^{1,1}(\operatorname{\mathbb{R}}^{d})$.

As illustrated in the next result, the Lévy integrability (L) draws the
borderline for which a space of type
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$ is trivial or not, see [Fog23,
Proposition 2.15] or [Fog20, Proposition 3.46] for a general setting.

######  Proposition 4.5.

Assume $\nu\geq 0$ is symmetric. The following assertions hold true.

  1. $(i)$

If $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})$, then
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})=L^{p}(\operatorname{\mathbb{R}}^{d})$
with equivalence in norms.

  2. $(ii)$

If $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p}\mathrm{d}h)$ and
$\Omega$ is bounded, then $W^{p}_{\nu}(\Omega)$ and
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ contain all bounded
Lipschitz functions.

  3. $(iii)$

Consider $C_{\delta}=\int_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h$. If $\nu$
is radial, $\Omega$ is connected and $C_{\delta}=\infty$ $\forall\,\delta>0$,
in particular $\nu\not\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p}\mathrm{d}h)$, then $u\in
W^{1,p}(\Omega)\cap W^{p}_{\nu}(\Omega)$ or $u\in C^{1}(\Omega)\cap
W^{p}_{\nu}(\Omega)$ if and only if $u=c$ is a constant function.

  4. $(iv)$

If $\nu$ is radial, then for any $u\in W^{1,p}(\operatorname{\mathbb{R}}^{d})$
there is $\delta=\delta(u)>0$, such that

$\displaystyle\frac{K_{d,p}}{2}C_{\delta}\|\nabla
u\|^{p}_{L^{p}(\operatorname{\mathbb{R}}^{d})}\leq|u|^{p}_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}\leq
2^{p}\|\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p}\mathrm{d}h)}\|u\|^{p}_{W^{1,p}(\operatorname{\mathbb{R}}^{d})}.$

##  5\. Nonlocal Trace Theorem

###  5.1. Weighted $L^{p}$-spaces

In order to set up the Dirichlet and Neumann problem in $L^{p}$-spaces over
$\operatorname{\mathbb{R}}^{d}$, we introduce Borel measures on
$\operatorname{\mathbb{R}}^{d}$ that capture the behavior of $\nu$ at
infinity. We will need the notion of unimodality.

######  Definition 5.1.

$\nu$ is called _unimodal_ if is radial with an almost decreasing profile,
i.e., there is $c>0$ such that $\nu(|x|)\geq c\,\nu(|y|)$ for all whenever
$|x|\leq|y|$.

######  Definition 5.2.

Let $\nu$ satisfies the $p$-Lévy integrability condition (L) and
$B\subset\operatorname{\mathbb{R}}^{d}$ be open with positive measure, i.e.,
$|B|>0$. Define the Borel measures
$\overline{\nu}_{B},\widetilde{\nu}_{B}:\operatorname{\mathbb{R}}^{d}\to[0,\infty)$
by

$\displaystyle\widetilde{\nu}_{B}(x)$
$\displaystyle=\int_{B}\left(1\wedge\nu(x-y)\right)\mathrm{d}y,$
$\displaystyle\overline{\nu}_{B}(x)$
$\displaystyle=\operatorname{ess}\inf\limits_{y\in B}\nu(x-y)\,.$ If $\nu$ is
a unimodal, then we define the Borel measure
$\widehat{\nu}_{R}:\operatorname{\mathbb{R}}^{d}\to[0,\infty)$ by
$\displaystyle\widehat{\nu}_{R}(x)$ $\displaystyle=\nu(R(1+|x|))\,,$

where $R>1$ is an arbitrary fixed number.

It is worthwhile noticing that $\nu$ needs not be unimodal for the definition
of $\widetilde{\nu}_{B}$ and $\overline{\nu}_{B}$. Let us discuss important
properties of the three measures $\widetilde{\nu}_{B}$, $\overline{\nu}_{B}$,
and $\widehat{\nu}_{R}$.

######  Proposition 5.3.

The following assertion are true.

  1. $(i)$

$\widehat{\nu}_{R},\,1\land\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$.

  2. $(ii)$

$\widetilde{\nu}_{B}\in
L^{1}_{\operatorname{loc}}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$. If $|B|<\infty$, then
$\widetilde{\nu}_{B}\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$.

  3. $(iii)$

$\overline{\nu}_{B}\in L^{1}(\operatorname{\mathbb{R}}^{d})$. If $\nu$
unimodal then $\overline{\nu}_{B}\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$.

  4. $(iv)$

$\overline{\nu}_{B}(x)=\widetilde{\nu}_{B}(x)=0$ for all $x\not\in
B_{\nu}:=B+\operatorname{supp}\nu$.

  5. $(v)$

For $B_{1}\subset B_{2}$ we have
$\overline{\nu}_{B_{2}}\leq\overline{\nu}_{B_{1}}$ and
$\widetilde{\nu}_{B_{1}}\leq\widetilde{\nu}_{B_{2}}$.

###### Proof.

The proof of $(i)-(iii)$ can be found in [Fog20, Chapter 3] or adapted from
[FK22, Section 2.3]. To prove $(iv)$, let $x\not\in B+\operatorname{supp}\nu$
then $x-y\not\in\operatorname{supp}\nu$ for all $y\in B$. In other words
$\nu(x-y)=0$ for almost all $y\in B.$ Whence,
$\overline{\nu}_{B}(x)=\widetilde{\nu}_{B}(x)=0$. $(v)$ is blatantly obvious.
∎

The weights $\widehat{\nu}_{R}$, $\widetilde{\nu}_{B}$ and
$\overline{\nu}_{B}$ turn out to be comparable when $\nu$ is unimodal and
satisfies the doubling scaling condition at infinity (5.2).

######  Definition 5.4.

We say that a radial kernel $\nu$ satisfies a doubling condition at infinity
if:

$\displaystyle\text{For every }\theta\geq 1\text{ there exist
}c_{1},c_{2}>0\text{ with }c_{1}\nu(r)\leq\nu(\theta r)\leq c_{2}\nu(r)\text{
for all }r\geq 1\,.$ (5.1)

Not that the property (5.1) is indeed equivalent to say that

$\displaystyle\text{ There exist }c_{1},c_{2}>0\text{ with
}c_{1}\nu(r)\leq\nu(2r)\leq c_{2}\nu(r)\text{ for all }r\geq 1.$ (5.2)

Obviously, the doubling condition is only relevant when the support of $\nu$
is large enough.

######  Theorem 5.5.

Assume $\nu$ is unimodal and $B$ is bounded, e.g. say $B\subset B_{R}(0)$ then

$\displaystyle\sigma^{*}_{R}C^{-1}(1\land\nu(2Rx))\leq\widehat{\nu}_{R}(x)\leq
C(1\land\nu(x)),$ $\displaystyle
C^{-1}\widehat{\nu}_{R}(x)\leq\overline{\nu}_{B}(x)\leq
C\widetilde{\nu}_{B}(x),$
$\displaystyle\widetilde{\nu}_{B}(x)\leq\sigma_{R}C(1\land\nu(\frac{x}{2})),$

where $\sigma_{R}=(1+\nu^{-1}(2R))$ and $\sigma^{*}_{R}=1\land\nu(2R)$. In
addition, if $\nu$ is of full support and satisfies the doubling condition
(5.1) then

$\displaystyle\widetilde{\nu}_{B}(x)\asymp\overline{\nu}_{B}(x)\asymp\widehat{\nu}_{R}(x)\asymp
1\land\nu(x).$

Here, the constant $C>0$ and the constants behind the relation $\asymp$ are
generic and depend on $B$, $R$ and $\nu$. Moreover, one notes that
$\sigma^{*}_{R}=\sigma^{-1}_{R}=0$ if $\nu(2R)=0$.

###### Proof.

Let us fix $x\in\operatorname{\mathbb{R}}^{d}$. Since $R\leq R(1+|x|)$ and
$|x|\leq R(1+|x|)$, by unimodality,

$\widehat{\nu}_{R}(x)\leq C1\land\nu(x)\quad$ with $\quad C=c(1+\nu(R))$.

For $|x|\leq 1$ we have $R(1+|x|)\leq 2R$, the unimodality implies

$c\widehat{\nu}_{R}(x)\geq\nu(2R)\geq\nu(2R)(1\land\nu(2Rx))\geq\sigma^{*}_{R}(1\land\nu(2Rx)).$

For $|x|\geq 1$ we have $R(1+|x|)\leq 2R|x|$ the unimodality implies
$c\widehat{\nu}_{R}(x)\geq\nu(2Rx)\geq\sigma^{*}_{R}(1\land\nu(2Rx)).$ In any
case, taking $C=c(1+\nu(R))$, we find that

$\displaystyle\sigma^{*}_{R}C^{-1}(1\land\nu(2Rx))\leq\widehat{\nu}_{R}(x)\leq
C(1\land\nu(x)).$

Next, according to Proposition 5.3 $\overline{\nu}_{B}$ is bounded, thus by
definition of $\overline{\nu}_{B}$

$\displaystyle\overline{\nu}_{B}(x)\leq(1+\|\overline{\nu}_{B}\|_{L^{\infty}(\operatorname{\mathbb{R}}^{d})})(1\land\nu(x-y))\quad\text{for
almost all $\,y\in B$}$

whence letting
$C=|B|^{-1}(1+\|\overline{\nu}_{B}\|_{L^{\infty}(\operatorname{\mathbb{R}}^{d})})$
we find that

$\displaystyle\overline{\nu}_{B}(x)\leq
C\int_{B}1\land\nu(x-y)\mathrm{d}y=C\widetilde{\nu}_{B}(x).$

We find that $|x-y|\leq R(1+|x|)$ for $y\in B$, the unimodality implies
$\widehat{\nu}_{R}(x)\leq c\nu(x-y)$. Hence we get

$\displaystyle c^{-1}\widehat{\nu}_{R}(x)\leq\operatorname*{ess\,inf}_{y\in
B}\nu(x-y)=\overline{\nu}_{B}(x).$

Now, for $\frac{|x|}{2}\leq 2R$ the unimodality implies $1\leq
c\nu^{-1}(2R)\nu(\frac{x}{2})\leq c\sigma_{R}\nu(\frac{x}{2})$. Since by
definition of $\widetilde{\nu}_{B}(x)$ we have
$\widetilde{\nu}_{B}(x)\leq\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}$,
we deduce that

$\displaystyle\widetilde{\nu}_{B}(x)\leq\sigma_{R}(c+1)\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}(1\land\nu(\frac{x}{2})).$

If $\frac{|x|}{2}\geq 2R$ then $\frac{|x|}{2}\leq|x-y|$ for all $y\in B\subset
B_{R}(0)$. By unimodality we get $\nu(x-y)\leq c\nu(\frac{x}{2})$ which
implies $1\land\nu(x-y)\leq(c+1)1\land\nu(\frac{x}{2})$ for all $y\in B$.
whence we get

$\displaystyle\widetilde{\nu}_{B}(x)=\int_{B}1\land\nu(x-y)\mathrm{d}y\leq\sigma_{R}(c+1)|B|(1\land\nu(\frac{x}{2})).$

In any case, taking
$C=(c+1)(|B|+\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})})$ we find
that

$\displaystyle\widetilde{\nu}_{B}(x)\leq\sigma_{R}C(1\land\nu(\frac{x}{2})).$

Last, assume that $\nu$ has full support and satisfies the doubling condition
(5.1). In view of the previous estimates, it sufficient to show that
$1\land\nu(\frac{x}{2})\asymp 1\land\nu(x)\asymp 1\land\nu(2Rx)$. For $|x|\leq
2$ then unimodality implies $c\nu(2Rx)\geq\nu(2R)\geq\nu(2R)(1\land\nu(x))$.
Analogously, $c\nu(x)\geq\nu(2)(1\land\nu(\frac{x}{2}))$. Thus, deduce that

$\displaystyle(1+c\nu^{-1}(2))^{-1}1\land\nu(\frac{x}{2})\leq
1\land\nu(x)\leq(1+c\nu^{-1}(2R))1\land\nu(2Rx).$

For $|x|\geq 2$, the doubling condition (5.1) implies
$c_{1}\nu(\frac{x}{2})\leq\nu(x)\leq c_{2}\nu(2Rx)$, so that

$\displaystyle(c_{1}^{-1}+1)^{-1}1\land\nu(\frac{x}{2})\leq
1\land\nu(x)\leq(c_{2}+1)1\land\nu(2Rx).$

Finally, we put $C=\max(c_{1}^{-1}+1,c_{2}+1,1+c\nu^{-1}(2R),1+c\nu^{-1}(2))$
so that $0<C<\infty$ since $\nu$ has full support and hence we get

$\displaystyle C^{-1}(1\land\nu(\frac{x}{2}))\leq 1\land\nu(x)\leq
C(1\land\nu(2Rx).$

This achieves the proof. ∎

######  Example 5.6.

Consider $\nu(h)=|h|^{-d-sp}$, $s\in(0,1)$ then $\widetilde{\nu}\asymp
1\land\nu$. In this case one can take $\widetilde{\nu}(h)=(1+|h|)^{-d-sp}$. In
this case space $W^{p}_{\nu}(\Omega)$ equals the classical Sobolev-Slobodeckij
space $W^{s,p}(\Omega)$. For the same choice of $\nu$ we define
$W^{s,p}(\Omega|\operatorname{\mathbb{R}}^{d})$ as the space
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$.

######  Lemma 5.7.

Assume that $B\subset\Omega$ where $B$ is bounded, say $B\subset B_{R}(0)$.
Let $\omega\in\\{\widetilde{\nu}_{B},\overline{\nu}_{B},\widehat{\nu}_{R}\\}$.
If $\nu$ is unimodal with full support then on
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, the following norms
$\|\cdot\|^{\\#}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{*}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are
equivalent.

$\displaystyle\|u\|^{*p}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{d}}|u(x)|^{p}\omega(x)\mathrm{d}x+\iint\limits\limits_{(\Omega^{c}\times\Omega^{c})^{c}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x\,,$
$\displaystyle\|u\|^{\\#p}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
$\displaystyle=\int_{\Omega}|u(x)|^{p}\omega(x)\mathrm{d}x+\iint\limits\limits_{(\Omega^{c}\times\Omega^{c})^{c}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x\,.$

Furthermore, if $\Omega$ is bounded then the norms
$\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{*}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are also
equivalent.

###### Proof.

In virtue of Theorem 5.5 we have $C^{-1}\widehat{\nu}_{R}(x)\leq\omega(x)\leq
C\widetilde{\nu}_{B}(x)$ for some $C>0$. Whence, since $B$ is bounded and
$\nu$ is of full support, the unimodality implies that there is $c^{\prime}>0$
such that $\omega(x)\geq c^{\prime}$ for almost all $x\in B$. This together
with the facts that $B\subset\Omega$ and
$\int_{\Omega^{c}}(1\land\nu(x-y))\mathrm{d}y\leq\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}$
implies

$\displaystyle\int_{\Omega}|u(x)|^{p}$
$\displaystyle\omega(x)\mathrm{d}x+\iint\limits_{\Omega\Omega^{c}}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle\geq
c^{\prime}\int_{B}|u(x)|^{p}\mathrm{d}x+\iint\limits_{\Omega\Omega^{c}}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle\geq
c^{\prime}\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}^{-1}\iint\limits_{B\Omega^{c}}|u(x)|^{p}(1\land\nu(x-y))\mathrm{d}y\,\mathrm{d}x+\iint\limits_{B\Omega^{c}}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle\geq(1\land
c^{\prime}\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}^{-1})\iint\limits_{\Omega^{c}B}\Big{[}|u(x)|^{p}+|u(x)-u(y)|^{p}\Big{]}1\land\nu(x-y)\mathrm{d}x\,\mathrm{d}y.$

Using $b^{p}\geq 2^{1-p}(a+b)^{p}-a^{p}$, $a,b\geq 0$ and $\omega(y)\leq
C\widetilde{\nu}_{B}(y)$ we get

$\displaystyle\int_{\Omega}|u(x)|^{p}$
$\displaystyle\omega(x)\mathrm{d}x+\iint\limits_{\Omega\Omega^{c}}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y\,\mathrm{d}x\geq
C_{1}\int_{\Omega^{c}}|u(y)|^{p}\omega(y)\mathrm{d}y\,$

with $C_{1}=\frac{C^{-1}}{2^{p-1}}(1\land
c^{\prime}\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}^{-1})$. This
clearly implies that
$\|u\|^{*}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\leq
C\|u\|^{\\#}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ for some
$C>0$. Thus the norms
$\|\cdot\|^{*}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{\\#}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are
equivalent.

Now if $\Omega$ is bounded, then by unimodality and boundedness of $\omega$
(see Proposition 5.3) we also have
$\|\omega\|_{L^{\infty}(\operatorname{\mathbb{R}}^{d})}\geq\omega(x)\geq
c^{\prime}$, $x\in\Omega$ for some $c^{\prime}>0$. The equivalence of the
norms $\|\cdot\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$and
$\|\cdot\|^{\\#}_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ readily
follows. ∎

The next result shows that functions in
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ possess certain weighted
integrability.

######  Theorem 5.8.

Assume $B\subset\Omega$ and $R\geq 1$. Let
$\omega\in\\{\widetilde{\nu}_{B},\overline{\nu}_{B},\widehat{\nu}_{R}\\}$.

  1. $(i)$

For $\omega\in\\{\widetilde{\nu}_{B},\overline{\nu}_{B}\\}$ we have the
continuous embedding

$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\operatorname{\mathbb{R}}^{d},\omega).$

The same holds for $\omega=\widehat{\nu}_{R}$ if in addition
$|B_{R}(0)\cap\Omega|>0$.

  2. $(ii)$

For $\omega\in\\{\overline{\nu}_{B},\widehat{\nu}_{R}\\}$ we have the
continuous embeddings

$\displaystyle L^{p}(\operatorname{\mathbb{R}}^{d},\omega)\hookrightarrow
L^{p-1}(\operatorname{\mathbb{R}}^{d},\omega)\cap
L^{1}(\operatorname{\mathbb{R}}^{d},\omega)$ (5.3) equivalently for any
$q\in[\min(1,p-1),p]$ we have $\displaystyle
L^{p}(\operatorname{\mathbb{R}}^{d},\omega)\hookrightarrow
L^{q}(\operatorname{\mathbb{R}}^{d},\omega)\hookrightarrow
L^{\min(1,p-1)}(\operatorname{\mathbb{R}}^{d},\omega).$ (5.4)

The same holds for $\omega=\widetilde{\nu}_{B}$ if in addition $|B|<\infty$.

###### Proof.

$(i)$ The proof of the embedding
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\operatorname{\mathbb{R}}^{d},\omega)$ can be adapted from [FK22,
Section 2.3] and [Fog20, Lemma 3.24].

$(ii)$ For
$\omega\in\\{\widetilde{\nu}_{B},\overline{\nu}_{B},\widehat{\nu}_{R}\\}$ by
assumptions, Proposition 5.3 implies $\omega\in
L^{1}(\operatorname{\mathbb{R}}^{d})$. Thus Hölder’s inequality implies
$\|u\|_{L^{r}(\operatorname{\mathbb{R}}^{d},\omega)}\leq\|\omega\|^{1/r-1/q}_{L^{1}(\operatorname{\mathbb{R}}^{d})}\|u\|_{L^{q}(\operatorname{\mathbb{R}}^{d},\omega)}$
whenever $q\geq r>0$. The desired embeddings immediately follow. ∎

######  Corollary 5.9.

Let $\omega\in\\{\widetilde{\nu}_{B},\overline{\nu}_{B},\widehat{\nu}_{R}\\}$
where $B\subset\Omega$. Assume $|B|<\infty$ and $|B_{R}(0)\cap\Omega|>0$,
$R\geq 1$. The following embeddings are continuous

$\displaystyle
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\operatorname{\mathbb{R}}^{d},\omega)\hookrightarrow
L^{p-1}(\operatorname{\mathbb{R}}^{d},\omega)\cap
L^{1}(\operatorname{\mathbb{R}}^{d},\omega).$

###### Proof.

This is a straightforward consequence of Theorem 5.8. ∎

######  Corollary 5.10.

Assume $\nu$ is unimodal with of full support and satisfies the doubling
condition (5.1). The following embeddings are continuous

$\displaystyle
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\operatorname{\mathbb{R}}^{d},1\land\nu)\hookrightarrow
L^{p-1}(\operatorname{\mathbb{R}}^{d},1\land\nu)\cap
L^{1}(\operatorname{\mathbb{R}}^{d},1\land\nu).$

###### Proof.

This is a direct consequence of Corollary 5.9 and Theorem 5.5. ∎

In the fractional setting $L^{p-1}(\operatorname{\mathbb{R}}^{d},\omega)$ is
also sometimes called the fraction tail space.

######  Corollary 5.11.

Let $\nu(h)=|h|^{-d-sp}$ with $s\in(0,1)$, we put
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})=W^{s,p}(\Omega|\operatorname{\mathbb{R}}^{d})$.
The following embeddings are continuous

$\displaystyle W^{s,p}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\operatorname{\mathbb{R}}^{d},(1+|h|)^{-d-sp}))\hookrightarrow
L^{p-1}(\operatorname{\mathbb{R}}^{d},(1+|h|)^{-d-sp}))\cap
L^{1}(\operatorname{\mathbb{R}}^{d},(1+|h|)^{-d-sp})).$

###### Proof.

This follows from Corollary 5.10 since $1\land\nu(h)\asymp(1+|h|)^{-d-sp}$. ∎

###  5.2. Trace space of $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$

The main goal of this part is to design an abstract notion trace space of
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ similarly as one does for
the space $W^{1,p}(\Omega)$. The nonlocal trace space of
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ assumes functions defined
on $\operatorname{\mathbb{R}}^{d}\setminus\Omega$, or strictly speaking on the
nonlocal boundary $\Omega_{e}=\Omega_{\nu}\setminus\Omega$ of $\Omega$ with
respect to $\nu$. The main reason is that elements of
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ are essentially defined on
$\operatorname{\mathbb{R}}^{d}$, or strictly speaking on the nonlocal hull
$\Omega_{\nu}=\Omega+\operatorname{supp}\nu$. This contrasts with the local
situation, where the trace space of $W^{1,p}(\Omega)$ (with $\Omega$ smooth
enough) are elements defined on the boundary $\partial\Omega$.

######  Definition 5.12 (Trace space of
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$).

The trace space of $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ denoted
$T^{p}_{\nu}(\Omega^{c})$ is the space of restrictions to
$\operatorname{\mathbb{R}}^{d}\setminus\Omega$ of functions of
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. More precisely,

$\displaystyle
T^{p}_{\nu}(\Omega^{c})=\\{v:\Omega^{c}\to\mathbb{R}\leavevmode\nobreak\
\text{meas.}\leavevmode\nobreak\ \leavevmode\nobreak\ \hbox{ such that
}\leavevmode\nobreak\ \leavevmode\nobreak\
v=u|_{\Omega^{c}}\leavevmode\nobreak\ \leavevmode\nobreak\ \hbox{with
}\leavevmode\nobreak\ \leavevmode\nobreak\ u\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\\}.$

We endow $T^{p}_{\nu}(\Omega^{c})$ with its natural norm,

$\displaystyle\|v\|_{T^{p}_{\nu}(\Omega^{c})}=\inf\\{\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}:\leavevmode\nobreak\
\leavevmode\nobreak\ u\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
\leavevmode\nobreak\ \hbox{ with }\leavevmode\nobreak\ \leavevmode\nobreak\
v=u|_{\Omega^{c}}\\}.$

If we identify $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\equiv
W^{p}_{\nu}(\Omega|\Omega_{\nu})$ then we can also identify
$T^{p}_{\nu}(\Omega^{c})\equiv T^{p}_{\nu}(\Omega_{e})$ where
$\Omega_{e}=\Omega_{\nu}\setminus\Omega$ and

$\displaystyle
T^{p}_{\nu}(\Omega_{e})=\\{v:\Omega_{e}\to\mathbb{R}\leavevmode\nobreak\
\text{meas.}\leavevmode\nobreak\ \leavevmode\nobreak\ \hbox{ such that
}\leavevmode\nobreak\ \leavevmode\nobreak\
v=u|_{\Omega_{e}}\leavevmode\nobreak\ \leavevmode\nobreak\ \hbox{with
}\leavevmode\nobreak\ \leavevmode\nobreak\ u\in
W^{p}_{\nu}(\Omega|\Omega_{\nu})\\}.$

######  Theorem 5.13.

If the space $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a Banach
space then so is the space $T^{p}_{\nu}(\Omega^{c})$.

###### Proof.

Noting that $T^{p}_{\nu}(\Omega^{c})$ and the quotient space
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})/W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$
are identical with equal norm in space and that
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a closed
subspace of $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}),$ one concludes
that $T^{p}_{\nu}(\Omega^{c})$ is complete. For a more detailed proof see
[Fog20, Chapter 3]. ∎

######  Theorem 5.14 (Nonlocal Trace Theorem).

Let
$\omega\in\\{\widetilde{\nu}_{\Omega},\overline{\nu}_{\Omega},\widehat{\nu}_{R}\\}$
where $|B_{R}(0)\cap\Omega|>0$ $($see Definition 5.2$)$. Define the trace
operator $u\mapsto\operatorname{Tr}(u)=u\mid_{\Omega^{c}}$,

  1. $(i)$

$\ker(\operatorname{Tr})=W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$
and
$\operatorname{Tr}(W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}))=T^{p}_{\nu}(\Omega^{c})$.

  2. $(ii)$

The mappings
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\xrightarrow{\operatorname{Tr}}T^{p}_{\nu}(\Omega^{c})\xrightarrow{\operatorname{Id}}L^{p}(\Omega^{c},\omega)$
are continuous

###### Proof.

It is easy to check that
$\operatorname{Tr}(W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}))=T^{p}_{\nu}(\Omega^{c})$
and
$\ker(\operatorname{Tr})=W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$.
It immediately follows from the definition of
$\|\cdot\|_{T^{p}_{\nu}(\Omega^{c})}$ that
$\|\operatorname{Tr}(u)\|_{T^{p}_{\nu}(\Omega^{c})}\leq\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
for all $u\in W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ whereas, by
Theorem 5.8 there exists $C>0$ such that

$\displaystyle\|\operatorname{Tr}(u)\|_{L^{p}(\Omega^{c},\omega)}\leq\|u\|_{L^{p}(\operatorname{\mathbb{R}}^{d},\omega)}\leq
C\|u\|_{W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\qquad\text{for all
}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ u\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}).$

Inasmuch as the above estimate is true for $v\in
W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ we deduce
$\|u\|_{L^{p}(\Omega^{c},\omega)}\leq C\|u\|_{T^{p}_{\nu}(\Omega^{c})}$ for
all $u\in T^{p}_{\nu}(\Omega^{c})$. ∎

One may view the objects $L^{p}(\Omega^{c},\omega)$,
$T^{p}_{\nu}(\Omega^{c})$, $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$
and $W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ respectively as
the nonlocal counterpart of the local objects $L^{p}(\partial\Omega)$,
$W^{1-1/p,p}(\partial\Omega)$, $W^{1,p}(\Omega)$ and $W^{1,p}_{0}(\Omega)$.

######  Theorem 5.15 (Classical Trace Theorem, see [BF13, Chap III]).

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is bounded Lipschitz.
There exists a linear and continuous trace operator
$\gamma_{0}:W^{1,p}(\Omega)\to L^{p}(\partial\Omega)$ such that
$\gamma_{0}u=u|_{\partial\Omega}$ for all $u\in C^{1}(\overline{\Omega})\cap
W^{1,p}(\Omega)$. Moreover, $\ker(\gamma_{0})=W^{1,p}_{0}(\Omega)$ and
$\gamma_{0}(W^{1,p}(\Omega))=W^{1,1-1/p}(\partial\Omega)$ (by definition).

######  Remark 5.16.

Let us emphasize our nonlocal trace operator $\operatorname{Tr}$ does not need
any special construction via functional analysis and density argument. Since
$\Omega^{c}$ is still a d-dimensional manifold. Then it makes sense to
consider hardcore restriction of functions on $\Omega^{c}.$ Moreover no
regularity on $\Omega$ is required nor on $u$. Whereas in the local situation
(see Theorem 5.15), the trace of a Sobolev function $u$ on the boundary
$\partial\Omega$ requires the smoothness of both $u$ and $\partial\Omega$.

It is natural to ask the following question: Can the space
$T^{p}_{\nu}(\Omega^{c})$ be self defined with an intrinsic norm preserving
its initial Banach structure in a way that its trivial connection to
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is less seeable? In the
local situation, it is possible to define a scalar product on the space
$H^{1/2}(\partial\Omega)$ when $\Omega$ is a special Lipschitz domain
(see[Din96]). This question is discussed in [FK22, GH22] when $p=2$, using the
main result from [BGPR20]. Another treatment of nonlocal trace operator for
the special fractional kernel $\nu(h)=|h|^{-d-sp}$ is encapsulated in [DK19,
Theorem 3].

##  6\. Compact embeddings

In this section we prove compact embeddings of the spaces
$W^{p}_{\nu}(\Omega)$, $W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and
$W_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ into $L^{p}(\Omega)$.
Our result on global compactness Theorem 6.6 requires some extra regularity
assumptions on $\Omega$ compatible with $\nu$. We exploit some recent ideas
from [JW20, DMT18]. However, extended details and discussions for the global
compactness can be found in [Fog20, FK22].

###  6.1. Local and global compactness results

Given a Banach space $X$, we say that an operator $T:X\to
L^{p}_{\operatorname{loc}}(\Omega)$ is compact if for any compact set
$K\subset\Omega$, the operator $R_{K}T:X\to L^{p}_{\operatorname{loc}}(K)$,
$R_{K}Tu=Tu|_{K}$ is compact.

######  Lemma 6.1 ([Bre11, Corollary 4.28]).

Let $w\in L^{1}(\operatorname{\mathbb{R}}^{d})$. The convolution operator
$T_{w}:L^{p}(\operatorname{\mathbb{R}}^{d})\to
L^{p}(\operatorname{\mathbb{R}}^{d})$, $T_{w}u=w*u$ is continuous with
$\|T_{w}\|_{\mathcal{L}(L^{p}(\operatorname{\mathbb{R}}^{d}),L^{p}(\operatorname{\mathbb{R}}^{d}))}\leq\|w\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}$.
Moreover, $R_{K}T_{w}:L^{p}(\operatorname{\mathbb{R}}^{d})\to L^{p}(K)$ is
compact for any measurable set $K\subset\operatorname{\mathbb{R}}^{d}$ with
$|K|<\infty$.

######  Theorem 6.2.

If $\nu$ satisfies (L). The following assertions are equivalent.

  1. $(i)$

$\nu$ is not integrable, i.e., $\nu\not\in
L^{1}(\operatorname{\mathbb{R}}^{d})$.

  2. $(ii)$

The embedding $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(K)$ is compact, for any measurable set with $|K|<\infty$.

  3. $(iii)$

The embedding $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}_{\operatorname{loc}}(\operatorname{\mathbb{R}}^{d})$ is compact.

  4. $(iv)$

The embedding $W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(B(0,1))$ is compact,

###### Proof.

We only prove $(iv)\implies(i)$ and $(i)\implies(ii)$ as the other
implications are straightforward. Assume $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d})$ and put $B=B(0,1)$ then
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})=L^{2}(\operatorname{\mathbb{R}}^{d})$
and hence the mappings
$L^{p}(B)\xrightarrow{E}W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\xrightarrow{R_{B}}L^{p}(B)$,
where $Eu=\overline{u}$ is the zero extension of $u$ and $R_{B}u=u|_{B}$, are
continuous. Since is the identity map $I=R_{B}\circ E:L^{p}(B)\to L^{p}(B)$ is
not compact, necessarily, $R$ is not compact. Now assume $\nu\not\in
L^{1}(\operatorname{\mathbb{R}}^{d})$ and $\delta>0$ sufficiently small such
that $0<\|\nu_{\delta}\|_{L^{1}(\mathbb{R}^{d})}$. Consider
$w_{\delta}(h)=\nu_{\delta}(h)\|\nu_{\delta}\|^{-1}_{L^{1}(\mathbb{R}^{d})}$
so that $\|w_{\delta}\|_{L^{1}(\mathbb{R}^{d})}=1$. Define
$T_{w_{\delta}}=w_{\delta}*u$, $u\in L^{p}(\mathbb{R}^{d})$. By the symmetry
of $\nu$ we have, for all $x\in\operatorname{\mathbb{R}}^{d}$

$\displaystyle
T_{w_{\delta}}u(x)=\int_{\mathbb{R}^{d}}w_{\delta}(y)u(x-y)\mathrm{d}y=\int_{\mathbb{R}^{d}}w_{\delta}(y)u(x+y)\mathrm{d}y\,.$

The Jensen’s inequality implies

$\displaystyle\|u-T_{w_{\delta}}u\|^{p}_{L^{p}(\mathbb{R}^{d})}$
$\displaystyle=\int_{\mathbb{R}^{d}}\Big{|}\int_{\mathbb{R}^{d}}[u(x)-u(x+h)]w_{\delta}(h)\mathrm{d}\,h\Big{|}^{p}\mathrm{d}x$
$\displaystyle\leq\|\nu_{\delta}\|^{-1}_{L^{1}(\mathbb{R}^{d})}\iint\limits_{\mathbb{R}^{d}\mathbb{R}^{d}}|u(x)-u(x+h)|^{p}\nu(h)\mathrm{d}\,h\mathrm{d}x\leq\|\nu_{\delta}\|^{-1}_{L^{1}(\mathbb{R}^{d})}\|u\|^{p}_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}\,.$

Accordingly, since $\nu\not\in L^{1}(\operatorname{\mathbb{R}}^{d})$, for a
subset $K\subset\mathbb{R}^{d}$ with $|K|<\infty$ we find that

$\displaystyle\|R_{K}-R_{K}T_{w_{\delta}}\|_{\mathcal{L}\big{(}W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d}),\,L^{p}(K)\big{)}}\leq\|\nu_{\delta}\|^{-1/p}_{L^{1}(\mathbb{R}^{d})}\xrightarrow{\delta\to
0}0\,.$

It follows that the operator
$R_{K}:W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\to L^{p}(K)$ with
$R_{K}u=u|_{K}$ is compact since each operator $R_{K}T_{w_{\delta}}$ is
compact (by Lemma 6.1) and the set of compact operator is closed. ∎

Another version of Theorem 6.2 is proved in [JW20, Theorem 1.1] for the case
$p=2$. It is worth mentioning that earlier analogous results are provided in
[PZ17, Proposition 6] and [BJ17, Proposition 1] for periodic functions on the
torus. This technique of killing the singularity is also used in [BJ13, Lemma
3.1].

######  Corollary 6.3.

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and $\nu\not\in
L^{1}(\operatorname{\mathbb{R}}^{d})$. The embedding
$W^{p}_{\nu}(\Omega)\hookrightarrow L^{p}_{\operatorname{loc}}(\Omega)$ is
compact. Moreover, if $|\Omega|<\infty$ then the embedding
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\Omega)$, is compact.

###### Proof.

For fixed $\varphi\in C_{c}^{\infty}(\Omega)$ the operator
$J_{\varphi}:W^{p}_{\nu}(\Omega)\to
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$, $J_{\varphi}u=u\varphi$ is
continuous. By Theorem 6.2 the embedding
$W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow L^{p}(K)$,
$K=\operatorname{supp}\varphi\subset\Omega$, is compact. It follows that the
embedding $W^{p}_{\nu}(\Omega)\hookrightarrow L^{p}(K)$ is compact. Hence the
first claim is proved. The embeddings
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})\hookrightarrow L^{p}(\Omega)$ are
continuous and, by Theorem 6.2, the last one is compact when
$|\Omega|<\infty$. Whence the second claim follows. ∎

Another consequence of Theorem 6.2 is the local compactness of
$W^{p}_{\nu}(\Omega)$ in $L^{p}(\Omega)$.

######  Corollary 6.4.

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open and $\nu\not\in
L^{1}(\operatorname{\mathbb{R}}^{d})$. The following assertions hold.

  1. $(i)$

For a bounded sequence $(u_{n})_{n}$ in $W^{p}_{\nu}(\Omega)$ there exists
$u\in W^{p}_{\nu}(\Omega)$ and subsequence $(u_{n_{j}})_{j}$ converging to $u$
in $L^{p}_{\operatorname{loc}}(\Omega)$. Moreover,

$\displaystyle|u|_{W^{p}_{\nu}(\Omega)}\leq\liminf_{n\to\infty}|u_{n}|_{W^{p}_{\nu}(\Omega)}.$

  2. $(ii)$

Assume $|\Omega|<\infty$ then for a bounded sequence $(u_{n})_{n}$ in
$W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ there exists $u\in
W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ and subsequence
$(u_{n_{j}})_{j}$ converging to $u$ in $L^{p}(\Omega)$. Moreover,

$\displaystyle|u|_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}\leq\liminf_{n\to\infty}|u_{n}|_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}.$

###  6.2. Global compactness.

Let us introduce conditions on $\Omega$ and $\nu$ yielding global compactness
results.

######  Definition 6.5.

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and bounded, and
$\nu\in L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{p})$ and $\nu\not\in
L^{1}(\operatorname{\mathbb{R}}^{d})$. We say that $(\nu,\Omega)$ is in the
class $\mathscr{A}_{i}$, $i\in\\{1,2,3\\}$, if in addition

  * ($\mathbf{\mathscr{A}_{1}}$)

$\ldots$ there exists an $W^{p}_{\nu}(\Omega)$-extension operator
$E:W^{p}_{\nu}(\Omega)\to W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})$, i.e.,
there is $C(\nu,\Omega,d)>0$ such that for every $u\in W^{p}_{\nu}(\Omega)$,
$\|u\|_{W_{\nu}^{p}(\operatorname{\mathbb{R}}^{d})}\leq
C\|u\|_{W^{p}_{\nu}(\Omega)}$ and $Eu|_{\Omega}=u$.

  * ($\mathbf{\mathscr{A}_{2}}$)

$\ldots$ $\Omega$ has Lipschitz boundary, $\nu$ is radial and
$q(\delta)\xrightarrow[]{\delta\to 0}\infty$ where

$\displaystyle
q(\delta):=\frac{1}{\delta^{p}}\int\limits_{B_{\delta}(0)}|h|^{p}\nu(h)\mathrm{d}h\,.$
(6.1)

  * ($\mathbf{\mathscr{A}_{3}}$)

$\ldots$ the following condition holds true:
$\widetilde{q}(\delta)\xrightarrow[]{\delta\to 0}\infty$ where

$\displaystyle\widetilde{q}(\delta):=\inf_{a\in\partial\Omega}\int_{\Omega_{\delta}}\nu(h-a)\mathrm{d}h$
(6.2)

with
$\Omega_{\delta}=\\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)>\delta\\}$.

Here is our global compactness result; see [FK22] for the case $p=2$ or
[Fog20, Theorem 3.89].

######  Theorem 6.6.

If the couple $(\nu,\Omega)$ belongs to one of the class
$\mathscr{A}_{i},\leavevmode\nobreak\ i=1,2,3$ then the embedding
$W^{p}_{\nu}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact. In particular,
the embedding
$W^{p}_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{p}(\Omega)$ is compact.

######  Remark 6.7.

The well known Rellich-Kondrachov’s compact embeddings
$W^{1,p}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)$ and
$W^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)$ when $\Omega$ is Lipschitz,
respectively can be derived from Theorem 6.2 combined with the embedding
$W^{1,p}_{0}(\Omega)\hookrightarrow
W^{p}_{\nu,\Omega}(\Omega|\operatorname{\mathbb{R}}^{d})$ and from Theorem 6.6
combined with the embedding $W^{1,p}(\Omega)\hookrightarrow
W^{p}_{\nu}(\Omega)$ when $\Omega$ is Lipschitz.

##  7\. Nonlocal Poincaré type inequalities

###  7.1. Nonlocal Poincaré-Friedrichs inequality

In this section $1\leq p<\infty$, $\Omega\subset\operatorname{\mathbb{R}}^{d}$
is open and that
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is symmetric
satisfying $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d}\setminus B(0,r))$ for
every $r>0$. In particular, the latter holds true if $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{\gamma}\mathrm{d}h)$ for any
$\gamma\geq 0$. Let us define the space
$L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})$ by

$\displaystyle L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})=\\{u\in
L^{p}(\operatorname{\mathbb{R}}^{d}):\,\text{$u=0$ a.e on $\Omega^{c}$}\\}.$

Next we use a more refined argument to prove the above Poincaré-Friedrichs
type inequalities with strictly positive constants, in a general setting only
by assuming that $\nu$ is nontrivial that is $\nu\not\equiv 0$. We follow the
strategy from[JW20] using the $2^{m}$-folded convolutions which is also used
in [FKV15, proof of Lemma 2.7].

######  Lemma 7.1.

Let $q\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ with $q\geq 0$ be symmetric, i.e.,
$q(h)=q(-h)$ and nontrivial, i.e., $\big{|}\\{q>0\\}\big{|}>0$. Define
$2^{m}$-fold convolution of $q$ as follows $q_{0}=q$,
$q_{m}=q_{m-1}*q_{m-1}=\underbrace{q*\cdots*q}_{2^{m}-\text{times}}$. For
every $m\geq 1$, the following assertions are true.

  1. $(i)$

Each $q_{m}$ belongs to $L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ and is uniformly continuous.

  2. $(ii)$

There exists $\delta>0$, such that $\inf_{B_{\delta_{m}}(0)}q_{m}>0$ with
$\delta_{m}=(\frac{7}{4})^{m}\delta$ so that
$\delta_{m}\xrightarrow{m\to\infty}\infty$.

  3. $(iii)$

For $u:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}$ measurable,
we have

$\displaystyle\mathcal{Q}_{m}(u)\leq B_{m}\mathcal{Q}(u),\qquad
B_{m}=2^{pm}\|q\|^{2^{m}-1}_{L^{1}(\operatorname{\mathbb{R}}^{d})},$ (7.1)

where we set $\mathcal{Q}(u)=\mathcal{Q}_{0}(u)$ and

$\displaystyle\mathcal{Q}_{m}(u):=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p}q_{m}(x-y)\mathrm{d}y\,\mathrm{d}x.$

###### Proof.

(i) Since $q_{m}=q_{m-1}*q_{m-1}$, by the Young’s inequalities $q_{m}\in
L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ and

$\displaystyle\|q_{m}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\leq\|q_{m-1}\|^{2}_{L^{1}(\operatorname{\mathbb{R}}^{d})}\leq\|q\|^{2^{m}}_{L^{1}(\operatorname{\mathbb{R}}^{d})},$
(7.2)
$\displaystyle\|q_{m}\|_{L^{\infty}(\operatorname{\mathbb{R}}^{d})}\leq\|q_{m-1}\|_{L^{\infty}(\operatorname{\mathbb{R}}^{d})}\|q_{m-1}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\leq\|q\|_{L^{\infty}(\operatorname{\mathbb{R}}^{d})}\|q\|^{2^{m}-1}_{L^{1}(\operatorname{\mathbb{R}}^{d})}.$
(7.3)

The uniform continuity $q_{m}$ readily follows from the continuity of the
shift in $L^{1}(\operatorname{\mathbb{R}}^{d})$.

(ii) Since $q$ is not identically vanishing, we have

$\displaystyle
q_{1}(0)=q*q(0)=\int_{\operatorname{\mathbb{R}}^{d}}|q(h)|^{2}\mathrm{d}h>0$

wherefrom, the continuity of $q_{1}$ at $0$ implies that
$\theta=\inf_{B_{\delta}(0)}q_{1}>0$ for some $\delta>0$. We claim that
$q_{2}(x)=q_{1}*q_{1}(x)>0$ whenever $|x|\leq\frac{7\delta}{4}$. Indeed, note
that $B_{\frac{\delta}{16}}(\frac{x}{2})\subset B_{\delta}(0)\cap
B_{\delta}(x)$ since for $z\in B_{\frac{\delta}{16}}(\frac{x}{2})$ we have

$\displaystyle\big{|}z-\frac{x}{2}\pm\frac{x}{2}\big{|}\leq\big{|}z-\frac{x}{2}\big{|}+\big{|}\frac{x}{2}\big{|}\leq\frac{15\delta}{16}<\delta.$

Given that $q_{1}(h)q_{1}(x-h)>\theta^{2}$ for all $h\in B_{\delta}(0)\cap
B_{\delta}(x)$ we finally get

$\displaystyle
q_{2}(x)=q_{1}*q_{1}(x)\geq\theta^{2}|B_{\frac{\delta}{16}}(0)|>0\quad\text{for
all $|x|\leq\frac{7\delta}{4}$}.$

Repeating this process, one easily retrieves that for
$\theta_{m-1}=\inf_{B_{\delta_{m-1}}(0)}q_{m-1}(x)>0$

$\displaystyle
q_{m}(x)\geq\theta^{2}_{m-1}|B_{\frac{\delta_{m-1}}{16}}(0)|>0\quad\text{for
all $|x|\leq\delta_{m}.$}$

(iii) Using the inequality $|a+b|^{p}\leq 2^{p-1}(|a|^{p}+|b|^{p})$ and
Fubini’s theorem yields

$\displaystyle\mathcal{Q}_{m}(u)$
$\displaystyle=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(x+h)|^{p}\int_{\operatorname{\mathbb{R}}^{d}}q_{m-1}(z)q_{m-1}(h-z)\mathrm{d}z\mathrm{d}h\mathrm{d}x$
$\displaystyle\leq
2^{p-1}\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\big{(}|u(x)-u(x+z)|^{p}+|u(x+z)-u(x+h)|^{p}\big{)}\int_{\operatorname{\mathbb{R}}^{d}}q_{m-1}(z)q_{m-1}(h-z)\mathrm{d}z\mathrm{d}h\mathrm{d}x$
$\displaystyle=2^{p-1}\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(x+z)|^{p}\Big{(}\int_{\operatorname{\mathbb{R}}^{d}}q_{m-1}(\xi)\mathrm{d}\xi\Big{)}q_{m-1}(z)\mathrm{d}z\mathrm{d}x$
$\displaystyle+2^{p-1}\int_{\operatorname{\mathbb{R}}^{d}}q_{m-1}(z)\mathrm{d}z\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x+z)-u(x+z+\xi)|^{p}q_{m-1}(\xi)\mathrm{d}\xi\mathrm{d}x\quad(\text{fix
$z$ and put $\xi=h-z$})$
$\displaystyle=2^{p}\|q_{m-1}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\mathcal{Q}_{m-1}(u).$

In short, this combined with inequality (7.2) implies

$\displaystyle\mathcal{Q}_{m}(u)\leq
2^{p}\|q_{m-1}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\mathcal{Q}_{m-1}(u)\leq\Big{(}\prod_{k=1}^{m}2^{p}\|q\|^{2^{k-1}}_{L^{1}(\operatorname{\mathbb{R}}^{d})}\Big{)}\mathcal{Q}(u)=2^{pm}\|q\|^{2^{m}-1}_{L^{1}(\operatorname{\mathbb{R}}^{d})}\mathcal{Q}(u).$

∎

It is decisive to keep in mind that for $u\in
L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})$ we have

$\displaystyle\mathcal{E}(u,u)$
$\displaystyle=\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x=\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}|u(x)-u(y)|^{p}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x.$

######  Theorem 7.2 (Poincaré-Friedrichs inequality III).

Assume $\nu\not\equiv 0$, i.e., $\big{|}\\{\nu>0\\}\big{|}>0$,
$\nu(h)=\nu(-h)$ and $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d}\setminus
B(0,r))$ for every $r>0$. Assume $\Omega$ is bounded one direction, i.e.,
there exist $R>0$ and $e\in\operatorname{\mathbb{R}}^{d}$, $|e|=1$ such that
$\Omega\subset H_{R}$ with $H_{R}=\\{z\in\operatorname{\mathbb{R}}^{d}:|z\cdot
e|\leq R\\}$. Then for $m\in\mathbb{N}$ large, there is
$0<C_{R,m}:=C(d,p,R,m,\nu)<\infty$ such that

$\displaystyle\|u\|^{p}_{L^{p}(\Omega)}\leq
C_{R,m}\mathcal{E}(u,u)\,\qquad\text{for all $u\in
L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})$}.$ (7.4)

Moreover, with the notation $q_{m}=q_{m-1}*q_{m-1}$, $q_{0}=q$, $q:=1\land\nu$
we can choose

$\displaystyle
C_{R,m}=2^{pm}\|q\|^{2^{m}-1}_{L^{1}(\operatorname{\mathbb{R}}^{d})}\Big{(}\int_{H^{c}_{2R}}q_{m}(h)\mathrm{d}h\Big{)}^{-1}.$

###### Proof.

We have $H_{R}\subset
H_{2R}(x)=\\{z\in\operatorname{\mathbb{R}}^{d}:|(z-x)\cdot e|\leq 2R\\}$ for
$x\in H_{R}$. Accordingly, for $u\in
L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})$ there holds

$\displaystyle\mathcal{Q}_{m}(u)$ $\displaystyle\geq
2\int_{\Omega}|u(x)|^{p}\mathrm{d}x\int_{H_{R}^{c}}q_{m}(x-y)\mathrm{d}y$
$\displaystyle\geq\int_{\Omega}|u(x)|^{p}\mathrm{d}x\int_{H_{2R}^{c}(x)}q_{m}(x-y)\mathrm{d}y=\|u\|^{p}_{L^{p}(\Omega)}\int_{H_{2R}^{c}}q_{m}(h)\mathrm{d}h.$

The fact that $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d}\setminus B(0,r))$
implies $q=1\land\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$. In addition, $\nu\equiv 0$ if and
only if $q\equiv 0$. Thus, $q=1\land\nu\leq\nu$, the estimate above and Lemma
7.1 (iii) yield

$\displaystyle\operatorname{\mathcal{E}}(u,u)\geq\mathcal{Q}(u)\geq
B^{-1}_{m}\mathcal{Q}_{m}(u)\geq C^{-1}_{R,m}\|u\|^{p}_{L^{p}(\Omega)}.$

According to Lemma 7.1 (i), $q_{m}\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ and by Lemma 7.1 (ii) for $m$ large
we have $\inf_{B_{\delta_{m}}(0)}q_{m}>0$ and $|B_{\delta_{m}}(0)\setminus
H^{c}_{2R}|>0$ where we recall $\delta_{m}\xrightarrow{m\to\infty}\infty$.
Hence

$\displaystyle 0<|B_{\delta_{m}}(0)\setminus
H^{c}_{2R}|\big{(}\inf_{B_{\delta_{m}}(0)}q_{m}\big{)}\leq\int_{H^{c}_{2R}}q_{m}(h)\mathrm{d}h\leq\|q_{m}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}<\infty,$

wherefrom we deduce that $0<C_{R,m}<\infty$. ∎

Next we want o consider the case where $|\Omega|<\infty$. We need the
following Lemma.

######  Lemma 7.3.

Let $E\subset\operatorname{\mathbb{R}}^{d}$ be measurable with $|E|<\infty$.
Define $r_{E}=\big{(}\frac{|E|}{|B_{1}|}\big{)}^{1/d}$, then we have

$\displaystyle\operatorname*{ess\,inf}_{x\in\operatorname{\mathbb{R}}^{d}}\int_{E^{c}}\nu(x-y)\mathrm{d}y\geq\nu^{\\#}(|E|),\quad\text{with}\quad\nu^{\\#}(|E|)=\int_{\\{\nu<\nu^{*}(r_{E})\\}}\nu(h)\mathrm{d}h,$
(7.5)

where $\nu^{*}$ is the symmetric rearrangement of $\nu$ defined by

$\displaystyle\nu^{*}(x)=\nu^{*}(|x|)=\int_{0}^{\infty}\mathds{1}_{\\{|\nu|>s\\}^{*}}(x)\mathrm{d}s=\inf\\{s>0:\leavevmode\nobreak\
|\\{|\nu|>s\\}|\leq c_{d}|x|^{d}\\}.$ (7.6)

It is worth noting that for the case $\nu(h)=|h|^{-d-sp},$ $h\neq 0$ we have
$\nu^{*}=\nu$ and

$\displaystyle\nu^{\\#}(|E|)=\int_{\\{\nu<\nu^{*}(r_{E})\\}}\nu(h)\mathrm{d}h=\int_{|h|>r_{E}}|h|^{-d-sp}\mathrm{d}h=\frac{1}{sp}\Big{(}\frac{|E|}{|B_{1}(0)|}\Big{)}^{-sp}|\mathbb{S}^{d-1}|.$

Accordingly, Lemma 7.3 implies the following; see also [DNPV12, Lemma 6.1]

######  Theorem 7.4 (Poincaré-Friedrichs inequality IV).

Assume $\nu\not\equiv 0$, i.e., $\big{|}\\{\nu>0\\}\big{|}>0$ and $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d}\setminus B(0,r))$ for every $r>0$. Assume
$|\Omega|<\infty$. Then for every $m\in\mathbb{N}$

$\displaystyle\|u\|^{p}_{L^{p}(\Omega)}\leq
C^{\\#}_{m}\mathcal{E}(u,u)\,\qquad\text{for all $u\in
L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})$},$ (7.7)

where, for sufficiently large $m$, we have
$0<C^{\\#}_{m}=C(d,p,|\Omega|,m,\nu)<\infty$ with

$\displaystyle
C^{\\#}_{m}=2^{pm}\|q\|^{2^{m}-1}_{L^{1}(\operatorname{\mathbb{R}}^{d})}\Big{(}q^{\\#}_{m}(|\Omega|)\Big{)}^{-1}.$

Here, $q_{m}=q_{m-1}*q_{m-1}$, $q_{0}=q$ with $q:=1\land\nu=\min(1,\nu)$ and
$q_{m}^{\\#}$ is defined as in (7.5).

###### Proof.

Accordingly, for $u\in L^{p}_{\Omega}(\operatorname{\mathbb{R}}^{d})$ there
holds that

$\displaystyle\mathcal{Q}_{m}(u)$ $\displaystyle\geq
2\int_{\Omega}|u(x)|^{p}\mathrm{d}x\int_{\Omega^{c}}q_{m}(x-y)\mathrm{d}y\geq\|u\|^{p}_{L^{p}(\Omega)}q^{\\#}_{m}(|\Omega|).$

The assumptions on $\nu$ imply that $q=1\land\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ and $\nu\equiv 0$ if and only if
$q\equiv 0$. Moreover, $q=1\land\nu\leq\nu$, the above estimate and Lemma 7.1
(iii) yield

$\displaystyle\operatorname{\mathcal{E}}(u,u)\geq\mathcal{Q}(u)\geq
B^{-1}_{m}\mathcal{Q}_{m}(u)\geq(C^{\\#}_{m})^{-1}\|u\|^{p}_{L^{p}(\Omega)}.$

By Lemma 7.1 (i), $q_{m}\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$ and hence
$q^{\\#}_{m}(|\Omega|)<\infty$. By Lemma 7.1 (ii) we have
$\delta_{m}>|\Omega|$ for $m$ large so that
$q^{\\#}_{m}(|\Omega|)\geq\inf_{B_{\delta_{m}}(0)}q_{m}>0$. Therefore,
$0<C^{\\#}_{m}<\infty$. ∎

###  7.2. Nonlocal Poincaré inequality

In this section $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and
bounded, $1\leq p<\infty$ and
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is symmetric.
Our goal in this section is to find some conditions on $\Omega$ and $\nu$
under which the following Poincaré inequality holds true, i.e., we can find a
constant $C=C(d,p,\Omega,\nu)>0$ such that

$\displaystyle\|u-\mbox{$\fint_{\Omega}$}u\|^{p}_{L^{p}(\Omega)}\leq
C\operatorname{\mathcal{E}}_{\Omega}(u,u)\,\quad\text{for all $u\in
L^{p}(\Omega)$}.$ ($P$)

Here and in what follows the notation
$\mbox{$\fint_{E}$}u=\frac{1}{|E|}\int_{E}u(x)\mathrm{d}x$. Recall that we
define

$\displaystyle\operatorname{\mathcal{E}}_{\Omega}(u,u)$
$\displaystyle=\iint\limits_{\Omega\Omega}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y\,\mathrm{d}x.$

So that since
$\operatorname{\mathcal{E}}_{\Omega}(u,u)\leq\operatorname{\mathcal{E}}(u,u)$,
($P$) would imply

$\displaystyle\|u-\mbox{$\fint_{\Omega}$}u\|^{p}_{L^{p}(\Omega)}\leq
C\operatorname{\mathcal{E}}(u,u)\,\quad\text{for all $u\in L^{p}(\Omega)$}.$
(7.8)

The following Lemma 7.5 is probably known in the literature. We however give a
simple proof for the convenience of the reader.

######  Lemma 7.5 (Finite chain covering).

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is connected. For every
$r>0$ there is a finite family of balls $(B_{i})_{1\leq i\leq n}$,
$B_{i}=B(x_{i},r)$ covering $\Omega$ with $x_{i}\in\Omega$ such that
$B_{i-1}\cap B_{i}\neq\emptyset$, $i=2,3,\cdots,n$.

###### Proof.

It is readily seen that the balls $B(x,r)$, $x\in\Omega$ covers the compact
set $\overline{\Omega}$ (the closure of $\Omega$). Thus, there is a finite
sub-cover $\mathcal{B}=\\{B_{i}\,:\,i\in\\{1,2,\cdots,n\\}\\}$ of $\Omega$
with $B_{i}=B(x_{i},r),x_{i}\in\Omega$. Next, let us write $B_{i-1}\sim B_{i}$
to indicate that $B_{i-1}\cap B_{i}\neq\emptyset$. Since $\Omega$ is
connected, there are $B,B^{\prime}\in\mathcal{B}$ such that $B\cap
B^{\prime}\neq\emptyset$. Up to a relabeling, denote the chain
$C_{2}=B_{1}\sim B_{2}$ with $B_{1}=B,B_{2}=B^{\prime}$. Assume, up to
relabeling the indices, there is a chain

$\displaystyle C_{i}=B_{1}\sim B_{2}\sim\cdots\sim B_{i}.$

Given that $\Omega$ is connected, there is
$B\in\mathcal{B}\setminus\bigcup_{j=1}^{i}B_{j}$ such that
$B\cap\bigcup_{j=1}^{i}B_{j}\neq\emptyset$. Thus, we can consider
$j_{0}=\max\\{j\in\\{1,2,\cdots,i\\}\,:B_{j}\sim B\\}$ and define a chain
$C_{i+1}$ as follows

$\displaystyle C_{i+1}$ $\displaystyle=B_{1}\sim\cdots\sim B_{j_{0}-1}\sim
B\sim B_{j_{0}}\sim\cdots\sim B_{i}$ $\displaystyle\quad\text{if $j_{0}\neq
1$},$ $\displaystyle C_{i+1}$ $\displaystyle=B\sim B_{1}\sim
B_{2}\sim\cdots\cdots\sim B_{i}$ $\displaystyle\quad\text{if $j_{0}=1$}.$

In this manner, up to relabeling the indices, one gets a chain $C_{n}$
containing the whole $\mathcal{B}$. ∎

We also need to consider the following generalization of the connected sets.

######  Definition 7.6.

We say that $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is $\rho-$connected,
$\rho\geq 0$, if $\Omega=\Omega_{1}\cup\Omega_{2}\cup\cdots\cup\Omega_{m}$,
$m\geq 1,$ where each $\Omega_{i}$ is open and connected such that
$\operatorname{dist}(\Omega_{i},\Omega_{i+1})<\rho$, $i=1,2,\cdots,n-1$.

It is worthwhile to observe that every connected set is $0$-connected and the
converse is not true. Consider
$B_{\pm}=B_{1}(0)\cap\\{x=(x^{\prime},x_{d}):\pm x_{d}>0\\}$ which are
connected. Therefore, since $\operatorname{dist}(B_{-},B_{+})=0$ we see that
$\Omega=B_{-}\cup B_{+}$ is $0$-connected but not connected. Another
observation is that if $\operatorname{dist}(\Omega_{i},\Omega_{i+1})<\rho$ one
finds $a_{i}\in\Omega_{i},b_{i}\in\Omega_{i+1}$ such that
$|a_{i}-b_{i}|<\rho$.

######  Lemma 7.7.

Assume $\nu>0$ a.e. on $B_{r}(0)$ for some $r>0$. Assume that one of the
following conditions is true.

  1. $(i)$

$r\geq\operatorname{diam}(\Omega)$.

  2. $(ii)$

$\Omega$ is connected.

  3. $(iii)$

$\Omega$ is $\rho$-connected with $0\leq\rho<r$.

There holds $\operatorname{\mathcal{E}}_{\Omega}(u,u)=0$ if and only if
$u=\lambda$ a.e. in $\Omega$, $\lambda\in\operatorname{\mathbb{R}}$.

###### Proof.

Clearly, if $u=\lambda$ a.e. in $\Omega$ then
$\operatorname{\mathcal{E}}_{\Omega}(u,u)=0$. Conversely, if
$\operatorname{\mathcal{E}}_{\Omega}(u,u)=0$, there is a null set
$A\subset\Omega$ such that

$\displaystyle\int_{\Omega}|u(x)-u(y)|^{p}\nu(x-y)\mathrm{d}y=0\qquad\text{for
all $x\in\Omega\setminus A$}.$

For each $x\in\Omega\setminus A$ there is a null set $A_{x}\subset\Omega$ such
that

$\displaystyle|u(x)-u(y)|^{p}\nu(x-y)=0\qquad\text{for all
$y\in\Omega\setminus A_{x}$}.$

Since $\nu>0$ a.e. in $B_{r}(0)$, we can assume up to renaming the null set
$A_{x}$ that

$\displaystyle u(y)=u(x)\qquad\text{for $y\in B_{r}(x)\cap(\Omega\setminus
A_{x}),\,\,x\in\Omega\setminus A$}.$ (7.9)

$(i)$ If $r\geq\operatorname{diam}(\Omega)$ then for fixed
$x_{0}\in\Omega\setminus A$ we have $\Omega\subset B_{r}(x_{0})$ so that
$u(x)=u(x_{0})$ for all $x\in\Omega\setminus A_{x_{0}}$. Hence, $u=u(x_{0})$
almost everywhere in $\Omega$.

$(ii)$ Assume $\Omega$ is connected. By Lemma 7.5, we can cover $\Omega$ with
a family of balls $(B_{i})_{1\leq i\leq n}$, $B_{i}=B(x_{i},r/2)$ such that
$B_{i-1}\cap B_{i}\neq\emptyset,\,\,i=2,3,\cdots,n.$ Thus, there is
$z_{i}\in(B_{i-1}\cap B_{i})\setminus A$. Consider the null set
$A^{\prime}=A_{z_{2}}\cup A_{z_{3}}\cup\cdots\cup A_{z_{n}}$. Given that
$B_{i-1}\cup B_{i}\subset B_{r}(z_{i})$, according to (7.9) we get
$u(x)=u(z_{i})$ for all $x\in(B_{r}(z_{i})\cap\Omega)\setminus A_{z_{i}}$. In
particular, we have $u(x)=u(z_{i})$ for all $x\in(B_{i-1}\cup
B_{i})\cap(\Omega\setminus A^{\prime})$. By the same token, $u(x)=u(z_{i+1})$
for all $x\in(B_{i}\cup B_{i+1})\cap(\Omega\setminus A^{\prime})$. It turns
out that $u(x)=u(z_{i})=u(z_{i+1})$ for all $x\in B_{i}\cap(\Omega\setminus
A^{\prime})$. Finally, we deduce that $u(z_{2})=u(z_{3})=\cdots
u(z_{n})=\lambda$. Since $\Omega=\bigcup_{i=1}^{n}B_{i}\cap\Omega$, we can
conclude that $u(x)=\lambda$ for all $x\in\Omega\setminus A^{\prime}$ where we
recall that $A^{\prime}$ is a null set.

$(iii)$ Now assume $\Omega$ is $\rho$-connected with $0\leq\rho<r$, i.e.
$\Omega=\Omega_{1}\cup\Omega_{2}\cdots\Omega_{m}$ where each $\Omega_{i}$ is
open bounded, connected and
$\operatorname{dist}(\Omega_{i},\Omega_{i+1})<\rho$. The previous step implies
that $u=\lambda_{i}$ a.e. on $\Omega_{i}$, $i=1,2,\cdots,m$. On the other
hand, considering $0<\varepsilon<r-\rho$ such that
$0\leq\rho<\rho+2\varepsilon<r$, we can find $a_{i}\in\Omega_{i}$ and
$b_{i}\in\Omega_{i+1}$ such that

$\operatorname{dist}(\Omega_{i},\Omega_{i+1})\leq|a_{i}-b_{i}|<\rho+\varepsilon<r.$

It follows that $B(a_{i},\varepsilon)\cap\Omega_{i}\neq\emptyset$ and
$B(a_{i},\rho+\varepsilon)\cap\Omega_{i+1}\neq\emptyset$. Given that $A$ is a
null set, we can find $\xi_{i}\in(\Omega_{i+1}\cap
B(a_{i},\rho+\varepsilon))\setminus A$. Since $u=\lambda_{i+1}$ a.e. in
$\Omega_{i+1}$, by (7.9) we get

$u(x)=u(\xi_{i})=\lambda_{i+1}$ for all $x\in
B_{r}(\xi_{i})\cap(\Omega\setminus A_{\xi_{i}}).$

Note that, for $z\in B_{\varepsilon}(a_{i})$ since $\xi_{i}\in
B(a_{i},\rho+\varepsilon)$ we have

$|z-\xi_{i}|\leq|z-a_{i}|+|a_{i}-\xi_{i}|<\rho+2\varepsilon<r.$

This implies $B_{\varepsilon}(a_{i})\subset B_{r}(\xi_{i})$. Since
$u=\lambda_{i}$ a.e. in $\Omega_{i}$, in particular we have

$\displaystyle\text{$\lambda_{i}=u(x)=u(\xi_{i})=\lambda_{i+1}$\quad for
all\quad$x\in B_{\varepsilon}(a_{i})\cap(\Omega_{i}\setminus
A_{\xi_{i}})\subset B_{r}(\xi_{i})\cap(\Omega\setminus A_{\xi_{i}})$}.$

This shows that $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{n},$ and hence it
follows that $u=\lambda_{1}$ a.e. in $\Omega$. ∎

The following result is some general nonlocal Poincaré inequality.

######  Theorem 7.8.

Assume that there is $r>0$ such that

$\displaystyle\kappa:=\operatorname*{ess\,inf}_{h\in B_{r}(0)}\nu(h)>0.$
(7.10)

If $r\geq\operatorname{diam}(\Omega)$ or else $\Omega$ is $\rho$-connected
with $r>\rho\geq 0$ then there is $C=C(d,p,\Omega,\nu)>0$ such that

$\displaystyle\|u-\mbox{$\fint_{\Omega}$}u\|^{p}_{L^{p}(\Omega)}\leq
C\operatorname{\mathcal{E}}_{\Omega}(u,u)\,\qquad\text{for all $u\in
L^{p}(\Omega)$}.$ (7.11)

###### Proof.

If $r\geq\operatorname{diam}(\Omega)$ then $\nu(x-y)\geq\kappa$ for all
$x,y\in\Omega$. Hence Jensen’s inequality yields

$\displaystyle\operatorname{\mathcal{E}}_{\Omega}(u,u)\geq\kappa\int_{\Omega}\int_{\Omega}|u(x)-u(y)|^{p}\mathrm{d}y\mathrm{d}x\geq\kappa|\Omega|\|u-\mbox{$\fint_{\Omega}u$}\|^{p}_{L^{p}(\Omega)}.$

Now assume that $0<r<\operatorname{diam}(\Omega)$ and $\Omega$ is
$\rho$-connected with $0<\rho\leq r$. Assume such $C$ does not exist. We can
find a sequence $(u_{n})_{n}\subset L^{p}(\Omega)$ such that
$\fint_{\Omega}u_{n}=0$, $\|u_{n}\|_{L^{p}(\Omega)}=1$ and
$\operatorname{\mathcal{E}}_{\Omega}(u_{n},u_{n})\leq\frac{1}{2^{n}}$ and
hence

$\displaystyle\kappa\iint\limits_{\Omega\Omega}|u_{n}(x)-u_{n}(y)|^{p}\mathds{1}_{B_{r}(0)}(x-y)\mathrm{d}y\,\mathrm{d}x\leq\iint\limits_{\Omega\Omega}|u_{n}(x)-u_{n}(y)|^{p}\nu(x-y)\mathrm{d}y\,\mathrm{d}x\leq\frac{1}{2^{n}}\,.$

Since $(u_{n})_{n}$ is bounded in $L^{p}(\Omega)$, passing through a
subsequence if necessary we can assume that $(u_{n})_{n}$ weakly converges in
$L^{p}(\Omega)$ to some $u$, where $u\in L^{p}(\Omega)$ $p>1$ and $u$ is a
signed Radon measure on $\Omega$ for $p=1$. In particular,
$\fint_{E}u_{n}(x)\mathrm{d}x\to\fint_{E}u(x)\mathrm{d}x$ for every Borel set
$E\subset\Omega$. In addition, we can assume there is a null set
$A\subset\Omega$ such that

$\displaystyle\int_{B_{r}(x)\cap\Omega}|u_{n}(x)-u_{n}(y)|^{p}\mathrm{d}y\xrightarrow{n\to\infty}0\quad\text{for
all $x\in\Omega\setminus A$}.$ (7.12)

Moreover, for each $x\in\Omega\setminus A$ there is another null set
$A_{x}\subset\Omega$ such that

$\displaystyle u_{n}(x)-u_{n}(y)\xrightarrow{n\to\infty}0\quad\text{for all
$y\in(B_{r}(x)\cap\Omega)\setminus A_{x}$}.$ (7.13)

Consider $(B_{r/2}(x_{i}))_{1\leq i\leq n}$ a finite cover of $\Omega$ with
$x_{i}\in\Omega$. Choose $\xi_{i}\in(B_{r/2}(x_{i})\cap\Omega)\setminus A$.
Hence $(E_{\xi_{i}})_{1\leq i\leq n}$ with
$E_{\xi_{i}}=B_{r}(\xi_{i})\cap\Omega$ is also a finite cover of $\Omega$.
Note that

$\displaystyle\Big{|}u_{n}(\xi_{i})-\fint_{E_{\xi_{i}}}u(y)\mathrm{d}y\Big{|}^{p}$
$\displaystyle\leq
2^{p-1}\fint_{E_{\xi_{i}}}\big{|}u_{n}(\xi_{i})-u_{n}(y)\big{|}^{p}\mathrm{d}y+2^{p-1}\Big{|}\fint_{E_{\xi_{i}}}u_{n}(y)\mathrm{d}y-\fint_{E_{\xi_{i}}}u(y)\mathrm{d}y\Big{|}^{p}.$

This together with the weak convergence and the convergence in (7.12) imply
that

$\displaystyle
u_{n}(\xi_{i})\xrightarrow{n\to\infty}\lambda_{\xi_{i}}=\fint_{E_{\xi_{i}}}u(y)\mathrm{d}y.$