# A general framework for nonlocal Neumann problems

Guy Foghem† Technische Universität Dresden, Fakultät für Mathematik, Institut
für Wissenschaftliches Rechnen, Zellescher Weg 23-25 01217, Dresden, Germany.
<EMAIL_ADDRESS>and  Moritz Kassmann‡
https://dx.doi.org/10.4310/CMS.2024.v22.n1.a2 Universität Bielefeld, Fakultät
für Mathematik, Postfach 100131, 33501 Bielefeld, Germany
<EMAIL_ADDRESS>
###### Abstract.

Within the framework of Hilbert spaces, we solve nonlocal problems in bounded
domains with prescribed conditions on the complement of the domain. Our main
focus is on the inhomogeneous Neumann problem in a rather general setting. We
also study the transition from complement value problems to local boundary
value problems. Several results are new even for the fractional Laplace
operator. The setting also covers relevant models in the framework of
peridynamics.

###### Key words and phrases:

Neumann problem, nonlocal Sobolev spaces, integro-differential operators,
integro-differential equations, Dirichlet forms

###### 2020 Mathematics Subject Classification:

28A80, 35J20, 35J92, 46B10, 46E35, 47A07, 49J40, 49J45

†,‡Financial support by the DFG via IRTG 2235: “Searching for the regular in
the irregular: Analysis of singular and random systems” is gratefully
acknowledged. †Guy Foghem gratefully acknowledges financial support by the DFG
via the Research Group 3013: “Vector-and Tensor-Valued Surface PDEs”.

##  1\. Introduction

###  1.1. Main Results

Over the last years, there have been several studies of nonlocal Neumann
problems of the following type: Given a bounded open set
$\Omega\subset\operatorname{\mathbb{R}}^{d}$, one is interested in well-
posedness for

$\displaystyle Lu=f\text{ in }\Omega,\qquad\mathcal{N}u=g\text{ on
}\operatorname{\mathbb{R}}^{d}\setminus\Omega\,,$ ($N$)

where $L$ is an integral or integro-differential operator and
$\operatorname{\mathcal{N}\\!\\!}$ is a related integral operator, which plays
the role of some kind of normal derivative on
$\operatorname{\mathbb{R}}^{d}\setminus\Omega$. The main goal of this article
is to prove well-posedness results for ($N$) in a general setting. We assume:

$\displaystyle Lu(x)$
$\displaystyle=\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}\big{(}u(x)-u(y)\big{)}k(x,y)\mathrm{d}y\qquad$
$\displaystyle(x\in\operatorname{\mathbb{R}}^{d})\,,$
$\displaystyle\mathcal{N}u(y)$
$\displaystyle=\int_{\Omega}(u(y)-u(x))k(x,y)\mathrm{d}x\qquad$
$\displaystyle(y\in\Omega^{c})\,.$

Here,
$k:\operatorname{\mathbb{R}}^{d}\times\operatorname{\mathbb{R}}^{d}\setminus\operatorname{diag}\to[0,\infty)$
is measurable and satisfies

$\displaystyle\Lambda^{-1}\nu(y-x)\leq
k(x,y)\leq\Lambda\nu(y-x)\quad(x,y\in\operatorname{\mathbb{R}}^{d})\,,$ (E)

where $\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is the
density of a symmetric Lévy measure, i.e., $\nu$ satisfies

$\displaystyle\nu(h)=\nu(-h)\text{ for all }h\neq 0\,\,\,\text{ and
}\,\,\,\int_{\operatorname{\mathbb{R}}^{d}}\big{(}1\land|h|^{2}\big{)}\nu(h)\mathrm{d}h<\infty\,.$
(L)

The main new contributions of the present article include the following:

  1. (a)

Extending previous results, e.g. from [DROV17], [MPL19], [DTZ22], we treat the
inhomogeneous problem for natural choices of data $g$. The corresponding
results are new for the fractional Laplace operator.

  2. (b)

We provide a general framework that includes integrable and singular kernels
at the same time.

  3. (c)

We show that the trace spaces introduced in [DK19] and [BGPR20] coincide.

  4. (d)

We introduce a new Dirichlet-to-Neumann operator based on the operator
$\operatorname{\mathcal{N}\\!\\!}$.

  5. (e)

Our well-posedness results are aligned with classical results for second order
partial differential operators. We show convergence of nonlocal to local
problems, where we treat singular and bounded kernels together.

Let us explain condition (E). We denote $a\land b=\min(a,b)$ for
$a,b\in\operatorname{\mathbb{R}}$. In the case $k(x,y)=\nu(y-x)$ with $\nu$ as
above, the operator $L$ is translation invariant and generates a symmetric
Lévy process. The density $\nu$ defines the “order” of the operator $L$, which
becomes apparent in the case of $\nu(h)=C_{d,\alpha}|h|^{-d-\alpha}$ for
$h\neq 0$ where $\alpha\in(0,2)$ is fixed and $C_{d,\alpha}$ is an appropriate
constant. The resulting operator is the so-called fractional Laplace operator
$(-\Delta)^{\alpha/2}$. The choice of $C_{d,\alpha}$ ensures the relation
$\widehat{(-\Delta)^{\alpha/2}u}(\xi)=|\xi|^{\alpha}\widehat{u}(\xi)$ for all
functions $u$ in $C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$. Let us
mention that asymptotically one has $C_{d,\alpha}\asymp\alpha(2-\alpha)$. This
will play an important role for our analysis. Further details about the
fractional Laplacian $(-\Delta)^{\alpha/2}$ and the constant $C_{d,\alpha}$
can be found in [AAS67, NPV12, FG20]. Finally, let us mention that, the
assumptions (E) and (L) are not sufficient in order to guarantee the existence
of the pointwise expression $Lu(x)$ in the general case, even if $u$ is
smooth. The Hilbert space approach used in Section 4 avoids this issue because
we only deal with the corresponding quadratic forms. It is worth to mention
that the nonlocal operator $\mathcal{N}$ was initially introduced by [DROV17]
and is called the _nonlocal normal derivative operator_ across the boundary of
$\Omega$ with respect to $\nu$. Another type of such an operator appeared
earlier in the literature see for instance [DGLZ12].

Let us quickly review the classical Neumann problem, for the reader’s
convenience. Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be a bounded
open subset whose boundary $\partial\Omega$ is sufficiently regular. Given
$f:\Omega\to\mathbb{R}$ and $g:\partial\Omega\to\mathbb{R}$ measurable, the
classical inhomogeneous Neumann problem associated to the data $f$ and $g$
consists in finding a function $u:\Omega\to\mathbb{R}$ satisfying

$\displaystyle-\Delta u=f\quad\text{in}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Omega\quad\quad\text{ and
}\quad\quad\frac{\partial u}{\partial n}=g\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{on}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \partial\Omega.$ (1.1)

Here $\frac{\partial u}{\partial n}$ denotes the outward normal derivative of
$u$ on $\partial\Omega$. From a weak formulation point of view, $u$ is said to
be a weak solution of (1.1) if $u\in H^{1}(\Omega)$ satisfies

$\displaystyle\int_{\Omega}\nabla u(x)\cdot\nabla
v(x)\mathrm{d}x=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\partial\Omega}g(y)v(y)\mathrm{d}\sigma(y),\quad\mbox{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ v\in H^{1}(\Omega)\,.$

The Neumann boundary problem has received considerably less attention in the
literature compared to the Dirichlet boundary problem. Classical textbooks
like [Mik78] treat the basic aspects. A rigorous treatment including
regularity up to the boundary, Schauder estimates, $L^{p}$ estimates and the
variational formulation can be found in the lecture notes [Leo13]. A recent
article covering classical results for elliptic equations in divergence form
is [DV09].

Following [FKV15, SV14, SV13] we introduce a bilinear form
$\operatorname{\mathcal{E}}$ by

$\displaystyle\mathcal{E}(u,v)=\frac{1}{2}\\!\\!\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u(x)-u(y)\big{)}\big{(}v(x)-v(y)\big{)}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y$
(1.2)

for all smooth functions with compact support. As in the local case, a main
tool in the study of Neumann problems, is a Gauss-Green type formula for
$u,v\in C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$, see A.5:

$\displaystyle\int_{\Omega}Lu(x)v(x)\mathrm{d}x=\mathcal{E}(u,v)-\int_{\Omega^{c}}\mathcal{N}u(y)v(y)\mathrm{d}y.$
(1.3)

Relation (1.3) motivates us to introduce an energy space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ as the vector space of all
measurable functions
$u:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}$ such that the
restriction $u|_{\Omega}$ belongs to $L^{2}(\Omega)$ and
$\operatorname{\mathcal{E}}(u,u)$ is finite. See Section 2.2 for more details.
The energy space $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ can be seen
as a nonlocal analog of $H^{1}(\Omega)$. Let us make an interesting
observation. Let $f\in L^{2}(\Omega)$ and $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ be a minimizer of the
functional
$v\mapsto\frac{1}{2}\operatorname{\mathcal{E}}(v,v)-\int_{\Omega}fv$ in the
space $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. Then
$\operatorname{\mathcal{E}}(u,v)=0$ for all smooth functions with compact
support in $\operatorname{\mathbb{R}}^{d}\setminus\overline{\Omega}$. Since
$u,v\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, the Fubini theorem
implies

$\displaystyle\int_{\Omega^{c}}\operatorname{\mathcal{N}\\!\\!}u(y)v(y)\mathrm{d}y=\operatorname{\mathcal{E}}(u,v)=0\,,$

which implies $\operatorname{\mathcal{N}\\!\\!}u=0$ in $\Omega^{c}$, see
Corollary 4.8. On the one hand, this observation is aligned with the classical
theory where the normal derivative appears naturally when minimizing the
energy. On the other hand, and this is interesting, here we do not need to
assume any regularity of the kernel $k(x,y)$ and the boundary
$\partial\Omega$.

Let us summarize the main results of this work.

  1. (i)

The first step is to define a base space
$L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})$, in which we can define
the complement value problems. We define $\widetilde{\nu}$ and two alternative
options $\overline{\nu},\nu^{*}$ in 2.15. In Section 2.2 we study embedding
results of corresponding function spaces.

  2. (ii)

The next step is to introduce $T_{\nu}(\Omega^{c})$ as the trace space of
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in Section 2.6. In this
section, we study equivalent norms of the trace space and a density result.

  3. (iii)

An important tool in the proof of well-posedness results is the compact
embedding $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\Omega)$, which is a core result in Section 3, see Theorem 3.10.

  4. (iv)

Section 4 is dedicated to well-posedness results. We focus on the Neumann
problem in Section 4.1. An existence result for problem ($N$) is given in
Theorem 4.11. We also discuss a more general Robin-type complement value
problem.

  5. (v)

The setup of this work allows to define a fully nonlocal Dirichlet-to-Neumann
map with the help of the nonlocal Neumann-type derivative
$\operatorname{\mathcal{N}\\!\\!}$. For
$\Omega\subset\operatorname{\mathbb{R}}^{d}$, the Dirichlet data are given on
$\Omega^{c}$ and mapped to $\operatorname{\mathcal{N}\\!\\!}u$ on
$\Omega^{c}$, where $u$ satisfies the nonlocal equation in $\Omega$. Thus,
this map can be viewed as a nonlocal analog of the well-known Dirichlet-to-
Neumann operator given in [CS07]. Basic properties are formulated in Theorem
4.20 together with spectral properties in Theorem 4.23.

  6. (vi)

The analogy between the classical Neumann problem and problem ($N$) leads to a
convergence result when considering a sequence of complement value problems
for the fractional Laplace operator $(-\Delta)^{\alpha/2}$ where $\alpha\to
2$. Theorem 5.4 establishes the convergence of the corresponding sequence of
solutions $u_{\alpha}$ as $\alpha\to 2$.

###  1.2. Related literature

Nonlocal complements value problems have been studied in several works. In
particular, the Dirichlet problem is studied in many articles. For translation
invariant problems, see the survey [RO16, AFR20] for fine regularity results
and [FKV15, BV16] for the Hilbert space approach in a similar setting as in
this work.

An early contribution to nonlocal Neumann problems is [DROV17], where also the
Gauss-Green formula appears for a special case. There is a difference between
our approach and the one in [DROV17], which explains why we are able to study
the inhomogeneous Neumann problem. Let us explain our approach for the
simplest setup of the fractional Laplace operator $(-\Delta)^{\alpha/2}$,
i.e., $\nu(h)=C_{d,\alpha}|h|^{-d-\alpha}$. Given $f\in L^{2}(\Omega)$ and
$g\in L^{2}(\Omega^{c},(1+|x|)^{d+\alpha}\mathrm{d}x)$, motivated by the
Gauss-Green formula (1.3) we say that $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a weak solution or a
variational solution of the inhomogeneous Neumann problem ($N$) if

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

By testing ($V$) with $v=1$ gives the following necessary compatibility
condition

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

Equality ($V$) should be contrasted with the variational formulation of (1.1)
in the classical case: Given $f\in L^{2}(\Omega)$ and $g\in
L^{2}(\partial\Omega)$, find $u\in H^{1}(\Omega)$ such that

$\displaystyle\int_{\Omega}\nabla u(x)\cdot\nabla
v(x)\mathrm{d}x=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\partial\Omega}g(y)v(y)\mathrm{d}\sigma(y)\quad\text{
for all }v\in H^{1}(\Omega)\,.$

Note that [DROV17, Def. 3.6] and subsequent definitions like [MPL19,
Definition 2.7] look very similar to ($V$) at first glance. However, the norm
of the test space defined in [DROV17, Eq. (3.1)], [MPL19, Section 2] depends
on the Neumann data $g$, which is not natural. Our test space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in the weak formulation ($V$)
does not depend on the Neumann data $g$. For the general case, we refer the
reader to 4.3. It is worth mentioning that the weighted space
$L^{2}(\Omega^{c},(1+|x|)^{d+\alpha}\mathrm{d}x)$ is the natural function
space for the Neumann data $g$. In fact in Theorem 4.13 we are able to find
some $g$ not belonging to $L^{2}(\Omega^{c},(1+|x|)^{d+\alpha}\mathrm{d}x)$,
$\Omega=B_{1}(0)$ for which the variational Neumann problem ($V$) with $f=0$
does not have any weak solution in
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$.

The aforementioned issue does not show up for _homogeneous_ nonlocal Neumann
problems. For such problems, several results have been proved, e.g.,
regularity up to a boundary of a domain for the fractional Laplace operator in
[AFR20]. A particular observation linking the homogeneous Neumann problem to
the regional fractional Laplace operator is provided in [Aba20]. Eigenvalues
of nonlocal mixed problems are studied in [LMP+18]. Various nonlinear Neumann
problems are studied in [Che18, MPL19, CC20, ATL20, MPL21, BS21]. Some higher
order nonlocal Neumann problems are treated in [BMPS20]. The classical Neumann
problem is closely linked to reflected diffusions. It turns out to be a
challenging problem to establish a similar link between the nonlocal Neumann
problem and a Markov jump process together with its reflection. An attempt is
made in [Von21], which we comment on in detail in Remark 2.38.

###  1.3. Peridynamics and volume constraints on bounded sets

In the literature, several nonlocal problems are studied in the area of
peridynamics. Most of these models require complement conditions (a.k.a.
volume constraints) not necessarily on the whole complement of the domain
$\Omega$ but only often on a part of the complement. Here, we would like to
point out that our setting can be adapted to fit such requirements. Let us
exemplify this with a simple model. Consider the symmetric kernel of the form
$k(x,y)=\nu(x-y)$ with $\nu$ supported around the origin, say
$\operatorname{supp}\,\nu\subset B_{\delta}(0)$ for some $\delta>0$. A popular
example of a kernel in peridynamic models is given by
$\nu(h)=\mathds{1}_{B_{\delta}}(h)$. In this case it is natural to assume the
complement condition not on the whole complement
$\operatorname{\mathbb{R}}^{d}\setminus\Omega$ but only on
$\Omega(\delta)=\\{x\in\operatorname{\mathbb{R}}^{d}\setminus\Omega:\operatorname{dist}(x,\partial\Omega)<\delta\\}$.
A nonlocal problem of the form $Lu=f$ in $\Omega$ is then supplemented with a
complement condition prescribed on the volume constraint $\Omega(\delta)$,
e.g., $u=g$ on $\Omega(\delta)$ for a Dirichlet problem or $\mathcal{N}u=g$ on
$\Omega(\delta)$ for a Neumann problem. Here,
$f:\Omega\to\operatorname{\mathbb{R}}$ and
$g:\Omega(\delta)\to\operatorname{\mathbb{R}}$ are given data. Our approach
can easily be adopted to cover this case. In comparison to the weak
formulation ($V$), one would need to replace
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ by the space
$V_{\nu}(\Omega|E)$ defined as in (2.3), with $E=\Omega\cup\Omega(\delta)$ and
recall that $\nu(h)=\mathds{1}_{B_{\delta}}(h)$. Indeed, assuming for
simplicity that $\Omega$ is bounded Lipschitz and connected, the well-
posedness of the Neumann and the Dirichlet problem can be formulated as
follows.

######  Corollary 1.1.

Let $f\in L^{2}(\Omega)$ and $g\in L^{2}(\Omega(\delta))$. Then there is a
unique variational solution $u\in
V_{\nu}(\Omega|E)^{\perp}=V_{\nu}(\Omega|E)\cap\\{\int_{\Omega}u\mathrm{d}x=0\\}$
to the Neumann problem $Lu=f$ in $\Omega$ and $\mathcal{N}u=g$ on
$\Omega(\delta)$, i.e.,

$\displaystyle\mathcal{E}(u,v)=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\Omega(\delta)}g(y)v(y)\mathrm{d}y\quad\text{
for all }v\in V_{\nu}(\Omega|E)^{\perp}\,.$ (1.4)

Moreover, there is a constant $C>0$ independent of $f$ and $g$ such that

$\displaystyle\|u\|_{V_{\nu}(\Omega|E)}\leq
C\big{(}\|f\|_{L^{2}(\Omega)}+\|g\|_{L^{2}(\Omega(\delta))}\big{)}.$

######  Corollary 1.2.

Let $f\in L^{2}(\Omega)$ and $g\in V_{\nu}(\Omega|E)$. Then there is a unique
variational solution $u\in V_{\nu}(\Omega|E)$ to the Dirichlet problem $Lu=f$
in $\Omega$ and $u=g$ on $\Omega(\delta)$, i.e.,

$\displaystyle u-g\in
V_{\nu,0}(\Omega|E)\qquad\text{and}\qquad\mathcal{E}(u,v)=\int_{\Omega}f(x)v(x)\mathrm{d}x\quad\text{
for all }v\in V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})\,.$ (1.5)

Here we denote
$V_{\nu,0}(\Omega|E)=V_{\nu}(\Omega|E)\cap\\{u|_{\Omega(\delta)}=0\\}$.
Moreover, there is a constant $C>0$ independent of $f$ and $g$ such that

$\displaystyle\|u\|_{V_{\nu}(\Omega|E)}\leq
C\big{(}\|f\|_{L^{2}(\Omega)}+\|g\|_{V_{\nu}(\Omega|E)}\big{)}.$

The proofs of 1.1 and 1.2 are analogous to the ones of Theorem 4.11 and
Theorem 4.18; we also refer to [Fog23a] for a more general setting. Both
results are well known for special cases of $\nu$ in the area of peridynamics,
see [DTZ22, Section 3.2] and [KMS19]. We refer the reader to the exposition
and the references in [Du19]. Let us mention some related results. An early
work is [BT10] where several nonlocal complement value problems are studied
for integrable kernels with fixed support (horizon). Problems for nonlocal
nonlinear problems involving nonlocal operators of regional type are studied
in [BM14, BMCP15]. Nonlocal Dirichlet problems driven by nonsymmetric singular
kernels are considered in [FKV15] for scalar functions and in [KMS19] for
vector-valued functions. The vanishing-horizon limit has been considered in
several works, see Section 5. For references related to numerical results see
[DY21]. We provide $\Gamma$-convergence results for vanishing horizons in 5.2
and mention related results from peridynamics. Our systematic approach in
terms of functional analysis allows to treat general cases of $\nu$ resp.
general data $g$ in comparison with [DTZ22].

The paper is organized as follows. In Section 2 we introduce some nonlocal
function spaces and the corresponding trace spaces that will be used in the
sequel. In Section 3 we establish global compact embedding of nonlocal Sobolev
spaces in to $L^{2}$-spaces. This allows us to prove Poincaré type
inequalities for various ranges of Lévy integrable kernel $\nu$. The Section 4
is devoted to the study of the well-posedness of nonlocal problems with
Dirichlet, Neumann and Robin conditions associated with the Lévy operator $L$.
Afterwards, we investigate the Dirichlet-to-Neumann map for the Lévy operator
$L$. In Section 5 we show that local elliptic problems can be viewed as the
limit of the nonlocal ones. Last, in Appendix A we highlight some elementary
properties of the Lévy operator $L$.

_Acknowledgment:_ During his PhD studies Guy Foghem has spent a research stay
at Seoul National University in the framework of the International Research
Training Group 2235 “Searching for the regular in the irregular: Analysis of
random and singular systems” between Bielefeld University and Seoul National
University. The authors thank his host, Prof. Ki-Ahm Lee, for helpful
discussions on trace spaces.

_Further remarks:_ Several results of this work are based on the PhD thesis of
the first author [FG20]. Related research questions are subject to an ongoing
PhD project of Michael Vu from Trier University. Equivalent norms of the trace
space $T_{\nu}(\Omega^{c})$ in case of the fractional Laplace operator have
recently been identified by F. Grube and Th. Hensiek together with convergence
results that recover $H^{1/2}(\partial\Omega)$ in the limit $\alpha\to 2^{-}$.

##  2\. Lévy measures and 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
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and nonlocal trace spaces
$T_{\nu}(\Omega^{c})$. The function spaces are tailor-made for nonlocal
elliptic complement value problems including the Neumann problem. We prove the
existence of an embedding of $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$
into weighted spaces $L^{2}(\operatorname{\mathbb{R}}^{d},\nu^{\prime})$ for
different measures $\nu^{\prime}$ and into the nonlocal trace space
$T_{\nu}(\Omega^{c})$. We are able to compare $T_{\nu}(\Omega^{c})$ with known
trace spaces, see 2.32. A main result of this section is Theorem 2.26, which
proves that the bilinear form
$(\operatorname{\mathcal{E}},V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}))$
is a regular Dirichlet form on
$L^{2}(\operatorname{\mathbb{R}}^{d},\nu^{\prime})$. This result allows to
construct jump processes with some sort of reflection.

Throughout this work, let $\nu$ be a Lévy measure whose density is a
measurable symmetric function
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ satisfying
(L). We will impose further conditions on $\nu$ where needed. For simplicity,
we assume in our main results that $\nu$ has full support. See Section 1.3 for
a discussion on how this can be relaxed.

###  2.1. Lévy condition and energy forms

Before we begin, let us make an observation that nicely links (L) with
nonlocal energies.

######  Theorem 2.1.

Assume $\nu:\mathbb{R}^{d}\setminus\\{0\\}\to[0,\infty)$ is measurable and
radial 111Note added in proof: As we learned from Florian Grube, the assertion
of the theorem holds true for any Borel measure $\nu$. The $L^{p}$-setting is
treated in [Fog23a].. Then the energy

$\displaystyle\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\big{(}u(y)\\!-\\!u(x)\big{)}^{2}\nu(x\\!-\\!y)\;\mathrm{d}y\mathrm{d}x$

is finite for every $u\in C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$ if
and only if $\nu$ satisfies (L).

###### Proof.

By (2.7), it is easy to see that condition (L) implies finiteness of the
energy for $u\in C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$. For the
converse, let $u\in C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$ be
nontrivial and $\varepsilon>0$, then there is $\delta>0$ such that

$\displaystyle\|\nabla u(\cdot+h)-\nabla
u\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})}<\varepsilon\quad\text{ if
}|h|\leq\delta\,.$

Using the fundamental theorem of calculus, $\frac{1}{2}b^{2}\leq
a^{2}+(b-a)^{2}$ and polar coordinates yields

$\displaystyle\begin{split}\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}&\big{(}u(y)\\!-\\!u(x)\big{)}^{2}\nu(x\\!-\\!y)\;\mathrm{d}y\mathrm{d}x\geq\int_{\operatorname{\mathbb{R}}^{d}}\int_{B_{\delta}(0)}\Big{|}\int_{0}^{1}\nabla
u(x+th)\cdot h\mathrm{d}t\Big{|}^{2}\nu(h)\mathrm{d}h\,\mathrm{d}x\\\
&\geq\frac{1}{2}\int_{\operatorname{\mathbb{R}}^{d}}\int_{\mathbb{S}^{d-1}}|\nabla
u(x)\cdot
w|^{2}\mathrm{d}\sigma_{d-1}(w)\Big{(}\int_{0}^{\delta}r^{d+1}\nu(r)\mathrm{d}r\Big{)}\mathrm{d}x-\varepsilon\int_{B_{\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h\\\
&\geq\Big{(}\frac{1}{2}K_{d,2}\|\nabla
u\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})}-\varepsilon\Big{)}\int_{B_{\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h.\end{split}$

Recall that, invariance of the Lebesgue measure under rotations implies for
all $z\in\operatorname{\mathbb{R}}^{d}$

$\displaystyle\fint_{\mathbb{S}^{d-1}}|w\cdot
z|^{2}\mathrm{d}\sigma_{d-1}(w)=K_{d,2}|z|^{2}\quad\text{with}\quad
K_{d,2}=\fint_{\mathbb{S}^{d-1}}|w\cdot
e|^{2}\mathrm{d}\sigma_{d-1}(w)=\frac{1}{d}.$

Therefore, choosing $\varepsilon=\frac{1}{4}K_{d,2}\|\nabla
u\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})},$ we obtain

$\displaystyle\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\big{(}u(y)\\!-\\!u(x)\big{)}^{2}\nu(x\\!-\\!y)\;\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq\frac{1}{4}K_{d,2}\|\nabla
u\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})}\int_{B_{\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h.$

It remains to show that $\nu$ is integrable away from the origin. Consider
another $u\in C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$ with
$\operatorname{supp}u\subset B_{\tau}(0)$ and $0<\tau<\delta/2$. For all $x\in
B_{\tau}$ we have $B_{\tau}(0)\subset B_{\delta}(x)$ and hence

$\displaystyle\iint_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\big{(}u(y)\\!-\\!u(x)\big{)}^{2}\nu(x\\!-\\!y)\;\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq
2\int_{B_{\tau}(0)}|u(x)|^{2}\mathrm{d}x\int_{\operatorname{\mathbb{R}}^{d}\setminus
B_{\tau}(0)}\nu(x-y)\mathrm{d}y$ $\displaystyle\geq
2\int_{B_{\tau}(0)}|u(x)|^{2}\mathrm{d}x\int_{\operatorname{\mathbb{R}}^{d}\setminus
B_{\delta}(0)}\nu(h)\mathrm{d}h.$

This together with the previous estimate implies that $\nu\in
L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{2}\mathrm{d}h)$, i.e., $\nu$
satisfies condition $\eqref{eq:levy-cond}$. ∎

###  2.2. Sobolev-Slobodeckij-like spaces

Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open. Define the space
$H_{\nu}(\Omega)$ by

$\displaystyle H_{\nu}(\Omega)=\Big{\\{}u\in
L^{2}(\Omega)\,:\,\operatorname{\mathcal{E}}_{\Omega}(u,u)<\infty\Big{\\}}\,,$

equipped with the norm
$\|u\|^{2}_{H_{\nu}(\Omega)}=\|u\|^{2}_{L^{2}(\Omega)}+\operatorname{\mathcal{E}}_{\Omega}(u,u)$,
where

$\displaystyle\operatorname{\mathcal{E}}_{\Omega}(u,v)=\iint\limits_{\Omega\Omega}\big{(}u(x)-u(y)\big{)}\big{(}v(x)-v(y)\big{)}\,\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\,.$
(2.1)

When $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})$, e.g., in the case
$\nu(h)=\mathbbm{1}_{B_{1}}(h)$, the space $H_{\nu}(\Omega)$ equals
$L^{2}(\Omega)$.

Following [FKV15] we introduce the vector space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ as follows:

$\displaystyle
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})=\Big{\\{}u:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}\text{
meas.}:\,u|_{\Omega}\in
L^{2}(\Omega),|u|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}<\infty\Big{\\}},$

where the seminorm is defined by

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

We endow the vector space $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ with
the norm $\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ given by

$\displaystyle\|u\|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=\|u\|^{2}_{L^{2}(\Omega)}+|u|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\,.$

Next, given functions $u,v\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$,
we define a bilinear form $\operatorname{\mathcal{E}}$ by

$\displaystyle\mathcal{E}(u,v)=\frac{1}{2}\\!\\!\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u(x)-u(y)\big{)}\big{(}v(x)-v(y)\big{)}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y\,.$

######  Lemma 2.2.

We have
$|u|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\leq\mathcal{E}(u,u)\leq
2|u|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ for any measurable
function $u$.

###### Proof.

On the one hand, the inequality

$\displaystyle\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}\\!\\!\big{(}u(x)-u(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x\
\leq\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\big{(}u(x)-u(y)\big{)}^{2}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x=2\operatorname{\mathcal{E}}(u,u)$

holds trivially true. On the other hand,

$\displaystyle\mathcal{E}(u,u)$
$\displaystyle=\frac{1}{2}\\!\\!\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\\!\\!\big{(}u(x)-u(y)\big{)}^{2}\,\big{[}\mathds{1}_{\Omega}(x)\lor\mathds{1}_{\Omega}(y)\big{]}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{2}\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\\!\\!\big{(}u(x)-u(y)\big{)}^{2}\,\big{[}\mathds{1}_{\Omega}(x)+\mathds{1}_{\Omega}(y)\big{]}\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}\big{(}u(x)-u(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x\,,$

which completes the proof. ∎

Some authors find it convenient to work with the smaller space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d})$ equipped with its corresponding norm.
For the study of nonlocal Dirichlet problems this restriction is not
necessary, though. On the other hand, the requirement $u|_{\Omega}\in
L^{2}(\Omega)$ for $u\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is
natural as shown by the following observation.

######  Proposition 2.3.

Let $\nu$ be a unimodal Lévy measure and
$\Omega\subset\operatorname{\mathbb{R}}^{d}$ be a bounded open set. Assume
$\Omega\subset B_{R/2}(0)$ for some $R\geq 1$ with $\nu(R)\neq 0$. Then
$\operatorname{\mathcal{E}}(u,u)<\infty$ implies $u|_{\Omega}\in
L^{2}(\Omega)$.

The condition that $\nu$ is unimodal is not restrictive at all and we recall
its definition for the readers’ convenience.

######  Definition 2.4.

A Lévy density $\nu$ is called _unimodal_ if it is radial with an almost
decreasing profile, i.e., there is a constant $c\geq 1$ such that $\nu(r)\leq
c\,\nu(s)$ for all $r,s>0$ with $s\leq r$.

######  Remark 2.5.

There are radial Lévy measures, which are not almost decreasing such as

$\nu(x)=|x|^{-d-1}\big{(}\tfrac{2+\cos(|x|)}{3}\big{)}^{|x|^{4}}\,.$

###### Proof of 2.3.

First, since $\Omega\subset B_{R/2}(0)$, then for all $x,y\in\Omega$ we have
$\nu(x\\!-\\!y)\geq c^{\prime}$ with $c^{\prime}=c\nu(R)>0$. By Jensen’s
inequality, we have

$\displaystyle\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\big{(}u(x)-u(y)\big{)}^{2}\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\geq
c^{\prime}\iint\limits_{\Omega\Omega}(|u(x)|-|u(y)|)^{2}\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\geq
c^{\prime}|\Omega|\int_{\Omega}\Big{(}|u(x)|-\hbox{$\fint_{\Omega}|u|$}\Big{)}^{2}\mathrm{d}x.$

This shows that the mean value $\fint_{\Omega}|u|$ is finite. We conclude
$u\in L^{2}(\Omega)$ because of

$\int_{\Omega}|u(x)|^{2}\mathrm{d}x\leq
2\int_{\Omega}\Big{(}|u(x)|-\hbox{$\fint_{\Omega}$}|u|\Big{)}^{2}\mathrm{d}x+2|\Omega|\Big{(}\hbox{$\fint_{\Omega}|u|$}\Big{)}^{2}\,.$

∎

######  Definition 2.6.

We define $V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ as follows:

$\displaystyle\begin{split}V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})&=\\{u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
|\leavevmode\nobreak\ u=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{a.e.
on }\operatorname{\mathbb{R}}^{d}\setminus\Omega\\}\\\ &=\\{u\in
H_{\nu}(\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
|\leavevmode\nobreak\ u=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{a.e.
on }\operatorname{\mathbb{R}}^{d}\setminus\Omega\\}\,.\end{split}$ (2.2)

######  Remark 2.7.

The nonlocal Sobolev spaces $H_{\nu}(\Omega)$,
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ are well suited for nonlocal
linear problems. Interested readers may consult [FG20, Fog23b, Fog21, GH22,
Fog23a, GK23] for further expositions on this type of nonlocal function spaces
including the $L^{p}$-setting and trace resp. extension results.

######  Remark 2.8.

The function space $\big{(}H_{\nu}(\Omega),\|\cdot\|_{H_{\nu}(\Omega)}\big{)}$
is a separable Hilbert space, see [FKV15, FKV20]. The norms
$\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|_{H_{\nu}(\operatorname{\mathbb{R}}^{d})}$ agree on
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ and
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a closed subspace of
$H_{\nu}(\operatorname{\mathbb{R}}^{d})$, hence a Hilbert space.

######  Proposition 2.9.

If $\nu$ has full support, then
$\big{(}V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}),\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\big{)}$
is a separable Hilbert space.

We refer the reader to [FKV15, DROV17] for a proof in a special setting and to
[FG20, Thm 3.46] for the general case.

It is worthwhile noticing that
$\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ is always a norm
on $V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$, but not in general a
norm on $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ 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
$\|u\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=0$ whereas $u\neq 0$.
With regard to this comment and the discussion in Section 1.3 let us define
$V_{\nu}(\Omega|E)$ for $\Omega\subset E\subset\operatorname{\mathbb{R}}^{d}$:

$\displaystyle V(\Omega|E)=\Big{\\{}u:E\to\operatorname{\mathbb{R}}\text{
meas.}:\,u|_{\Omega}\in
L^{2}(\Omega),|u|^{2}_{V_{\nu}(\Omega|E)}<\infty\Big{\\}},$ (2.3)

where the seminorm is defined by

$\displaystyle|u|^{2}_{V_{\nu}(\Omega|E)}=\iint\limits_{\Omega
E}\big{(}u(x)-u(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}y\,\mathrm{d}x<\infty\,.$

$V_{\nu}(\Omega|E)$ is a seminormed space with respect to the seminorm
$\|u\|^{2}_{V_{\nu}(\Omega|E)}=\|u\|^{2}_{L^{2}(\Omega)}+|u|^{2}_{V_{\nu}(\Omega|E)}$.
We refer to [Fog23a] for the proof of the next result.

######  Proposition 2.10.

If $E=\Omega+\operatorname{supp}(\nu)$ and $0\in\operatorname{supp}(\nu)$ then
$\big{(}V_{\nu}(\Omega|E),\|\cdot\|_{V_{\nu}(\Omega|E)}\big{)}$ is a separable
Hilbert space, whenever and $\omega>0$ a.e. on $E\setminus\Omega$ where we put

$\displaystyle\omega(x)=\int_{\Omega}(1\land\nu(x-y))\mathrm{d}y.$

As already seen in Theorem 2.1, the condition (L) is important for the
properties of the spaces $H_{\nu}(\Omega)$ and
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$.

######  Proposition 2.11.

Let $\nu:\operatorname{\mathbb{R}}^{d}\to[0,\infty]$ be symmetric. The
following assertions hold true.

  1. $(i)$

If $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})$, then
$H_{\nu}(\operatorname{\mathbb{R}}^{d})=L^{2}(\operatorname{\mathbb{R}}^{d})$
with equivalence in norm.

  2. $(ii)$

If $\nu\in L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{2}\mathrm{d}h)$ and
$\Omega$ is bounded, then $H_{\nu}(\Omega)$ and
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ contain all bounded Lipschitz
functions.

  3. $(iii)$

If $\nu$ is radial and $\int_{B_{1}}|h|^{2}\nu(h)\mathrm{d}h=\infty$ and
$\int_{B_{1}\setminus B_{\delta}}\nu(h)\mathrm{d}h<\infty$ for every
$\delta>0$, then $u\in C^{1}(\operatorname{\mathbb{R}}^{d})\cap
H_{\nu}(\operatorname{\mathbb{R}}^{d})$ implies that $u$ is constant.

  4. $(iv)$

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

$\displaystyle\frac{1}{4d}\Big{(}\int_{B_{\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h\Big{)}\|\nabla
u\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})}\leq|u|^{2}_{H\nu(\operatorname{\mathbb{R}}^{d})}\leq
4\|\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{2}\mathrm{d}h)}\|u\|^{2}_{H^{1}(\operatorname{\mathbb{R}}^{d})}.$
(2.4)

The proof is analogous to the one of Theorem 2.1. See also [Fog23b,
Proposition 2.14] or [FG20, Proposition 3.46] for a general setting.

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 2.12.

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

  1. $(i)$

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

  2. $(ii)$

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

  3. $(iii)$

If $\Omega$ has a compact continuous boundary $\partial\Omega$, then
$C_{c}^{\infty}(\Omega)$ is dense in
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$.

  4. $(iv)$

If $\Omega$ has a compact Lipschitz boundary $\partial\Omega$, then
$C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$ is dense in
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ with respect to the norms
$\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
with
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}=\|u\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})}+|u|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$.

The proofs of the first and second statement can be found in [FG20] and
[DK21]. The first statement is similar to a Meyers-Serrin density type result.
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, Gri11] for a special choice of $\nu$ and in
[FG20], [BGPR20] for the general case. The proof of the fourth assertion is
given in [FKV20].

######  Remark 2.13.

Concerning the question, whether is is necessary to assume the continuity of
$\partial\Omega$ for the density of $C_{c}^{\infty}(\Omega)$ in
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ or not, it is interesting to
compare [FSV15, Remark 7] with [CF12, Theorem 3.3.9].

######  Remark 2.14.

For the kernel $\nu(h)=|h|^{-d-\alpha}$, $\alpha\in(0,2)$, let
$V^{\alpha/2}(\Omega|\operatorname{\mathbb{R}}^{d})$,
$V^{\alpha/2}_{0}(\Omega|\operatorname{\mathbb{R}}^{d})$ and
$H^{\alpha/2}(\Omega)$ be the spaces
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$,
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ and $H_{\nu}(\Omega)$
respectively. It is worth nothing that, see [Gri11], if $\Omega$ has a compact
Lipschitz boundary and $\alpha\neq 1$ then

$\displaystyle
V^{\alpha/2}_{0}(\Omega|\operatorname{\mathbb{R}}^{d})=\overline{C^{\infty}_{c}(\Omega)}^{V^{\alpha/2}(\Omega|\operatorname{\mathbb{R}}^{d})}=\overline{C^{\infty}_{c}(\Omega)}^{H^{\alpha/2}(\operatorname{\mathbb{R}}^{d})}=\overline{C^{\infty}_{c}(\Omega)}^{H^{\alpha/2}(\Omega)},$

where the first and the second equality follow from Theorem 2.12 $(ii)$.
Furthermore, if $0<\alpha<1$ then we also have
$H^{\alpha/2}(\Omega)=\overline{C^{\infty}_{c}(\Omega)}^{H^{\alpha/2}(\Omega)}.$

###  2.3. Weighted $L^{2}$-spaces

In order to set up the Dirichlet problem in $L^{2}$-spaces over
$\operatorname{\mathbb{R}}^{d}$, we define a Borel measure on
$\operatorname{\mathbb{R}}^{d}$ that captures the behavior of $\nu$ at
infinity. There are several possibilities.

######  Definition 2.15.

Let $\nu$ satisfies (L) with full support and
$B\subset\operatorname{\mathbb{R}}^{d}$ be non-empty and open. Define the
weights
$\overline{\nu},\widetilde{\nu}:\operatorname{\mathbb{R}}^{d}\to[0,\infty]$ by

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

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

######  Example 2.16.

Let $0<\alpha<2$ and $\nu(h)=|h|^{-d-\alpha}$ for $h\neq 0$. Let
$B\subset\operatorname{\mathbb{R}}^{d}$ be open and bounded, and $R>1$. Then

$\displaystyle\widetilde{\nu}(x)\asymp\overline{\nu}(x)\asymp\nu^{*}(x)\asymp(1+|x|)^{-d-\alpha}\,,$

where the constants behind the relation $\asymp$ depend on the choice of $B$
and $R$. See Theorem 2.22 for the general case.

Let us discuss important properties of the three measures $\widetilde{\nu}$,
$\overline{\nu}$, and $\nu^{*}$. The following lemmas show that it is possible
to define certain norms on $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$
(with nice properties), which are equivalent to the norm
$\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$.

######  Lemma 2.17 (Properties of $\widetilde{\nu}$).

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

  1. $(i)$

We have $\widetilde{\nu}\in L^{\infty}(\operatorname{\mathbb{R}}^{d})$.
Moreover, if $|B|<\infty$, then $\widetilde{\nu}\in
L^{1}(\operatorname{\mathbb{R}}^{d})$.

  2. $(ii)$

The embedding $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})$ is continuous. If
$|B|<\infty$, then
$L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})\hookrightarrow
L^{1}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})$ is continuous.

  3. $(iii)$

If $\nu$ is unimodal and $B$ is bounded, then on
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, the norms
$\|\cdot\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are
equivalent, where

$\displaystyle\|u\|^{*2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{d}}|u(x)|^{2}\widetilde{\nu}(x)\mathrm{d}x+\iint\limits\limits_{(\Omega^{c}\times\Omega^{c})^{c}}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\,,$
$\displaystyle\|u\|^{\\#2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
$\displaystyle=\int_{\Omega}|u(x)|^{2}\widetilde{\nu}(x)\mathrm{d}x+\iint\limits\limits_{(\Omega^{c}\times\Omega^{c})^{c}}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\,.$

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

###### Proof.

Firstly, we observe $1\land\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})$. It
follows
$\widetilde{\nu}(x)\leq\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}$
for almost every $x\in\operatorname{\mathbb{R}}^{d}$. If $|B|<\infty$, then
Fubini’s theorem implies

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}\widetilde{\nu}(x)\mathrm{d}x\leq|B|\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}<\infty\,.$

The continuous embedding
$L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})\hookrightarrow
L^{1}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})$ follows directly. The
continuity of the embedding
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})$ is obtained as follows

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}|u(x)|^{2}\widetilde{\nu}(x)\mathrm{d}x$
$\displaystyle\leq
2\int_{B}|u(y)|^{2}\left(\int_{\operatorname{\mathbb{R}}^{d}}1\land\nu(x-y)\mathrm{d}x\right)\mathrm{d}y+2\iint\limits_{B\operatorname{\mathbb{R}}^{d}}(u(x)-u(y))^{2}1\land\nu(x-y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq
C_{1}\int_{\Omega}|u(y)|^{2}\mathrm{d}y+C_{1}\iint\limits_{\Omega\operatorname{\mathbb{R}}^{d}}(u(x)-u(y))^{2}\nu(x-y)\mathrm{d}x\,\mathrm{d}y=C_{1}\|u\|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}.$

Here $C_{1}=2\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}+2$. The
following inequalities obviously hold

$\displaystyle\sqrt{C_{1}+1}\,\|u\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\geq\|u\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\geq\|u\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}.$

Next if $\nu$ is unimodal and $B$ is bounded, then there is a constant
$c^{\prime}>0$ such that $\widetilde{\nu}(x)\geq c^{\prime}$ for all $x\in B$.
The following estimates hold

$\displaystyle\int_{\Omega}|u(x)|^{2}$
$\displaystyle\widetilde{\nu}(x)\mathrm{d}x+\iint\limits_{\Omega\Omega^{c}}(u(x)-u(y))^{2}\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq
c^{\prime}\int_{B}|u(x)|^{2}\mathrm{d}x+\iint\limits_{\Omega\Omega^{c}}(u(x)-u(y))^{2}\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq
c^{\prime}\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}^{-1}\iint\limits_{B\Omega^{c}}|u(x)|^{2}1\land\nu(x-y)\mathrm{d}y\mathrm{d}x+\iint\limits_{B\Omega^{c}}(u(x)-u(y))^{2}\nu(x-y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq(1\land
c^{\prime}\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}^{-1})\iint\limits_{\Omega^{c}B}\Big{[}|u(x)|^{2}+(u(x)-u(y))^{2}\Big{]}1\land\nu(x-y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\geq\frac{1}{2}(1\land
c^{\prime}\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}^{-1})\int_{\Omega^{c}}u^{2}(y)\widetilde{\nu}(y)\mathrm{d}y\,.$

The first and the last line imply that
$\|u\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\leq
C\|u\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ for some
constant $C>0$. Thus the norms
$\|\cdot\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are
equivalent. If in addition $\Omega$ is bounded, then
$\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\geq\widetilde{\nu}(x)\geq
c^{\prime}$ for all $x\in\Omega$ for some $c^{\prime}>0$. The equivalence of
the norms $\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$and
$\|\cdot\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ is thus
proved. ∎

######  Lemma 2.18 (Properties of $\overline{\nu}$).

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open. Let $\nu$
satisfies (L) with full support. Assume $B\subset\Omega$ is open and nonempty.

  1. $(i)$

$\overline{\nu}\in L^{1}(\operatorname{\mathbb{R}}^{d})$ and, if $\nu$ is
unimodal, then $\overline{\nu}\in L^{\infty}(\operatorname{\mathbb{R}}^{d})$.

  2. $(ii)$

The embeddings $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d},\overline{\nu})\hookrightarrow
L^{1}(\operatorname{\mathbb{R}}^{d},\overline{\nu})$ are continuous.

  3. $(iii)$

If $\nu$ is unimodal and $B$ is bounded then the norms
$\|\cdot\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are
equivalent, where $\|u\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
and $\|u\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are defined
as in 2.17(iii) with $\widetilde{\nu}$ replaced by $\overline{\nu}$.

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

###### Proof.

To prove $(i)$, select $x_{0}\in B$ and $r>0$ such that $B_{8r}(x_{0})\subset
B$. Note that, for $x\in B_{2r}(x_{0})$ and $y\in B^{c}_{4r}(x_{0})\cap B$ or
for $x\in B^{c}_{2r}(x_{0})$ and $y\in B_{r}(x_{0})$ we have $|h|\geq r$ where
$h=x-y$. Therefore, for almost all $\xi\in B^{c}_{4r}(x_{0})$ and $z\in
B_{r}(x_{0})$ we find that

$\displaystyle\int_{B_{2r}(x_{0})}\underset{y\in B^{c}_{4r}(x_{0})\cap
B}{\operatorname*{ess\,inf}}\nu(x-y)\mathrm{d}x$
$\displaystyle\leq\int_{|x-y|\geq r}\nu(x-\xi)\mathrm{d}x=\int_{|h|\geq
r}\nu(h)\mathrm{d}h,$ $\displaystyle\int_{B^{c}_{2r}(x_{0})}\underset{y\in
B_{r}(x_{0})}{\operatorname*{ess\,inf}}\hskip 5.69054pt\nu(x-y)\mathrm{d}x$
$\displaystyle\leq\int_{|x-z|\geq r}\nu(x-z)\mathrm{d}x=\int_{|h|\geq
r}\nu(h)\mathrm{d}h.$

The integrability of $\overline{\nu}$ follows since

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}\overline{\nu}(x)\mathrm{d}x$
$\displaystyle\leq\int_{B_{2r}(x_{0})}\underset{y\in B^{c}_{4r}(x_{0})\cap
B}{\operatorname*{ess\,inf}}\nu(x-y)\mathrm{d}x+\int_{B^{c}_{2r}(x_{0})}\underset{y\in
B_{r}(x_{0})}{\operatorname*{ess\,inf}}\nu(x-y)\mathrm{d}x\leq 2\int_{|h|\geq
r}\nu(h)\mathrm{d}h<\infty.$

Analogously, assume $\nu$ is unimodal. Since for $(x,y)\in B_{2r}(x_{0})\times
B^{c}_{4r}(x_{0})\cap B$ or for $(x,y)\in B^{c}_{2r}(x_{0})\times
B_{2r}(x_{0})$ we have $|x-y|\geq r$, it follows that
$\overline{\nu}_{B}(x)\leq\nu(x-y)\leq c\nu(r)$.

Next, consider $K^{\prime}\subset B$ to be a measurable subset such that
$0<|K^{\prime}|<\infty$, then we have

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}|u(x)|^{2}\overline{\nu}(x)\mathrm{d}x$
$\displaystyle\leq
2|K^{\prime}|^{-1}\|\overline{\nu}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\int_{K^{\prime}}|u(y)|^{2}\mathrm{d}y+2|K^{\prime}|^{-1}\iint\limits_{K^{\prime}\operatorname{\mathbb{R}}^{d}}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq
C\|u\|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})},\quad\qquad$

where
$C=2|K^{\prime}|^{-1}(\|\overline{\nu}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}+1).$
This together with the previous step imply the continuity of the embeddings
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d},\overline{\nu})\hookrightarrow
L^{1}(\operatorname{\mathbb{R}}^{d},\overline{\nu})$. The rest of the proof is
analogous to that of 2.17. ∎

######  Lemma 2.19 (Properties of $\nu^{*}$).

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and $R>1$
satisfies $|B_{R}(0)\cap\Omega|>0$ and $|B_{R}(0)\cap\Omega^{c}|>0$. Let $\nu$
satisfies (L) with full support.

  1. $(i)$

$\nu^{*}\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$.

  2. $(ii)$

The embeddings $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d},\nu^{*})\hookrightarrow
L^{1}(\operatorname{\mathbb{R}}^{d},\nu^{*})$ are continuous.

  3. $(iii)$

On $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, the norms
$\|\cdot\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ and
$\|\cdot\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are
equivalent, where $\|u\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
and $\|u\|^{\\#}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ are defined
as in 2.17(iii) with $\widetilde{\nu}$ replaced by $\nu^{*}$.

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

The proof of 2.19 is analogous to that of 2.17 and can be found in [FKV20] or
[FG20, Lemma 3.24]. In order to show $\nu^{*}\in
L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$, one notes that for all
$h\in\operatorname{\mathbb{R}}^{d}$ we have $R\leq R(1+|h|)$ and $|h|\leq
R(1+|h|)$ and hence $\nu^{*}(h)\leq C(1\land\nu(h))$. This implies the claim
because of $1\land\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})\cap
L^{\infty}(\operatorname{\mathbb{R}}^{d})$.

The measures $\nu^{*}$, $\overline{\nu}$ and $\widetilde{\nu}$ turn out to be
comparable if $\nu$ satisfies a certain doubling condition at infinity.

######  Definition 2.20.

A radial Lévy density $\nu$ satisfies a doubling condition at infinity if:

$\displaystyle\text{For every }\theta\geq 1\text{ there exist
}c_{1},c_{2}>0\text{ with }c_{1}\nu(r)\leq\nu(\theta r)\leq c_{2}\nu(r)\text{
for all }r\geq 1\,.$ (2.5)

Not that the property (2.5) is indeed equivalent to saying that

$\displaystyle\text{ There exist }c_{1},c_{2}>0\text{ with
}c_{1}\nu(r)\leq\nu(2r)\leq c_{2}\nu(r)\text{ for all }r\geq 1.$ (2.6)

######  Remark 2.21.

The doubling condition at infinity imposes some decay of $\nu$ at infinity.
The example $\nu(h)=|h|^{-d-1}\mathds{1}_{\\{|h|\leq 7\\}}$ satisfies 2.4 but
not (2.5). Unimodality bounds one-sided oscillations of $\nu$ for all values
of $|x|$. Fix $0<\alpha<\beta<2$. Define $\nu_{1}(r)=r^{-d-\alpha}$ for $r\geq
1$. Define $\nu_{1}(r)=r^{-d-\beta}$ for $\frac{1}{2k+1}\leq r<\frac{1}{2k}$
and $\nu_{1}(r)=r^{-d-\alpha}$ for $\frac{1}{2k+2}\leq r<\frac{1}{2k+1}$ for
$k\in\mathbb{N}$. Then $\nu_{1}$ is not unimodal but it trivially does satisfy
(2.5).

######  Theorem 2.22.

Assume that $\nu$ is unimodal and satisfies (2.5). For
$B\subset\operatorname{\mathbb{R}}^{d}$ is bounded (e.g. B is a ball) and
$R\geq 1$ we have
$\widetilde{\nu}(x)\asymp\overline{\nu}(x)\asymp\nu^{*}(x)\asymp
1\land\nu(x)$.

###### Proof.

Let us observe that, $\widetilde{\nu},\overline{\nu},\nu^{*}$ and $1\land\nu$
are all bounded above. Indeed, for $x\in\operatorname{\mathbb{R}}^{d}$ we have
$R\leq R(1+|x|)$ and $|x|\leq R(1+|x|)$ and hence $\nu^{*}(x)\leq
C(1\land\nu(x))\leq C$. Obviously,
$\widetilde{\nu}(x)\leq\|1\land\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}$.
Now, let $r>0$ sufficiently small and $x_{0}\in B$ such that
$B_{8r}(x_{0})\subset B$. If $x\in B_{2r}(x_{0})$ and $y\in
B^{c}_{4r}(x_{0})\cap B$ or if $x\in B^{c}_{2r}(x_{0})$ and $y\in
B_{2r}(x_{0})$ then $|x-y|\geq r$. In both cases,
$\overline{\nu}(x)\leq\nu(x-y)\leq c\nu(r)$.

Next, there is no loss of generality if we assume that $B\subset B_{R}.$
Assume $|x|\leq 4R$ then $|x-y|\leq 5R$ for all $y\in B$. The unimodality and
the foregoing boundedness imply that $c^{-1}\nu(R(1+R))\leq\nu(R(1+|x|))\leq
C$, $c^{-1}\nu(5R)\leq\overline{\nu}(x)$, $1\land\nu(4R)\leq c1\land\nu(x)\leq
C$ and that $1\land\nu(5R)\leq c1\land\nu(x-y)$ that is $|B|1\land\nu(5R)\leq
c\widetilde{\nu}(x)\leq C.$ Thus
$\widetilde{\nu}(x)\asymp\overline{\nu}(x)\asymp\nu^{*}(x)\asymp 1\land\nu(x)$
for $|x|\leq 4R$.

Now assume that $|x|\geq 4R,$ then we have $1\leq\frac{|x|}{2}\leq|x-y|\leq
2|x|$ for all $y\in B$ and $|x|\leq R(1+|x|)\leq 2R|x|$. The doubling
condition (2.5) implies that for some constants $0<c_{1}<1<c_{2}$, we have
$c_{1}\nu(x)\leq\nu(R(1+|x|))\leq c_{2}\nu(x)$ and
$c_{1}\nu(x)\leq\nu(x-y)\leq c_{2}\nu(x)$ for all $y\in B$. We get
$c_{1}\nu(x)\leq\overline{\nu}(x)\leq c_{2}\nu(x)$ and integrating over $B$
implies that $c_{1}|B|1\land\nu(x)\leq c\widetilde{\nu}(x)\leq
c_{2}|B|1\land\nu(x).$ Together with the boundedness implies
$\widetilde{\nu}(x)\asymp\overline{\nu}(x)\asymp\nu^{*}(x)\asymp 1\land\nu(x)$
for $|x|\geq 4R$.

∎

######  Example 2.23.

As in Example 2.16, let $0<\alpha<2$ and $\nu(h)=|h|^{-d-\alpha}$ for $h\neq
0$. Then $\widetilde{\nu}(x)\asymp 1\land\nu(x)\asymp(1+|h|)^{-d-\alpha}$ and
the space $H_{\nu}(\Omega)$ equals the classical Sobolev-Slobodeckij space
$H^{\alpha/2}(\Omega)$. In this case, we denote the space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ by
$V^{\alpha/2}(\Omega|\operatorname{\mathbb{R}}^{d})$. We have
$V^{\alpha/2}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d},(1+|h|)^{-d-\alpha})$.

######  Remark 2.24.

Note that 2.17 $(i)$, 2.18 $(i)$ and 2.19 $(i)$ imply that the weights
$\widetilde{\nu},\nu^{*},\overline{\nu}$ respectively define Radon measures on
$\operatorname{\mathbb{R}}^{d}$.

###  2.4. Dirichlet forms

The discussion of the $L^{2}$-spaces related to
$\widetilde{\nu},\nu^{*},\overline{\nu}$ together with density results in
Theorem 2.12 allows us to define a new interesting Dirichlet form. We refer to
[FOT11] for the general theory of Dirichlet forms and their corresponding
Markov processes.

The following well-known result is a direct consequence of Theorem 2.12, (ii)
and (iii) .

######  Proposition 2.25.

Let $\Omega$ be open and bounded with a continuous boundary. Let $\nu$ be any
Lévy measure. Then each of the three bilinear forms
$(\operatorname{\mathcal{E}},V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d}))$,
$(\operatorname{\mathcal{E}}_{\Omega},\overline{C^{\infty}_{c}(\Omega)}^{H_{\nu}(\Omega)})$
and $(\operatorname{\mathcal{E}}_{\Omega},H_{\nu}(\Omega))$ is a regular
Dirichlet form on $L^{2}(\Omega)$. The corresponding Markov processes are
often called _killed_ , _censored_ resp. _reflected_ Lévy process.

An important side result of our work is the following theorem, which implies
the existence of a strong Markov process, which can be seen as another kind of
_reflected_ jump process with regards to $\Omega$. The theorem is an
improvement over [Von21, Theorem 4.4], see Remark 2.38. Its proof follows from
Theorem 2.12, (iv).

######  Theorem 2.26.

Let $\nu$ be unimodal with full support and
$\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open and bounded with a
Lipschitz-continuous boundary. Let $\nu^{\prime}$ be any of the measures
$\widetilde{\nu},\nu^{*},\overline{\nu}$ on $\operatorname{\mathbb{R}}^{d}$.
Then the bilinear form
$(\operatorname{\mathcal{E}},V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}))$
is a regular Dirichlet form on
$L^{2}(\operatorname{\mathbb{R}}^{d},\nu^{\prime})$.

###  2.5. Classical Sobolev spaces

Let us comment on the connection of the spaces under consideration with
classical Sobolev spaces. Recall that for an open set
$\Omega\subset\operatorname{\mathbb{R}}^{d}$, $H^{1}(\Omega)$ denotes the
classical Sobolev space endowed with the norm

$\|u\|^{2}_{H^{1}(\Omega)}=\|u\|^{2}_{L^{2}(\Omega)}+\|\nabla
u\|^{2}_{L^{2}(\Omega)}\,.$

######  Proposition 2.27.

The following embeddings hold true:

$\displaystyle H^{1}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
H_{\nu}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
H_{\nu}(\Omega)\hookrightarrow L^{2}(\Omega)\,.$

Here we equip $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ with the norm
$\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$.

###### Proof.

The proof is standard. For $u\in H^{1}(\operatorname{\mathbb{R}}^{d})$ and
$h\in\operatorname{\mathbb{R}}^{d}$ we have

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}(u(x+h)-u(x))^{2}\mathrm{d}x=\|u(\cdot+h)-u(\cdot)\|^{2}_{L^{2}(\operatorname{\mathbb{R}}^{d})}\leq
4(1\land|h|^{2})\|u\|^{2}_{H^{1}(\operatorname{\mathbb{R}}^{d})}\,.$

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

$\displaystyle\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\leq
4\|\nu\|_{L^{1}(\operatorname{\mathbb{R}}^{d},1\land|h|^{2}\mathrm{d}h)}\|u\|^{2}_{H^{1}(\operatorname{\mathbb{R}}^{d})}\,.$
(2.7)

This proves the first embedding, the remaining ones are trivial. ∎

Recall that $H^{1}_{0}(\Omega)$ is the closure of $C_{c}^{\infty}(\Omega)$
with respect to the $H^{1}(\Omega)$. The space $H^{1}_{0}(\Omega)$ also
coincides with the closure of $C_{c}^{\infty}(\Omega)$ in
$H^{1}(\operatorname{\mathbb{R}}^{d})$. In addition, the zero extension to
$\operatorname{\mathbb{R}}^{d}$ of any function in $H_{0}^{1}(\Omega)$ belongs
to $H^{1}(\operatorname{\mathbb{R}}^{d})$. Recall the definition of
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ from (2.2).

######  Proposition 2.28.

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

$\displaystyle H^{1}_{0}(\Omega)\hookrightarrow
V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
H_{\nu}(\Omega)\,$ where elements of $H_{0}^{1}(\Omega)$ are extended by zero
off $\Omega$. If additionally $\partial\Omega$ is continuous, $\displaystyle
V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow\overline{C^{\infty}_{c}(\Omega)}^{H_{\nu}(\Omega)}\hookrightarrow
L^{2}(\Omega)\,.$

It is worth noticing that not every function
$u\in\overline{C^{\infty}_{c}(\Omega)}^{H_{\nu}(\Omega)}$ has its extension by
zero in $H_{\nu}(\operatorname{\mathbb{R}}^{d})$. Indeed for this to hold, one
would need

$\displaystyle\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{\Omega^{c}}\nu(x\\!-\\!y)\mathrm{d}y<\infty.$

This condition is not always true since the measure $\nu$ might be very
singular at the origin. This observation shows that for some appropriate
domain $\Omega$ and for some appropriate measure $\nu$, e.g.
$\nu(h)=|h|^{-d-3/2}$, the spaces
$\overline{C^{\infty}_{c}(\Omega)}^{H_{\nu}(\Omega)}$ and
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ are strictly different
although they both possess $C_{c}^{\infty}(\Omega)$ as dense subspace. This
effect is purely nonlocal. Recall that elements of $H_{0}^{1}(\Omega)$ can be
isometrically extended by zero on $\operatorname{\mathbb{R}}^{d}$ as functions
of $H^{1}(\operatorname{\mathbb{R}}^{d})$.

Now assume $\Omega$ is a Lipschitz domain (or more generally an
$H^{1}$-extension domain). Let $\overline{u}\in
H^{1}(\operatorname{\mathbb{R}}^{d})$ be an extension of a function $u\in
H^{1}(\Omega)$ with
$\|\overline{u}\|_{H^{1}(\operatorname{\mathbb{R}}^{d})}\leq
C\|u\|_{H^{1}(\Omega)}$ for some constant $C$ depending only on $\Omega$ and
$d$. Within the estimate (2.7) we easily get the following continuous
embedding

######  Proposition 2.29.

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is an $H^{1}$-extension
domain then

$\displaystyle H^{1}(\Omega)\hookrightarrow H_{\nu}(\Omega)\,.$

The latter embedding may fail when $\Omega$ is not an extension domain (see
[Fog23b, Counterexample 3.8] or [NPV12, Example 9.1]). Note that
$H^{1}(\Omega)$ can be viewed as limiting space of a sequence of nonlocal
spaces of type $H_{\nu}(\Omega)$ and
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ see [Fog23b, Fog21, Fog23a]
for additional results.

###  2.6. Trace space of $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$

The main goal of this subsection is to discuss the trace space for
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ similarly as one does for the
classical Sobolev space $H^{1}(\Omega)$. Note that elements of
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ are defined on the whole of
$\operatorname{\mathbb{R}}^{d}$, thus the trace space consists of functions
defined on $\Omega^{c}$. This contrasts with the local situation, where the
trace space of $H^{1}(\Omega)$ consists of functions defined on the boundary
$\partial\Omega$. Unless otherwise stated, we assume that $\nu$ is fully
supported on $\operatorname{\mathbb{R}}^{d}$ and
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is endowed with the norm
$\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$. Note that, when
studying the fractional Laplace operator, trace spaces related to
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ when $\nu(h)=|h|^{-d-\alpha}$
for $h\neq 0$ and $0<\alpha<2$, have already been considered in [DK19],
[BGPR20] and [DTWY21]. Below, we comment on how our general approach relates
to these studies.

######  Definition 2.30.

We define $T_{\nu}(\Omega^{c})$ as the vector space of restrictions to
$\operatorname{\mathbb{R}}^{d}\setminus\Omega$ of functions of
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, i.e.,

$\displaystyle T_{\nu}(\Omega^{c})=\\{v:\Omega^{c}\to\mathbb{R}\text{
measurable }\;|\;v=u|_{\Omega^{c}}\text{ with }u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\\}.$

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

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

######  Theorem 2.31.

The space $T_{\nu}(\Omega^{c})$ is a separable Hilbert space with the scalar
product

$\displaystyle(u,v)_{T_{\nu}(\Omega^{c})}=\frac{1}{4}\Big{(}\|u+v\|^{2}_{T_{\nu}(\Omega^{c})}-\|u-v\|^{2}_{T_{\nu}(\Omega^{c})}\Big{)}.$

###### Proof.

Since the norm $\|\cdot\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
verifies the parallelogram law so does $\|\cdot\|_{T_{\nu}(\Omega^{c})}$. Thus
$\big{(}\cdot,\cdot\big{)}_{T_{\nu}(\Omega^{c})}$ is a scalar product on
$T_{\nu}(\Omega^{c})$ with associated norm $\|\cdot\|_{T_{\nu}(\Omega^{c})}$.
Noting that $T_{\nu}(\Omega^{c})$ and the quotient space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})/V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$
are identical with equal norm in space and that
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a closed subspace of
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}),$ one concludes that
$T_{\nu}(\Omega^{c})$ is complete. ∎

The main question now is whether the same space $T_{\nu}(\Omega^{c})$ can be
defined intrinsically. In other words, given a measurable function
$v:\Omega^{c}\to\mathbb{R}$, how can one decide whether the function belongs
to $T_{\nu}(\Omega^{c})$ or not. In the local situation, it is possible to
define a scalar product on the space $H^{1/2}(\partial\Omega)$ when $\Omega$
is a Lipschitz domain, see [Din96] for a proof.

We study this question in two settings, the one of [DK19] and the one of
[BGPR20]. For the special case $\nu(h)=|h|^{-d-\alpha}$ for $h\neq 0$ with
$0<\alpha<2$, it is proved in [DK19, Theorem 3] that for $v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ it holds

$\displaystyle\iint\limits\limits_{\Omega^{c}\Omega^{c}}\frac{\big{(}v(x)-v(y)\big{)}^{2}}{(|x-y|+\delta_{x}+\delta_{y})^{d+\alpha}}\mathrm{d}x\mathrm{d}y<\infty\,.$
(2.8)

Moreover, it is shown that, if (2.8) holds true for $v=g$ on $\Omega^{c}$,
there exists $u_{g}\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ such
that $u_{g}|_{\Omega^{c}}=g$ and

$\displaystyle\iint\limits\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\frac{\big{(}u_{g}(x)-u_{g}(y)\big{)}^{2}}{|x-y|^{d+\alpha}}\mathrm{d}x\mathrm{d}y\asymp\iint\limits\limits_{\Omega^{c}\Omega^{c}}\frac{\big{(}v(x)-v(y)\big{)}^{2}}{(|x-y|+\delta_{x}+\delta_{y})^{d+\alpha}}\mathrm{d}x\mathrm{d}y$
(2.9)

with the constants independent of $g$ and $u_{g}$. Therefore we obtain in this
case

$\displaystyle
T_{\nu}(\Omega^{c})=\Big{\\{}v:\Omega^{c}\to\mathbb{R}\leavevmode\nobreak\
\text{meas.}\big{|}\,\iint\limits_{\Omega^{c}\Omega^{c}}\\!\\!\frac{\big{(}v(x)-v(y)\big{)}^{2}}{(|x-y|+\delta_{x}+\delta_{y})^{d+\alpha}}\,\mathrm{d}x\,\mathrm{d}y<\infty\Big{\\}}\,,$
(2.10)

which we will make use of in 2.32.

Next, let us summarize the results of [BGPR20], which are established under
the following condition: One assumes that $\nu$ is radial and its profile
$\nu\in C^{2}((0,\infty))$ satisfies for some $C_{1},C_{2}>0$ and
$\beta\in(0,2)$

$\displaystyle\begin{split}|\nu^{\prime}(r)|+|\nu^{\prime\prime}(r)|&\leq
C_{1}\nu(r)\qquad(r>1)\,,\\\ \nu(\lambda r)&\leq
C_{2}\lambda^{d-\beta}\nu(r)\qquad(0<r,\lambda\leq 1),\\\ \nu(r)&\leq
C_{2}\nu(r+1)\qquad(r\geq 1)\,.\end{split}$ ($A_{\nu}$)

Assume $\Omega^{c}$ satisfies the volume density condition (in some works,
$\Omega^{c}$ is called a $d$-set), i.e., there exists a constant $c>0$ such
that $|\Omega^{c}\cap B_{r}(x)|\geq cr^{d}$ for all $x\in\partial\Omega$ and
all $r>0$. By the Lebesgue density theorem, the latter condition automatically
implies that $|\partial\Omega|=0$. Then under ($A_{\nu}$) [BGPR20, Theorem
2.3] proves that, for any $g\in T_{\nu}(\Omega^{c})$ there exists a unique
$u_{g}\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ such that
$u_{g}|_{\Omega^{c}}=g$ with

$\displaystyle\mathcal{H}_{\Omega}(g,g):=\iint\limits_{\Omega^{c}\Omega^{c}}\\!\\!\big{(}g(x)-g(y)\big{)}^{2}\,\gamma_{\Omega}(x,y)\mathrm{d}x\,\mathrm{d}y=\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u_{g}(x)-u_{g}(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y.$
(2.11)

The function $u_{g}$ satisfies the weak formulation

$\displaystyle\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u_{g}(x)-u_{g}(y)\big{)}(\phi(x)-\phi(y))\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y=0\quad\text{
for all }\phi\in V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ (2.12)

and the interaction kernel $\gamma_{\Omega}(x,y)$ is given via the Poisson
kernel of $\Omega$ by the formula

$\displaystyle\gamma_{\Omega}(x,y)=\int_{\Omega}P_{\Omega}(x,z)\nu(z-y)\mathrm{d}z\,\qquad
x,y\in\Omega^{c}\,.$

Furthermore, a precise formula for $u_{g}$ in $\Omega$ is given by the Poisson
integral

$\displaystyle
u_{g}(x)=P_{\Omega}[g](x)=\int_{\Omega^{c}}g(y)P_{\Omega}(x,y)\mathrm{d}y\,\qquad
x\in\Omega\,.$

From this, it is easy to show that

$\displaystyle
T_{\nu}(\Omega^{c})=\Big{\\{}v:\Omega^{c}\to\mathbb{R}\leavevmode\nobreak\
\text{meas.}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{H}_{\Omega}(v,v)=\iint\limits_{\Omega^{c}\Omega^{c}}\\!\\!\big{(}v(x)-v(y)\big{)}^{2}\,\gamma_{\Omega}(x,y)\mathrm{d}x\,\mathrm{d}y<\infty\Big{\\}}\,$

which is precisely the exterior space introduced in [BGPR20]. With this
definition, the connection between $T_{\nu}(\Omega^{c})$ and
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is less visible. For $v\in
T_{\nu}(\Omega^{c})$, by definition of $\|\cdot\|_{T_{\nu}(\Omega^{c})}$, we
have

$\displaystyle\|v\|^{2}_{T_{\nu}(\Omega^{c})}$
$\displaystyle=\inf\\{\|u\|^{2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}:\,u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\text{ with
}v=u|_{\Omega^{c}}\\}$
$\displaystyle\geq\inf\Big{\\{}\int_{\Omega}|u(x)|^{2}\mathrm{d}x:\,u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\text{ with
}v=u|_{\Omega^{c}}\Big{\\}}+\mathcal{H}_{\Omega}(v,v)\,.$

It is rather challenging to find or to estimate the quantity

$\displaystyle\inf\Big{\\{}\int_{\Omega}|u(x)|^{2}\mathrm{d}x:\leavevmode\nobreak\
\leavevmode\nobreak\ u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
\leavevmode\nobreak\ \hbox{with}\leavevmode\nobreak\
v=u|_{\Omega^{c}}\Big{\\}}.$

Remember that our goal here is to explicitly define a norm which is equivalent
to $\|\cdot\|_{T_{\nu}(\Omega^{c})}$ and has less visible connection to
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. To this end, we bring into
play the norm $\|\cdot\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$
defined in 2.17.

######  Proposition 2.32.

Assume $\Omega$ is open and bounded, such that $\Omega^{c}$ satisfies the
volume density condition. Assume $\nu$ satisfies ($A_{\nu}$). Let
$\widetilde{\nu}$ and
$\|\cdot\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}$ be
respectively the measure and the norm given in 2.17. Then

$\displaystyle
T_{\nu}(\Omega^{c})=\Big{\\{}v:\Omega^{c}\to\mathbb{R}\leavevmode\nobreak\
\text{meas.}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{H}_{\Omega}(v,v)=\iint\limits_{\Omega^{c}\Omega^{c}}\\!\\!\big{(}v(x)-v(y)\big{)}^{2}\,\gamma_{\Omega}(x,y)\mathrm{d}x\,\mathrm{d}y<\infty\Big{\\}}\,$

and the norms $\|\cdot\|_{T_{\nu}(\Omega^{c})}$,
$\|\cdot\|^{*}_{T_{\nu}(\Omega^{c})}$ and
$\|\cdot\|^{\dagger}_{T_{\nu}(\Omega^{c})}$ are all equivalent, where

$\displaystyle\|v\|^{*}_{T_{\nu}(\Omega^{c})}$
$\displaystyle=\inf\\{\|u\|^{*}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}:\leavevmode\nobreak\
\leavevmode\nobreak\ u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\leavevmode\nobreak\
\leavevmode\nobreak\ \hbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
v=u|_{\Omega^{c}}\\}$ $\displaystyle\|v\|^{\dagger 2}_{T_{\nu}(\Omega^{c})}$
$\displaystyle=\int_{\Omega^{c}}|v(x)|^{2}\widetilde{\nu}(x)\mathrm{d}x+\iint\limits_{\Omega^{c}\Omega^{c}}\\!\\!\big{(}v(x)-v(y)\big{)}^{2}\,\gamma_{\Omega}(x,y)\mathrm{d}x\,\mathrm{d}y\,.$

Next, consider $\nu(h)=(2-\alpha)|h|^{-d-\alpha}$ for $h\neq 0$ with
$0<\alpha<2$ fixed and $\widetilde{\nu}(h)=\frac{1}{(1+|h|)^{d+\alpha}}$. Set
$\delta_{z}=\operatorname{dist}(z,\partial\Omega)$. Then

$\displaystyle
T_{\nu}(\Omega^{c})=\Big{\\{}v:\Omega^{c}\to\mathbb{R}\leavevmode\nobreak\
\text{meas.}\big{|}\;\iint\limits_{\Omega^{c}\Omega^{c}}\\!\\!\frac{\big{(}v(x)-v(y)\big{)}^{2}}{(|x-y|+\delta_{x}+\delta_{y})^{d+\alpha}}\,\mathrm{d}x\,\mathrm{d}y<\infty\Big{\\}}\,$

and the aforementioned norms are equivalent to the norm

$\displaystyle\|v\|^{{}^{\prime}2}_{T_{\nu}(\Omega^{c})}=\int_{\Omega^{c}}\frac{|v(x)|^{2}}{(1+|x|)^{d+\alpha}}\mathrm{d}x+\iint\limits\limits_{\Omega^{c}\Omega^{c}}\frac{\big{(}v(x)-v(y)\big{)}^{2}}{(|x-y|+\delta_{x}+\delta_{y})^{d+\alpha}}\mathrm{d}x\mathrm{d}y\,.$

######  Remark 2.33.

In the case $\nu_{\alpha}(h)=(2-\alpha)|h|^{-d-\alpha}$ for $h\neq 0$ with
given $\alpha\in(0,2)$, it is interesting to understand the limiting behaviour
of the comparability estimate for
$\|v\|^{{}^{\prime}}_{T_{\nu_{\alpha}}(\Omega^{c})}$ and
$\|v\|^{*}_{T_{\nu_{\alpha}}(\Omega^{c})}$ as $\alpha\to 2^{-}$. Recent
upcoming results of Th. Hensiek and F. Grube show that one can modify the norm
$\|v\|^{{}^{\prime}}_{T_{\nu}(\Omega^{c})}$ so that
$\|v\|^{{}^{\prime}}_{T_{\nu_{\alpha}}(\Omega^{c})}$ would converge to
$\|v\|_{H^{1/2}(\partial\Omega)}$ as $\alpha\to 2^{-}$.

###### Proof.

The equivalence between $\|\cdot\|_{T_{\nu}(\Omega^{c})}$ and
$\|\cdot\|^{*}_{T_{\nu}(\Omega^{c})}$ is an immediate consequence of 2.17. By
(2.11) it follows that,

$\displaystyle\|v\|^{*2}_{T_{\nu}(\Omega^{c})}$
$\displaystyle=\inf\\{\|u\|^{*2}_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}:\,u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\text{ with
}v=u|_{\Omega^{c}}\\}$
$\displaystyle\geq\inf\Big{\\{}\int_{\operatorname{\mathbb{R}}^{d}}|u(x)|^{2}\widetilde{\nu}(x)\mathrm{d}x:\,u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\text{ with
}v=u|_{\Omega^{c}}\Big{\\}}+\mathcal{H}_{\Omega}(v,v)$
$\displaystyle\geq\int_{\Omega^{c}}|v(x)|^{2}\widetilde{\nu}(x)\mathrm{d}x+\mathcal{H}_{\Omega}(v,v)\,,$

which establishes
$\|v\|^{\dagger}_{T_{\nu}(\Omega^{c})}\leq\|v\|^{*}_{T_{\nu}(\Omega^{c})}$.
Hence the identity
$I:(T_{\nu}(\Omega^{c}),\|\cdot\|^{*}_{T_{\nu}(\Omega^{c})})\to(T_{\nu}(\Omega^{c}),\|\cdot\|^{\dagger}_{T_{\nu}(\Omega^{c})})$
is continuous. The space
$(T_{\nu}(\Omega^{c}),\|\cdot\|^{*}_{T_{\nu}(\Omega^{c})})$ is a Hilbert space
since $\|\cdot\|_{T_{\nu}(\Omega^{c})}$ and
$\|\cdot\|^{*2}_{T_{\nu}(\Omega^{c})}$ are equivalent. Also, using the Fatou
lemma one can easily show that
$(T_{\nu}(\Omega^{c}),\|\cdot\|^{\dagger}_{T_{\nu}(\Omega^{c})})$ is a Hilbert
space. As consequence of the open mapping theorem the norms
$\|\cdot\|^{\dagger}_{T_{\nu}(\Omega^{c})}$ and
$\|\cdot\|^{*}_{T_{\nu}(\Omega^{c})}$ are equivalent.

Next, let us consider $\nu(h)=(2-\alpha)|h|^{-d-\alpha}$ for $h\neq 0$ with
$0<\alpha<2$ fixed. From [DK19, Theorem 3], see (2.9) and (2.10), we conclude
that there exists a constant $C>0$ such that for all $v\in
T_{\nu}(\Omega^{c})$, $\|v\|^{*}_{T_{\nu}(\Omega^{c})}\leq
C\|v\|^{{}^{\prime}}_{T_{\nu}(\Omega^{c})}$. The equivalence between
$\|\cdot\|^{*}_{T_{\nu}(\Omega^{c})}$ and
$\|\cdot\|^{{}^{\prime}}_{T_{\nu}(\Omega^{c})}$ follows once again by the open
mapping theorem. ∎

######  Remark 2.34.

We emphasize that the nonlocal trace does not need any special construction
via functional analysis or density arguments. Since $\Omega^{c}$ is a
$d$-dimensional manifold, it makes sense to consider the restriction of a
measurable function on $\Omega^{c}$. No regularity of $\Omega$ resp.
$\partial\Omega$ is required. In the classical local situation, the definition
of a trace of a Sobolev function $u$ on the boundary $\partial\Omega$ requires
some smoothness of both, $u$ and $\partial\Omega$.

Let us collect some basics results results concerning the trace space
$T_{\nu}(\Omega^{c})$. With the aid of 2.17 we get the following.

######  Proposition 2.35.

The trace map
$\operatorname{Tr}:V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\to
L^{2}(\Omega^{c},\widetilde{\nu})$ with
$u\mapsto\operatorname{Tr}(u)=u\mid_{\Omega^{c}}$ has the following
properties: (a)
$\operatorname{Tr}(V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}))=T_{\nu}(\Omega^{c})$,
(b) $\ker(\operatorname{Tr})=V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$
and (c) $\operatorname{Tr}$ is linear and continuous. Moreover,
$T_{\nu}(\Omega^{c})$ is dense in $L^{2}(\Omega^{c},\widetilde{\nu})$.

###### Proof.

This is indeed, is a direct consequence of 2.17 since $u\in
L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})$ for all $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ so that
$\operatorname{Tr}(u)\in L^{2}(\Omega^{c},\widetilde{\nu})$ in particular
$\operatorname{Tr}$ is well defined. Moreover, by 2.17 there exists a constant
$C>0$ such that,

$\displaystyle\|\operatorname{Tr}(u)\|_{L^{2}(\Omega^{c},\widetilde{\nu})}\leq\|u\|_{L^{2}(\operatorname{\mathbb{R}}^{d},\widetilde{\nu})}\leq
C\|u\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\qquad\text{for all
}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}).$ (2.13)

The zero extension to $\Omega$ of elements
$C_{c}^{\infty}(\overline{\Omega}^{c})$ are in
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. Thus
$C_{c}^{\infty}(\overline{\Omega}^{c})$ is contained in $T_{\nu}(\Omega^{c})$
which implies that $T_{\nu}(\Omega^{c})$ is dense in
$L^{2}(\Omega^{c},\widetilde{\nu})$ since
$C_{c}^{\infty}(\overline{\Omega}^{c})$ is dense in
$L^{2}(\Omega^{c},\widetilde{\nu})$ ∎

######  Remark 2.36.

One may view the objects $L^{2}(\Omega^{c},\widetilde{\nu})$,
$T_{\nu}(\Omega^{c})$, $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ respectively as the nonlocal
counterpart of $L^{2}(\partial\Omega)$, $H^{1/2}(\partial\Omega)$,
$H^{1}(\Omega)$ and $H^{1}_{0}(\Omega)$. Indeed, $(i)$ the classical trace
operator $\gamma_{0}:H^{1}(\Omega)\to L^{2}(\partial\Omega)$ whenever it
exists is linear continuous, $(ii)$
$\gamma_{0}(H^{1}(\Omega))=H^{1/2}(\partial\Omega)$ and $(iii)$
$\ker(\gamma_{0})=H^{1}_{0}(\Omega)$.

######  Proposition 2.37.

Let
$C^{\infty}_{c}(\overline{\Omega^{c}})=C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})|_{\Omega^{c}}$
be set of restrictions on $\Omega^{c}$ of $C^{\infty}$ functions on
$\operatorname{\mathbb{R}}^{d}$ with compact support. If $\Omega$ is bounded
and Lipschitz then $C^{\infty}_{c}(\overline{\Omega^{c}})$ is dense in
$T_{\nu}(\Omega^{c})$.

###### Proof.

For $v\in T_{\nu}(\Omega^{c})$ we write $v=u|_{\Omega^{c}}$ with $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$. From [FKV20] we know that
there exists $u_{n}\in C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d})$ such
that, $\|u_{n}-u\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\to 0.$ Put,
$v_{n}=u_{n}|_{\Omega^{c}}$ by (2.13) we get

$\displaystyle\|v_{n}-v\|_{T_{\nu}(\Omega^{c})}\leq\|u_{n}-u\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\to
0.$

∎

######  Remark 2.38.

Let us comment on certain function spaces that are introduced in [Von21] in
order to study some reflected jump Markov processes. Instead of the natural
energy space $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ from [FKV15], the
author considers $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d},m)$ with
$m(x)=\mathds{1}_{\Omega}(x)+\mu(x)\mathds{1}_{\Omega^{c}}(x)$ and
$\mu(x)=\int_{\Omega}\nu(x-y)\mathrm{d}y$ for $x\in\Omega^{c}$. It is proved
in [Von21, Lemma 2.2 (iii)] that $L^{2}(\Omega^{c},\mu)$ is the trace space of
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d},m)$. Note that
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d},m)$ and its trace space
$L^{2}(\Omega^{c},\mu)$ are much smaller than
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ resp. $T_{\nu}(\Omega^{c})$,
which leads to the following issues.

  * •

When $\Omega$ is bounded, constant functions belong to
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and do not belong to
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d},m)$ in general. Thus in term of trace,
$x\mapsto\mathbbm{1}_{\Omega^{c}}(x)$ belongs to $T_{\nu}(\Omega^{c})$ but not
necessarily to $L^{2}(\Omega^{c},\mu)$. Several natural Dirichlet problems,
e.g. for the fractional Laplace operator, cannot be formulated with the help
of $L^{2}(\Omega^{c},\mu)$.

  * •

See Theorem 2.26 for a regular Dirichlet form leading to the existence of
reflected jump processes.

  * •

Given a function $v\in L^{2}(\Omega^{c},\mu)$, its extension by zero
$v_{0}=v\mathds{1}_{\Omega^{c}}$ belongs to
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d},m)$ because of

$\displaystyle\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}v_{0}(x)-v_{0}(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y=2\int_{\Omega^{c}}|v(x)|^{2}\mu(x)\mathrm{d}x.$

Moreover, the space $L^{2}(\Omega^{c},\mu)$ is continuously embedded in
$T_{\nu}(\Omega^{c})$, indeed,

$\displaystyle\|v\|_{T_{\nu}(\Omega^{c})}\leq\|v_{0}\|_{V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\leq\sqrt{2}\|v\|_{L^{2}(\Omega^{c},\mu)}.$

The fact that the extension by zero belongs to the energy space for any given
function in the trace space, is rather particular.

Since $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d},m)$ resp. its trace space
$L^{2}(\Omega^{c},\mu)$ are small compared to the spaces
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ resp. $T_{\nu}(\Omega^{c})$,
the range of possible nonlocal Dirichlet and Neumann problems is rather small.

##  3\. Compact embeddings and Poincaré inequality

In this section we prove compact embeddings of the spaces $H_{\nu}(\Omega)$,
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ and
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ into $L^{2}(\Omega)$. Our
result on global compactness, Theorem 3.10, requires some regularity
assumptions on $\Omega$ and $\nu$, which we introduce and discuss in Section
3.1. In Section 3.2 we establish global compactness using ideas from [JW19]
and [DMT18]. Note that [CD18, Theorem 2.2] is a related result. However, the
proof therein seems to be valid only for domains that can be decomposed as a
finite union of cubes. We circumvent this issue by an approximation argument
near the boundary of $\Omega$.

###  3.1. Assumptions on the Lévy measure

The definitions and most of the results of Section 2.2 do not require
assumptions on the Lévy measure $\nu$ beyond the classical Lévy condition (L).
In this subsection we collect further conditions on $\nu$ required for the
compactness results in Section 3.2. Recall the concept of unimodality from
2.4. We will prove a Poincaré-inequality in Theorem 3.12 for unimodal Lévy
measures with full support.

######  Definition 3.1.

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and bounded, and
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ satisfies
(L). We say that $(\nu,\Omega)$ is in the class $\mathbf{\mathscr{A}_{0}}$ if

  * ($\mathbf{\mathscr{A}_{0}}$)

$\nu$ is unimodal and has full support.

Note that, in the class $\mathbf{\mathscr{A}_{0}},$ $\nu$ is not necessarily
singular near $0$. In order to establish compactness results in Section 3.2 we
will discuss different assumptions. Note that (L) and unimodality do not imply
any lower bound on $\nu$, even $\nu=0$ would be allowed. Furthermore, if
$\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})$, then the spaces
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\cap
L^{2}(\operatorname{\mathbb{R}}^{d})$ and
$H_{\nu}(\operatorname{\mathbb{R}}^{d})$ coincide with
$L^{2}(\operatorname{\mathbb{R}}^{d})$, which is not locally compactly
embedded into $L^{2}(\Omega)$. For the remainder of this section, we assume
that $\nu$ satisfies (L) and

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}}\nu(h)\mathrm{d}h=\infty.$
(I)

Under (L), condition (I) obviously follows if $|h|^{d}\nu(h)\to\infty$ as
$|h|\to 0$. As explained in Corollary 3.7, (L) and (I) imply local compactness
of $H_{\nu}(\Omega)$ and $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in
$L^{2}(\Omega)$. Let us introduce conditions on $\Omega$ and $\nu$ under which
we are able to establish global compactness results.

######  Definition 3.2.

Assume $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and bounded, and
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ satisfies (L)
and (I). We say that $(\nu,\Omega)$ is in the class $\mathscr{A}_{i}$
$(i=1,2,3)$, if

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

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

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

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

$\displaystyle
q(\delta):=\frac{1}{\delta^{2}}\int\limits_{B_{\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h\,.$
(3.1)

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

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

$\displaystyle\widetilde{q}(\delta):=\inf_{a\in\partial\Omega}\int_{\Omega_{\delta}}\nu(h-a)\mathrm{d}h$
(3.2)

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

Note that monotonicity of $\nu$ is not necessarily required by any of the
conditions above. The class $\mathbf{\mathscr{A}_{1}}$ is well studied in the
literature for the case of the fractional Laplace operator. For example, it is
shown in [Zho15] that $\Omega$ is an extension domain for
$H^{\alpha/2}(\Omega),\leavevmode\nobreak\ \alpha\in(0,2)$, if and only if
$\Omega$ is a $d$-set and thus, $(|\cdot|^{-d-\alpha},\Omega)$ is an element
of $\mathbf{\mathscr{A}_{1}}$.

The class $\mathbf{\mathscr{A}_{2}}$ is easy to understand because the
conditions on $\Omega$ and $\nu$ do not interact. If $\Omega$ is a bounded
Lipschitz domain and $\nu$ satisfies (L) and

$\displaystyle\lim_{|h|\to 0}|h|^{d}\nu(h)=\infty\,,$ ($I^{\prime}$)

then $(\nu,\Omega)$ is in the class $\mathbf{\mathscr{A}_{2}}$. Indeed, for
$R>0$ sufficiently large there is $\delta_{0}>0$ such that $|h|^{d}\nu(h)\geq
2R$ whenever $|h|\leq\delta_{0}$. Thus $q(\delta)\geq|\mathbb{S}^{d-1}|R$ if
$0<\delta<\delta_{0}$. This shows that (3.1) is verified.

The class $\mathbf{\mathscr{A}_{3}}$ and condition (3.2) are more involved due
to a certain correlation between $\Omega$ and the singularity of $\nu$ near
the origin. Let us first provide an example of $\nu$ and $\Omega$ such that
$\mathbf{\mathscr{A}_{3}}$ fails. In the Euclidean plane consider
$\nu(h)=|h|^{-2-\alpha}\mathds{1}_{V}(h)$ with
$V=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:\leavevmode\nobreak\ |x_{1}|<|x_{2}|\\}$
and $\Omega=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:\leavevmode\nobreak\
4|x_{2}-6|<x_{1},\leavevmode\nobreak\ 0<x_{1}<4\\}$ whose boundary is
continuous. Considering $a=(0,6)\in\partial\Omega$ one has
$V\cap(\Omega_{\delta}-a)=\emptyset$ for every $\delta>0$, see Figure 1.
Therefore
$\widetilde{q}(\delta)\leq\int_{\Omega_{\delta}}\nu(h-a)\mathrm{d}h=0$ and the
condition (3.2) fails.

$\Omega$$\Omega_{\delta}-a$$V$$V$$(0,0)$$a$$v$

Figure 1. Example of $(\nu,\Omega)\notin\mathbf{\mathscr{A}_{3}}$

Next, let us provide a positive result. Note that for every domain $\Omega$
and every $\delta>0$ we know $\widetilde{q}(\delta)<\infty$ because, for each
$a\in\partial\Omega$ and each $\delta>0$, $\Omega_{\delta}\subset
B^{c}_{\delta}(a)$, which implies by (L)
$\widetilde{q}(\delta)\leq\int_{B^{c}_{\delta}(0)}\nu(h)\mathrm{d}h<\infty$.
We will show that $\partial\Omega\in C^{1,1}$ is sufficient. Recall that
$\Omega$ is of class $C^{1,1}$ if for every $a\in\partial\Omega$ there is
$r>0$ for which $B_{r}(a)\cap\partial\Omega=\\{x=(x^{\prime},x_{d})\in
B_{r}(a)\leavevmode\nobreak\ :x_{d}=\gamma(x^{\prime})\\}$ represents the
graph of a $C^{1,1}$ function $\gamma:\mathbb{R}^{d-1}\to\mathbb{R}$. That is
to say $\gamma$ is a $C^{1}$ function whose gradient is Lipschitz. The main
result in [Bar09] shows that an open set $\Omega$ is $C^{1,1}$ if and only if
$\Omega$ satisfies the interior and exterior sphere condition. We say that
$\Omega$ satisfies the interior and exterior sphere condition at some scale
$r>0$ if for every $a\in\partial\Omega$ one can find $a^{\prime}\in\Omega$ and
$a^{\prime\prime}\in\overline{\Omega}^{c}$ for which
$B_{r}(a^{\prime})\subset\Omega$,
$B_{r}(a^{\prime\prime})\subset\overline{\Omega}^{c}$ and
$\overline{B_{r}(a^{\prime})}\cap\overline{B_{r}(a^{\prime\prime})}=\\{a\\}$.
Note that, the interior and exterior sphere condition holds for every scale
$r\in(0,r_{0})$ once it holds for $r_{0}$. This characterization entails that
a $C^{1,1}$ set $\Omega$ is a $d$-set (or _volume density condition_ according
to some authors): that is, there exist two positive constants $c>0$ and
$r_{0}>0$ such that for every $r\in(0,r_{0})$ and every $a\in\partial\Omega$

$\displaystyle|\Omega\cap B_{r}(a)|\geq cr^{d}.$

######  Proposition 3.3.

Assume $\nu$ satisfies (L) and ($I^{\prime}$). Assume that $\Omega$ satisfies
the following strong volume density condition: there exist positive constants
$\tau>1$, $\delta_{0}>0$ and $c>0$ such that for all $\delta\in(0,\delta_{0})$
and $a\in\partial\Omega$

$\displaystyle|\Omega_{\delta}\cap B_{\tau\delta}(a)|\geq c\delta^{d}.$

Then $(\nu,\Omega)\in\mathbf{\mathscr{A}_{3}}$.

######  Remark 3.4.

  1. (i)

Any bounded $C^{1,1}$-domain $\Omega\subset\operatorname{\mathbb{R}}^{d}$
satisfies the aforementioned strong volume density condition. Fix
$a\in\partial\Omega$, by the interior sphere condition, consider
$\delta\in(0,\delta_{0}/4)$ for some $\delta_{0}$ sufficiently small. Let
$x\in\Omega$ depending on $a$ and $\delta$ such that
$B_{2\delta}(x)\subset\Omega$,
$\operatorname{dist}(x,\partial\Omega)=|x-a|=2\delta$ and
$\overline{B_{2\delta}(x)}\cap\partial\Omega=\\{a\\}$ then obviously,
$B_{\delta}(x)\subset\Omega_{\delta}\cap
B_{2\delta}(x)\subset\Omega_{\delta}\cap B_{4\delta}(a)$. This yields

$\displaystyle|\Omega_{\delta}\cap B_{4\delta}(a)|\geq
c_{d}\delta^{d},\quad\text{with}\leavevmode\nobreak\ \leavevmode\nobreak\
c_{d}=|B_{1}(0)|.$ (3.3)

  2. (ii)

It is interesting to know whether for small $\delta>0$, $\Omega_{\delta}$
inherits the regularity of $\Omega$. As proven in [GT01, Section 6.14] if
$\Omega$ is of class $C^{k}$ with $k\geq 2$ then so is $\Omega_{\delta}$.

###### Proof.

Let $R>0$ and consider $\tau>1$, $\delta_{0}>0$ as above such that if
$0<\delta<\delta_{0}$ then $|h|^{d}\nu(h)\geq R$ for $|h|<\delta$. Fix
$a\in\partial\Omega$, since $|\Omega_{\delta}\cap B_{\tau\delta}(a)|\geq
c\delta^{d}$ for all $0<\delta<\delta_{0}$. Therefore, recalling that
$\nu(h-a)\geq R|h-a|^{-d}\geq\frac{R}{\tau^{d}\delta^{d}}$ when $h\in
B_{\tau\delta}(a)$ we have

$\displaystyle\int_{\Omega_{\delta}}\nu(h-a)\mathrm{d}h\geq\frac{R}{\tau^{d}\delta^{d}}\int_{\Omega_{\delta}\cap
B_{\tau\delta}(a)}\mathrm{d}h=\,\frac{R}{\tau^{d}\delta^{d}}|\Omega_{\delta}\cap
B_{\tau\delta}(a)|\geq\frac{c}{\tau^{d}}R.$

Finally,

$\displaystyle\widetilde{q}(\delta)\geq\frac{c}{\tau^{d}}R$

which means that (3.2) is verified since $R$ can be arbitrarily large. ∎

###  3.2. Local and global compactness results

Before citing a result on local compactness, we recall a well-known result
about convolutions.

######  Lemma 3.5 (Corollary 4.28 of [Bre11]).

Let $w\in L^{1}(\operatorname{\mathbb{R}}^{d})$. Then the convolution operator

$\displaystyle T_{w}:L^{2}(\operatorname{\mathbb{R}}^{d})\to
L^{2}(\operatorname{\mathbb{R}}^{d}),\qquad T_{w}u=w*u$

is continuous with
$\|T_{w}\|_{\mathcal{L}(L^{2}(\operatorname{\mathbb{R}}^{d}),L^{2}(\operatorname{\mathbb{R}}^{d}))}\leq\|w\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}$.
Moreover, $T_{w}:L^{2}(\operatorname{\mathbb{R}}^{d})\to L^{2}(K)$ is compact
for any compact subset $K\subset\operatorname{\mathbb{R}}^{d}$.

We present the local compactness result [JW19] that we are going to use in the
sequel.

######  Theorem 3.6.

Let $\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ be a
measurable symmetric function such that (I) holds and

$\displaystyle\int_{\operatorname{\mathbb{R}}^{d}\setminus
B_{\delta}(0)}\nu(h)\mathrm{d}h<\infty\leavevmode\nobreak\ \text{ for every
$\delta>0$}.$ (3.4)

Then the embedding $H_{\nu}(\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\operatorname{\mathbb{R}}^{d})$ is locally compact. As a consequence,
for $\Omega\subset\operatorname{\mathbb{R}}^{d}$ open and bounded, the
embedding $V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\Omega)$ is compact.

It is worth mentioning that an earlier analogous result is provided in [PZ17,
Proposition 6] and [BJ17, Proposition 1] for periodic functions on the torus.
This technique, which consist of killing the singularity, is also used in
[BJ13, Lemma 3.1]. The assertion of Theorem 3.6 is proved in [JW19, Theorem
1.1] under the additional assumption that $\nu$ satisfies (L). An analogous
result is also proved in [CD18, Theorem 2.1] under restrictive assumptions on
the kernel, using the Pego criterion for compact compactness in
$L^{2}(\operatorname{\mathbb{R}}^{d})$. Looking at the proof carefully one
sees that conditions (I) and (3.4) is sufficient. This would allow to consider
densities $\nu$ with a very strong singularity at the origin, e.g.,
$\nu(h)=|h|^{-d-\beta}$ for $h\neq 0$ with any $\beta>0$.

As a straightforward consequence of Theorem 3.6 we have the following local
compactness of $H_{\nu}(\Omega)$ in $L^{2}(\Omega)$.

######  Corollary 3.7.

Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open but not necessarily
bounded. Assume
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to\mathbb{R}$ fulfills
conditions (L) and (I). The embedding $H_{\nu}(\Omega)\hookrightarrow
L^{2}(\Omega)$ is locally compact. Furthermore, for every bounded sequence
$(u_{n})_{n}$ there exists $u\in H_{\nu}(\Omega)$ and subsequence
$(u_{n_{j}})_{j}$ converging to $u$ in $L^{2}_{\operatorname{loc}}(\Omega)$.

###### Proof.

There is no loss of generality if we assume that a function $u\in
H_{\nu}(\Omega)$ is extended by zero outside of $\Omega$. For $\varphi\in
C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$, with
$\operatorname{supp}\varphi\subset\Omega$, the map
$J_{\varphi}:H_{\nu}(\Omega)\to H_{\nu}(\operatorname{\mathbb{R}}^{d})$, with
$J_{\varphi}u=u\varphi$ is continuous and is thus locally compact by Theorem
3.6. Therefore the embedding $H_{\nu}(\Omega)\hookrightarrow L^{2}(\Omega)$ is
locally compact. Indeed, for $u\in H_{\nu}(\Omega)$ we have

$\displaystyle\big{[}u(x)\varphi(x)-u(y)\varphi(y)\big{]}^{2}$
$\displaystyle=\big{[}u(x)(\varphi(x)-\varphi(y))+\varphi(y)(u(x)-u(y))\big{]}^{2}$
$\displaystyle\leq
2\|\varphi\|^{2}_{W^{1,\infty}(\operatorname{\mathbb{R}}^{d})}\big{[}|u(x)|^{2}(1\land|x-y|^{2})+\mathds{1}_{\operatorname{supp}\varphi}(y)(u(x)-u(y))^{2}\big{]}.$

As $\operatorname{supp}\varphi\subset\Omega$ is compact, consider
$0<r\leq\operatorname{dist}(\operatorname{supp}\varphi,\partial\Omega)$. Then
integrating both sides of the above estimate over
$\Omega\times\operatorname{\mathbb{R}}^{d}$ with respect to the measure
$\nu(x\\!-\\!y)\mathrm{d}y\mathrm{d}x$, yields the continuity of $J_{\varphi}$
as follows

$\displaystyle\iint\limits_{\operatorname{\mathbb{R}}^{d}\operatorname{\mathbb{R}}^{d}}\big{[}u(x)\varphi(x)$
$\displaystyle-u(y)\varphi(y)\big{]}^{2}\nu(x\\!-\\!y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\leq
2\|\varphi\|^{2}_{W^{1,\infty}(\operatorname{\mathbb{R}}^{d})}\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{\operatorname{\mathbb{R}}^{d}}(1\land|h|^{2})\nu(h)\mathrm{d}h$
$\displaystyle+\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}y\mathrm{d}x+\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{B_{r}(0)}\nu(h)\mathrm{d}h$
$\displaystyle\leq C_{\varphi}\|u\|^{2}_{H_{\nu}(\Omega)}\,.$

Next, we prove the second statement. Consider
$\Omega^{\prime}_{\delta}=\big{\\{}x\in\Omega:|x|<\frac{1}{\delta},\,\,\operatorname{dist}(x,\partial\Omega)>\delta\big{\\}}=\Omega_{\delta}\cap
B_{\frac{1}{\delta}}(0)$ and define
$\varphi_{\delta}(x)=\eta_{\delta/4}*\mathds{1}_{\Omega^{\prime}_{\delta/2}}(x)$
for $\delta>0$ small enough, where
$\eta_{\delta}(x)=\frac{1}{\delta^{d}}\eta(\frac{x}{\delta})$ with $\eta\in
C_{c}^{\infty}(\operatorname{\mathbb{R}}^{d})$ is supported in the unit ball
$B_{1}(0)$, $\eta\geq 0$ and
$\int_{\operatorname{\mathbb{R}}^{d}}\phi(x)\mathrm{d}x=1$. So that
$\varphi_{\delta}\in C_{c}^{\infty}(\Omega)$, $\varphi_{\delta}=1$ on
$\Omega^{\prime}_{\delta}$, $0\leq\varphi_{\delta}\leq 1$ and
$|\nabla\varphi_{\delta}|\leq c/\delta$. Given a sequence $(u_{n})_{n}$ that
is bounded in $H_{\nu}(\Omega)$, the previous observation entails that for
each $\delta>0$, for the sequence $(u_{n}\varphi_{\delta})_{n}$ there exists a
subsequence $n_{j}=n_{j}(\delta),$ $j\geq 1$, and $u^{\delta}\in
L^{2}(\Omega^{\prime}_{\delta})$ such that the sequence $(u_{n})_{n}$, (as
$u_{n}\varphi_{\delta}=u_{n}$ in $\Omega^{\prime}_{\delta}$) converges to some
$u^{\delta}$ in $L^{2}(\Omega^{\prime}_{\delta})$ and almost everywhere in
$\Omega^{\prime}_{\delta}.$ Employing the standard the Cantor’s
diagonalization procedure with $\delta=\frac{1}{2^{k}}$, one can construct a
subsequence $(u_{n_{j}})_{j}$ converging subsequence in
$L_{\operatorname{loc}}^{2}(\Omega)$ and almost everywhere in $\Omega$ to some
function $u$. Fatou’s lemma implies that $u\in H_{\nu}(\Omega)$ since

$\displaystyle\|u\|_{H_{\nu}(\Omega)}\leq\liminf_{j\to\infty}\|u_{n_{j}}\|_{H_{\nu}(\Omega)}<\infty.$

∎

Let us turn to the result on global compactness. We will need some estimates
near the boundary $\partial\Omega$. We begin with the following estimate
involving cut-off functions.

######  Lemma 3.8.

Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open and bounded. Assume
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is even and
measurable. Let $0<\delta<\frac{1}{3}\operatorname{diam}(\Omega)$. Let
$\varphi\in C^{\infty}_{c}(\Omega)$ be such that 222Take
$\varphi=1-\varphi_{\delta}$ with
$\varphi_{\delta}=\eta_{\delta/4}*\mathds{1}_{\Omega^{\prime}_{\delta/2}}$ as
in the proof of Corollary 3.7. $\varphi=0$ on $\Omega_{\delta}$, $\varphi=1$
on $\Omega\setminus\Omega_{\delta/2}$, $0\leq\varphi\leq 1$ and
$|\nabla\varphi|\leq c/\delta$. For every $u\in H_{\nu}(\Omega)$, the
following estimate holds true

$\displaystyle\iint\limits_{\Omega\Omega}([u\varphi](x)-[u\varphi](y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\leq\frac{C}{\delta^{2}}\int\limits_{\Omega_{\delta/2}}|u(x)|^{2}\mathrm{d}x+8\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
(3.5)

where, $C=8c^{2}\int_{B_{R}(0)}|h|^{2}\nu(h)\mathrm{d}h$ and
$R=\operatorname{diam}(\Omega)$.

###### Proof.

Firstly, since $\varphi=1$ on $\Omega\setminus\Omega_{\delta/2}$ we have

$\displaystyle\iint\limits_{\Omega\setminus\Omega_{\delta/2}\leavevmode\nobreak\
\Omega\setminus\Omega_{\delta/2}}([u\varphi](x)-[u\varphi](y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle=\iint\limits_{\Omega\setminus\Omega_{\delta/2}\leavevmode\nobreak\
\Omega\setminus\Omega_{\delta/2}}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\,.$

In view of the fact that $0\leq\varphi\leq 1$ and $|\varphi(x)-\varphi(y)|\leq
c/\delta|x-y|$ for every $x,y\in\Omega$, we have

$\displaystyle([u\varphi](x)-[u\varphi](y))^{2}$
$\displaystyle=\big{(}\varphi(y)(u(x)-u(y))+u(x)(\varphi(x)-\varphi(y))\big{)}^{2}$
$\displaystyle\leq
2(u(x)-u(y))^{2}+\frac{2c^{2}}{\delta^{2}}|u(x)|^{2}|x-y|^{2}\,.$ (3.6)

Secondly, noticing that $\Omega\subset B_{R}(x)$ for all $x\in\Omega$ where
$R=\operatorname{diam}(\Omega)$ and integrating both sides of (3.6) over
$\Omega_{\delta/2}\times\Omega_{\delta/2}$ we obtain the following estimate

$\displaystyle\iint\limits_{\Omega_{\delta/2}\Omega_{\delta/2}}([u\varphi](x)-[u\varphi](y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq
2\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y+\frac{2c^{2}}{\delta^{2}}\int\limits_{\Omega_{\delta/2}}|u(x)|^{2}\mathrm{d}x\int\limits_{B_{R}(x)}|x-y|^{2}\nu(x\\!-\\!y)\mathrm{d}y$
$\displaystyle=2\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y+\frac{2c^{2}}{\delta^{2}}\Big{(}\int\limits_{B_{R}(0)}|h|^{2}\nu(h)\mathrm{d}h\Big{)}\int\limits_{\Omega_{\delta/2}}|u(x)|^{2}\mathrm{d}x\,.$

Likewise to the previous estimate, using (3.6) we get

$\displaystyle\iint\limits_{\Omega_{\delta/2}\times\Omega\setminus\Omega_{\delta/2}}([u\varphi](x)-[u\varphi](y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq
2\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y+\frac{2c^{2}}{\delta^{2}}\Big{(}\int\limits_{B_{R}(0)}|h|^{2}\nu(h)\mathrm{d}h\Big{)}\int\limits_{\Omega_{\delta/2}}|u(x)|^{2}\mathrm{d}x\,.$

Altogether, the desired estimate follows as claimed since by symmetry we can
use the split

$\displaystyle\iint\limits_{\Omega\times\Omega}=\iint\limits_{\Omega_{\delta/2}\times\Omega_{\delta/2}}+\,2\iint\limits_{\Omega_{\delta/2}\times\Omega\setminus\Omega_{\delta/2}}+\iint\limits_{\Omega\setminus\Omega_{\delta/2}\times\Omega\setminus\Omega_{\delta/2}}\,.$

∎

The next lemma plays a crucial role in the sequel.

######  Lemma 3.9.

Assume that $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is open and bounded,
and $\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ is
radial. There exists $C>0$ such that for every $u\in L^{2}(\Omega)$ and every
positive $\delta<\frac{1}{3}\operatorname{diam}(\Omega)$

$\displaystyle\int_{\Omega}|u(x)|^{2}\mathrm{d}x\leq\frac{C}{\delta^{2}\widetilde{q}(2\delta)}\int_{\Omega_{\delta/2}}|u(x)|^{2}\mathrm{d}x+\frac{8}{\widetilde{q}(2\delta)}\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\,,$
(3.7)

with $\widetilde{q}$ as is (3.2). Moreover, if $\Omega$ has a Lipschitz
boundary, then

$\displaystyle\int_{\Omega}|u(x)|^{2}\mathrm{d}x\leq\frac{C}{\delta^{2}q(2\delta)}\int_{\Omega_{\delta/2}}|u(x)|^{2}\mathrm{d}x+\frac{8}{q(2\delta)}\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\,,$
(3.8)

with $q$ as in (3.1).

###### Proof.

Let $\varphi$ be as in 3.8 and fix $a\in\partial\Omega$. A routine check
reveals that $\Omega_{2\delta}-a\subset\Omega_{\delta}-x$ for every
$x\in\Omega\cap B_{\delta}(a)$ which yields,

$\displaystyle\int_{\Omega\cap B_{\delta}(a)}$ $\displaystyle[\varphi
u]^{2}(x)\mathrm{d}x\int_{\Omega_{\delta}}\nu(x\\!-\\!y)\mathrm{d}y\geq\int_{\Omega\cap
B_{\delta}(a)}[\varphi
u]^{2}(x)\mathrm{d}x\int_{\Omega_{\delta}-x}\nu(h)\mathrm{d}h$
$\displaystyle\geq\int_{\Omega\cap B_{\delta}(a)}[\varphi
u]^{2}(x)\mathrm{d}x\int_{\Omega_{2\delta}-a}\nu(h)\mathrm{d}h\geq\widetilde{q}(2\delta)\int_{\Omega\cap
B_{\delta}(a)}[\varphi u]^{2}(x)\mathrm{d}x.$

By a compactness argument there exist $a^{1},a^{2},\cdots
a^{n}\in\partial\Omega$ such that
$\partial\Omega\subset\bigcup\limits_{i=1}^{n}B_{\delta/2}(a^{i})$. So that,
$\Omega\setminus\Omega_{\delta/2}\subset\bigcup\limits_{i=1}^{n}\Omega\cap
B_{\delta}(a^{i})\subset\Omega\setminus\Omega_{\delta}$. $\varphi u=0$ on
$\Omega_{\delta/2}$ trivially implies $\varphi u=0$ on $\Omega_{\delta}$.
Therefore with the aid of the above estimate we obtain the following estimate

$\displaystyle\iint\limits_{\Omega\Omega}\big{(}[\varphi u](x)-[\varphi
u](y)\big{)}^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\geq
2\iint\limits_{\Omega\setminus\Omega_{\delta}\leavevmode\nobreak\
\Omega_{\delta}}[\varphi u]^{2}(x)\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\geq\int\limits_{\bigcup\limits_{i=1}^{n}\Omega\cap
B_{\delta}(a^{i})}[\varphi
u]^{2}(x)\mathrm{d}x\int_{\Omega_{\delta}}\nu(x\\!-\\!y)\mathrm{d}y\geq
2\widetilde{q}(2\delta)\int\limits_{\bigcup\limits_{i=1}^{n}\Omega\cap
B_{\delta}(a^{i})}[\varphi u]^{2}(x)\mathrm{d}x$ $\displaystyle\geq
2\widetilde{q}(2\delta)\int\limits_{\Omega\setminus\Omega_{\delta/2}}[\varphi
u]^{2}(x)\mathrm{d}x=2\widetilde{q}(2\delta)\int\limits_{\Omega\setminus\Omega_{\delta/2}}|u(x)|^{2}(x)\mathrm{d}x\,.$

This combined with (3.5) gives (3.7). Next, let us assume that $\Omega$ is a
Lipschitz domain. Following the same procedure as in [Pon04, Eq. (22) and Eq.
(23)] one arrives at

$\displaystyle 2\delta^{2}\iint\limits_{\Omega\Omega}\big{(}[\varphi
u](x)-[\varphi
u](y)\big{)}^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\geq\Big{(}\int\limits_{B_{2\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h\Big{)}\int\limits_{\Omega\setminus\Omega_{\delta/2}}[\varphi
u]^{2}(x)\mathrm{d}x.$

that is,

$\displaystyle\iint\limits_{\Omega\Omega}\big{(}[\varphi u](x)-[\varphi
u](y)\big{)}^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\geq
2q(2\delta)\int\limits_{\Omega\setminus\Omega_{\delta/2}}[\varphi
u]^{2}(x)\mathrm{d}x,$

which combined with (3.5) implies (3.8). ∎

Here is our global compactness result.

######  Theorem 3.10.

Let $\Omega$ be an open bounded subset of $\operatorname{\mathbb{R}}^{d}$ and
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty]$ be a
measurable function. If the couple $(\nu,\Omega)$ belongs to the class
$\mathscr{A}_{i},\leavevmode\nobreak\ i=1,2,3$ then the embedding
$H_{\nu}(\Omega)\hookrightarrow L^{2}(\Omega)$ is compact. In particular, the
embedding $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
L^{2}(\Omega)$ is compact.

###### Proof.

Given the continuous embedding
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
H_{\nu}(\Omega)$, it will be sufficient only to prove that the embedding
$H_{\nu}(\Omega)\hookrightarrow L^{2}(\Omega)$ is compact. For $(\nu,\Omega)$
belonging to the class $\mathbf{\mathscr{A}_{1}}$ the result is a direct
consequence of Theorem 3.6. Now assume $(\nu,\Omega)$ belongs to the class
$\mathbf{\mathscr{A}_{2}}$ (resp. $\mathbf{\mathscr{A}_{3}}$) then for
$\varepsilon>0$ there is $\delta>0$ small enough such that
$8q^{-1}(2\delta)<\varepsilon$ (resp.
$8\widetilde{q}^{-1}(2\delta)<\varepsilon$) If $(u_{n})_{n}$ is a bounded
sequence of $H_{\nu}(\Omega)$ then Corollary 3.7 infers the existence of a
subsequence $(u_{n_{j}})_{j}$ of $(u_{n})_{n}$ converging to some $u\in
H_{\nu}(\Omega)$ in $L^{2}(\Omega_{\delta/2})$ i.e
$\|u_{n_{j}}-u\|_{L^{2}(\Omega_{\delta/2})}\to 0$ as $j\to\infty$. In any
case, in view of 3.9, passing to the limsup in (3.7) or in (3.8) applied to
$u_{n_{j}}-u$ we get

$\displaystyle\limsup_{j\to\infty}\int_{\Omega}|u_{n_{j}}(x)-u(x)|^{2}\mathrm{d}x\leq
M\varepsilon$

where

$M=2\|u\|^{2}_{H_{\nu}(\Omega)}+2\sup_{n}\|u_{n}\|^{2}_{H_{\nu}(\Omega)}<\infty.$

Finally, $\limsup\limits_{j\to\infty}\|u_{n_{j}}-u\|_{L^{2}(\Omega)}=0$ since
$\varepsilon>0$ is arbitrarily chosen, which achieves the proof. ∎

######  Remark 3.11.

A noteworthy consequence of what we have obtained so far is that, for an
appropriate choice of $\nu,$ the well-known Rellich-Kondrachev compact
embeddings $H^{1}_{0}(\Omega)\hookrightarrow L^{2}(\Omega)$ and
$H^{1}(\Omega)\hookrightarrow L^{2}(\Omega)$ when $\Omega$ is Lipschitz,
respectively derive from Theorem 3.6 combined with the continuous embedding
$H^{1}_{0}(\Omega)\hookrightarrow
V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ and from Theorem 3.10
combined with the continuous embedding $H^{1}(\Omega)\hookrightarrow
H_{\nu}(\Omega)$ when $\Omega$ is Lipschitz.

The efforts made to establish Theorem 3.10 will be rewarded for the
elaboration of the Poincaré type inequality which will be useful in the
forthcoming section.

######  Theorem 3.12 (Poincaré inequality).

Let $\Omega$ be an open bounded subset of $\operatorname{\mathbb{R}}^{d}$ and
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty]$ be a
measurable function with full support. Assume the couple $(\nu,\Omega)$
belongs to one of the class $\mathscr{A}_{i},\leavevmode\nobreak\ i=0,1,2,3$.
Then there is exists a positive constant $C=C(d,\Omega,\nu)$ depending only on
$d,\leavevmode\nobreak\ \Omega$ and $\nu$ such that

$\displaystyle\big{\|}u-\mbox{$\fint_{\Omega}$}u\big{\|}^{2}_{L^{2}(\Omega)}\leq
C\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\quad\text{for
all }\leavevmode\nobreak\ u\in L^{2}(\Omega)\,,$ (3.9)

and hence

$\displaystyle\big{\|}u-\mbox{$\fint_{\Omega}$}u\big{\|}^{2}_{L^{2}(\Omega)}\leq
C\mathcal{E}(u,u)\,\qquad\text{for all }\leavevmode\nobreak\ u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\,.$ (3.10)

###### Proof.

Assume such constant does not exist then we can find a sequence $(u_{n})_{n}$
elements of $H_{\nu}(\Omega)$ such that for every $n$,
$\fint_{\Omega}u_{n}=0$, $\|u_{n}\|_{L^{2}(\Omega)}=1$ and

$\displaystyle\iint\limits_{\Omega\Omega}(u_{n}(x)-u_{n}(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\leq\frac{1}{2^{n}}\,.$

The sequence $(u_{n})_{n}$ is thus bounded in $H_{\nu}(\Omega)$ which by
Theorem 3.10 is compactly embedded in $L^{2}(\Omega)$ whenever $(\nu,\Omega)$
is in the class $\mathscr{A}_{i},\leavevmode\nobreak\ i=1,2,3$. Therefore, if
it is the case, passing through a subsequence, $(u_{n})_{n}$ converges in
$L^{2}(\Omega)$ to some function $u$. Clearly it follows that
$\fint_{\Omega}u=0$ and $\|u\|_{L^{2}(\Omega)}=1$. Moreover, by Fatou’s Lemma
we have

$\displaystyle\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\leq\liminf_{n\to\infty}\iint\limits_{\Omega\Omega}(u_{n}(x)-u_{n}(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y=0$

which implies that $u$ equals the constant function
$x\mapsto\fint_{\Omega}u=0$ almost everywhere on $\Omega$. This goes against
the fact that $\|u\|_{L^{2}(\Omega)}=1$ hereby showing that our initial
assumption was wrong.

Next assume $(\nu,\Omega)$ belongs to the class $\mathbf{\mathscr{A}_{0}}$
then, as $\nu$ has full a support, is unimodal and $\Omega$ is bounded, there
is a constant $c>0$ such that $\nu(x\\!-\\!y)\geq c$ for all $x,y\in\Omega$.
Using this and Jensen’s inequality we obtain the desired inequality as
follows:

$\displaystyle\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y$
$\displaystyle\geq
c|\Omega|\int_{\Omega}\fint_{\Omega}(u(x)-u(y))^{2}\mathrm{d}x\mathrm{d}y$
$\displaystyle\geq
c|\Omega|\|u-\mbox{$\fint_{\Omega}u$}\|^{2}_{L^{2}(\Omega)}\,.$

The proof is complete because (3.10) is a consequence of (3.9). ∎

The above Poincaré inequality (3.9)-(3.10) can be seen as the nonlocal
counterpart of the classical Poincaré inequality which states that, for a
connected bounded Lipschitz domain $\Omega$, there is $C>0$ for which

$\displaystyle\big{\|}u-\mbox{$\fint_{\Omega}$}u\big{\|}_{L^{2}(\Omega)}\leq
C\|\nabla u\|_{L^{2}(\Omega)}\,,\qquad\text{for all }\leavevmode\nobreak\ u\in
L^{2}(\Omega)$

where by convention we assume $\|\nabla u\|_{L^{2}(\Omega)}=\infty$ if
$|\nabla u|$ is not in $L^{2}(\Omega)$. Alongside to this we also recall the
classical Poincaré-Friedrichs inequality: there is $C>0$ such that

$\displaystyle\|u\|_{L^{2}(\Omega)}\leq C\|\nabla
u\|_{L^{2}(\Omega)}\,\qquad\text{for all }\leavevmode\nobreak\ u\in
H_{0}^{1}(\Omega)\,.$

In the same spirit, as we will see below the corresponding nonlocal Poincaré-
Friedrichs inequality $V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ (which
we recall is the closure of the $C_{c}^{\infty}(\Omega)$ in
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$) is much more easier to obtain
and no compactness argument is required. This provides an easier alternative
proof to the Poincaré-Friedrichs inequality from [FKV15, Lemma 2.7].
Furthermore, under the condition that the embedding is
$V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow L^{2}(\Omega)$
is compact, a similar inequality is proved in [JW19] wherein the authors only
assume $\Omega$ to be bounded in one direction.

######  Theorem 3.13 (Poincaré-Friedrichs inequality).

Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be open and bounded. Let
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty)$ be a
symmetric function such that one of the two conditions holds true:

  1. (i)

$\nu\mathds{1}_{\operatorname{\mathbb{R}}^{d}\setminus B_{R}}$ is nontrivial
and integrable for $R=\operatorname{diam}(\Omega)$ ,

  2. (ii)

$\nu\in L^{1}(\operatorname{\mathbb{R}}^{d})$ and $|\\{\nu>0\\}|>0$.

Then for some constant $C=C(d,\Omega,\nu)>0$

$\displaystyle\|u\|^{2}_{L^{2}(\Omega)}\leq C\mathcal{E}(u,u)\quad\text{for
all $u\in V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$.}$ (3.11)

###### Proof.

Set $R=\operatorname{diam}(\Omega)$. Then for all $x\in\Omega$ we have
$B^{c}_{R}(x)\subset\Omega^{c}$. For $u\in
V_{\nu,0}(\Omega|\operatorname{\mathbb{R}}^{d})$ we recall that
$u=0\leavevmode\nobreak\ a.e$ on $\Omega^{c}$. Thus,

$\displaystyle\mathcal{E}(u,u)$
$\displaystyle=\frac{1}{2}\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y+\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{\Omega^{c}}\nu(x\\!-\\!y)\mathrm{d}y$
$\displaystyle\geq
2\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{B^{c}_{R}(x)}\nu(x\\!-\\!y)\mathrm{d}y=2\|\nu_{R}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})}\|u\|^{2}_{L^{2}(\Omega)}.$

Take C= $(2\|\nu_{R}\|_{L^{1}(\operatorname{\mathbb{R}}^{d})})^{-1}$ with
$\nu_{R}=\nu\mathds{1}_{\operatorname{\mathbb{R}}^{d}\setminus B_{R}(0)}$.
This settles the first case. The second case is treated in [FKV15, Lemma 2.7].
∎

######  Remark 3.14.

Note that a Poincaré-Friedrichs inequality of the form

$\displaystyle\|u\|^{2}_{L^{2}(\Omega)}\leq
C\iint\limits_{\Omega\Omega}(u(x)-u(y))^{2}\nu(x\\!-\\!y)\mathrm{d}x\mathrm{d}y\qquad(u\in
C_{c}^{\infty}(\Omega))\,.$ (3.12)

does not hold in general, independently of whether the embedding
$H_{\nu}(\Omega)\hookrightarrow L^{2}(\Omega)$ is compact or not. For example,
consider $\nu(h)=|h|^{-d-\alpha}$ with $0<\alpha<1$. Then,
$C_{c}^{\infty}(\Omega)$ is dense in $H^{\alpha/2}(\Omega)$ but
$H^{\alpha/2}(\Omega)$ contains all constant functions. Thus (3.12) fails.
Note that, in the case $\nu(h)=|h|^{-d-\alpha},\alpha\in(0,2)$, a necessary
and sufficient condition on $\Omega$ for (3.12) to hold is provided in [DK22].

##  4\. Existence of weak solutions and spectral decomposition

This section is devoted to the following results: well-posedness of the
Neumann problem in Theorem 4.11, the spectral decomposition of the
corresponding operator in Theorem 4.15, the Robin problem in Theorem 4.16, and
the definition of the nonlocal Dirichlet-to-Neumann map in Theorem 4.20
together with its spectral decomposition in Theorem 4.23. We refer the reader
to Section 1 for comments about related expositions in the literature.

Throughout this section, $\Omega\subset\operatorname{\mathbb{R}}^{d}$ is
assumed to be open. We recall that the function
$\nu:\operatorname{\mathbb{R}}^{d}\setminus\\{0\\}\to[0,\infty]$ is assumed to
be symmetric and to satisfy the Lévy integrability condition (L). Let
$k:\operatorname{\mathbb{R}}^{d}\times\operatorname{\mathbb{R}}^{d}\setminus\operatorname{diag}\to[0,\infty)$
be symmetric and measurable such that for some $\Lambda\geq 1$

$\displaystyle\Lambda^{-1}\nu(y-x)\leq
k(x,y)\leq\Lambda\nu(y-x)\quad(x,y\in\operatorname{\mathbb{R}}^{d}).$ (E)

We will formulate well-posedness results for equations $Lu=f$ in $\Omega$,
where

$\displaystyle
Lu(x)=\operatorname{\operatorname{p.\\!v.}}\int_{\operatorname{\mathbb{R}}^{d}}\big{(}u(y)-u(x)\big{)}k(x,y)\mathrm{d}y\,.$
(4.1)

Note that the expression $Lu(x)$ does not exist in general if $u$ is smooth.
One would require additional assumptions on $k$. Note that $L$ can be
understood as an integro-differential operator. The aforementioned phenomenon
is similar to the fact that expressions like
$\operatorname{div}\big{(}A(x)\nabla u(x)\big{)}$ do not exist in general for
smooth functions $u$ without further assumptions on the matrix $A(x)$. Given
functions $u,v\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$, we define a
bilinear form $\operatorname{\mathcal{E}}$ by

$\displaystyle\mathcal{E}(u,v)=\frac{1}{2}\\!\\!\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u(x)-u(y)\big{)}\big{(}v(x)-v(y)\big{)}\,k(x,y)\mathrm{d}x\,\mathrm{d}y\,.$
(4.2)

Note that under the condition (E) the expression $\mathcal{E}(u,v)$ is well
defined for $u,v\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$.

######  Definition 4.1.

We define a nonlocal operator $\operatorname{\mathcal{N}\\!\\!}$ acting on
functions $v:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}$ by

$\displaystyle\mathcal{N}v(y)$
$\displaystyle=\int_{\Omega}(v(y)-v(x))k(x,y)\mathrm{d}x\qquad(y\in\Omega^{c}).$
(4.3)

Note that (4.3) requires some integrability condition of $v$. If $\nu$ is a
unimodal Lévy measure, then $\mathcal{N}v(y)$ is well defined for $v\in
L^{1}(\operatorname{\mathbb{R}}^{d},\widehat{\nu})$, see the beginning of the
proof of A.2. Furthermore, the definition of $\mathcal{N}v(y)$,
$y\in\Omega^{c}$, does not require any principal value integral, because there
is a positive distance between $y$ and $\overline{\Omega}$.

######  Remark 4.2.

(i) We work under the assumption (E) in order to establish well-posedness for
complement value problem. One could replace this assumption by the assumption

$\displaystyle\begin{split}\Lambda^{-1}\\!\\!\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u(x)-u(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y\leq\mathcal{E}(u,u)\leq\Lambda\\!\\!\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}\\!\\!\big{(}u(x)-u(y)\big{)}^{2}\,\nu(x\\!-\\!y)\mathrm{d}x\,\mathrm{d}y\end{split}$
(E’)

for all functions $u\in L^{2}_{loc}(\operatorname{\mathbb{R}}^{d})$. This
assumption allows for many more general cases of $k$, see the discussions in
[DK20, CS20]. (ii)Throughout this section we will work with the weight
$\widetilde{\nu}$. Analogous results hold true when choosing $\overline{\nu}$
or $\nu^{*}$ from 2.15.

###  4.1. Neumann boundary condition

In light of the Gauss-Green formula (A.5) it is reasonable to define weak
solutions of the Neumann problem under consideration as follows. Assume
$\Omega\subset\operatorname{\mathbb{R}}^{d}$ is an open set. Let
$f:\Omega\to\mathbb{R}$ and
$g:\operatorname{\mathbb{R}}^{d}\setminus\Omega\to\mathbb{R}$ be two
measurable functions. The Neumann problem for the operator $L$ associated to
the data $f$ and $g$ is to find a measurable function
$u:\operatorname{\mathbb{R}}^{d}\to\mathbb{R}$ such that

$\displaystyle Lu=f\quad\text{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Omega\quad\quad\text{ and
}\quad\quad\mathcal{N}u=g\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{on}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \operatorname{\mathbb{R}}^{d}\setminus\Omega.$ ($N$)

######  Definition 4.3.

Let $f\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\prime}$ and $g\in
T_{\nu}(\Omega^{c})^{\prime}$. We say that $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a weak solution or a
variational solution of the inhomogeneous Neumann problem ($N$) if

$\displaystyle\mathcal{E}(u,v)=\langle f,v\rangle+\langle
g,v\rangle\quad\mbox{for all}\leavevmode\nobreak\ \leavevmode\nobreak\ v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\,,$ ($V^{\prime}$)

where we use the natural embedding
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\hookrightarrow
T_{\nu}(\Omega^{c})$. Note that the existence of a solution $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ implies the compatibility
condition $\langle f,1\rangle+\langle g,1\rangle=0$.

If, in particular, $f\in L^{2}(\Omega)$ and $g\in
L^{2}(\Omega^{c},\widetilde{\nu}^{-1})$, then $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is a weak solution of ($N$) if

$\displaystyle\mathcal{E}(u,v)=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\Omega^{c}}g(y)v(y)\mathrm{d}y,\quad\mbox{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})\,.$ ($V$)

In this case, the compatibility condition reads

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

######  Remark 4.4.

The compatibility condition ($C$) is an implicit necessary requirement. Recall
that the local counterpart of this compatibility condition associated with
(1.1), where $g$ is defined on $\partial\Omega$, is given by

$\displaystyle\int_{\Omega}f(x)\mathrm{d}x+\int_{\partial\Omega}g(y)\mathrm{d}\sigma(y)=0.$
(4.4)

######  Remark 4.5.

(i) Note that [DROV17, Def. 3.6] looks very similar to ($V$) at first glance.
However, the norm of the test space defined in [DROV17, Eq. (3.1)] depends on
the Neumann data $g$, which is not natural. Our test space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in the weak formulation ($V$)
does not depend on the Neumann data $g$. Moreover for the existence of weak
solutions to ($N$), it is sufficient to choose $f\in L^{2}(\Omega)$ and $g\in
L^{2}(\Omega^{c},\widetilde{\nu}^{-1})$, see Theorem 4.12. (ii) For non-
singular kernels, 4.3 coincides with the definition in [DTZ22, Section 3.2].

The next result shows that both problems ($N$) and ($V$) are related under
additional regularity assumption.

######  Proposition 4.6.

Let $\Omega$ be an open bounded subset of $\operatorname{\mathbb{R}}^{d}$ with
Lipschitz boundary. Assume $k(x,y)=\nu(y-x)$, i.e., $\Lambda=1$. Let $u\in
C^{2}_{b}(\operatorname{\mathbb{R}}^{d})$, $f\in L^{2}(\Omega)$ and $g\in
L^{2}(\Omega^{c},\widetilde{\nu}^{-1})$. Then $u$ satisfies ($N$) if and only
if $f$ and $g$ are compatible in the sense of ($C$) and $u$ satisfies ($V$).

###### Proof.

If $u$ solves ($N$) i.e. $Lu=f$ in $\Omega$ and $\mathcal{N}u=g$ on
$\Omega^{c}$, then by the Gauss-Green formula (A.5) we obtain the following

$\displaystyle\mathcal{E}(u,v)=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\Omega^{c}}g(y)v(y)\mathrm{d}y,\quad\mbox{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ v\in
C^{1}_{b}(\operatorname{\mathbb{R}}^{d}).$ (4.5)

As shown in (4.9)-(4.10) below, all terms involved in (4.5) are linear and
continuous on $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ with respect to
the variable $v$. Moreover smooth functions of compact support are dense in
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ hence the relation in (4.5)
remains true for functions $v$ in
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ that is ($V$) is satisfied. In
particular taking $v=1$ one gets the condition ($C$).

Conversely, assume $u$ solves ($V$) then inserting the Gauss-Green formula
(A.5) with $v\in C^{1}_{b}(\operatorname{\mathbb{R}}^{d})\subset
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ in (4.5) yields

$\displaystyle\int_{\Omega}Lu(x)v(x)\mathrm{d}x-\int_{\Omega}f(x)v(x)\mathrm{d}x=\int_{\Omega^{c}}g(y)v(y)\mathrm{d}y-\int_{\Omega^{c}}\mathcal{N}u(y)v(y)\mathrm{d}y,\quad\mbox{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ v\in
C^{1}_{b}(\operatorname{\mathbb{R}}^{d}).$

Specializing this relation for $v\in C^{\infty}_{c}(\Omega)$ and $v\in
C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d}\setminus\overline{\Omega})$
respectively we end up with

$\displaystyle\int_{\Omega}Lu(x)v(x)\mathrm{d}x-\int_{\Omega}f(x)v(x)\mathrm{d}x$
$\displaystyle=0\qquad$ $\displaystyle\mbox{ for all}\leavevmode\nobreak\
\leavevmode\nobreak\ v\in C^{\infty}_{c}(\Omega),$
$\displaystyle\int_{\Omega^{c}}g(y)v(y)\mathrm{d}y-\int_{\Omega^{c}}\mathcal{N}u(y)v(y)\mathrm{d}y$
$\displaystyle=0$ $\displaystyle\mbox{ for all}\leavevmode\nobreak\
\leavevmode\nobreak\ v\in
C^{\infty}_{c}(\operatorname{\mathbb{R}}^{d}\setminus\overline{\Omega}).$

Recall that, by A.1, $Lu$ is well defined and bounded. Hence, $Lu$ belongs to
$L^{2}(\Omega)$. Similarly $\mathcal{N}u$ is well defined and bounded, i.e.,
it belongs to $L^{\infty}(\Omega^{c})$. Thus, up to null sets, we conclude
from the above equations $Lu=f$ in $\Omega$ and $\mathcal{N}u=g$ on
$\operatorname{\mathbb{R}}^{d}\setminus\Omega$, which proves ($N$). ∎

Both integrodifferential operators $L$ and $\mathcal{N}$ annihilate additive
constants. Whence as long as $u$ is a solution to the system ($N$) or to the
variational problem ($V$) so is the function $\widetilde{u}=u+c$ for any
$c\in\mathbb{R}$. Accordingly, both problems are ill-posed in the sense of
Hadamard. The situation is likewise in the local setting with the operators
$L$ and $\mathcal{N}$ respectively replaced by the operators $-\Delta$ and
$\frac{\partial}{\partial n}$. In order to overcome this issue, it is common
to introduce an appropriate function space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$ as follows:

$\displaystyle
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}:=\Big{\\{}u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d}):\int_{\Omega}u(x)\mathrm{d}x=0\Big{\\}}.$

Assuming that $\Omega$ is bounded, the space
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$ endowed with the
scalar product of $V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})$ is Hilbert
space as well. Instead of ($V$) we need to consider the following weak
formulations:

$\displaystyle\mathcal{E}(u,v)$ $\displaystyle=\langle f,v\rangle+\langle
g,v\rangle$ $\displaystyle\text{ for all }v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}\,,$
($V^{{}^{\prime}\perp}$) $\displaystyle\mathcal{E}(u,v)$
$\displaystyle=\int_{\Omega}f(x)v(x)\mathrm{d}x+\int_{\Omega^{c}}g(y)v(y)\mathrm{d}y\quad$
$\displaystyle\text{ for all }v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}\,.$ ($V^{\perp}$)

In contrast to ($V$), the variational problem ($V^{\perp}$) possesses at most
one solution since $\mathcal{E}(\cdot,\cdot)$ defines a scalar product on
$V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$. Analogous
observations can be made in the local setting by introducing the space
$H^{1}(\Omega)^{\perp}=\big{\\{}u\in
H^{1}(\Omega):\int_{\Omega}u(x)\mathrm{d}x=0\big{\\}}$.

By standard procedure, a solution of the variational problem ($V$) is
characterized as a critical point (a minimizer) of the functional

$\displaystyle\mathcal{J}(v)$
$\displaystyle=\frac{1}{2}\mathcal{E}(v,v)-\int_{\Omega}fv\,\mathrm{d}x-\int_{\Omega^{c}}gv\mathrm{d}x$
(4.6)
$\displaystyle=\frac{1}{4}\iint\limits_{(\Omega^{c}\times\Omega^{c})^{c}}(v(x)-v(y))^{2}\nu(x\\!-\\!y)\,\mathrm{d}x\,\mathrm{d}y-\int_{\Omega}fv\,\mathrm{d}x-\int_{\Omega^{c}}gv\mathrm{d}x.$

######  Proposition 4.7.

Let $\Omega\subset\operatorname{\mathbb{R}}^{d}$ be an open set. Then a
function $u\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$ is a
solution to ($V^{\perp}$) if and only if $u$ is a solution of the minimization
problem

$\displaystyle\mathcal{J}(u)=\min_{v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}}\mathcal{J}(v).$
($M^{\perp}$)

Moreover, if $f:\Omega\to\mathbb{R}$ and $g:\Omega^{c}\to\mathbb{R}$ are
compatible in the sense of ($C$), $u\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$ solves ($V^{\perp}$) if
and only if for any $c\in\mathbb{R}$, $u+c$ solves the variational problem
($V$) and the latter problem is equivalent to the minimization problem

$\displaystyle\mathcal{J}(u)=\min_{v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})}\mathcal{J}(v).$ ($M$)

###### Proof.

Let $u\in V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$ so that
($V^{\perp}$) holds true for all $v\in
V_{\nu}(\Omega|\operatorname{\mathbb{R}}^{d})^{\perp}$. Employing Cauchy-
Schwartz inequality yields

$\displaystyle\mathcal{E}(u,v)$
$\displaystyle\leq\frac{1}{2}\mathcal{E}(u,u)+\frac{1}{2}\mathcal{E}(v,v)=\mathcal{E}(u,u)-\frac{1}{2}\mathcal{E}(u,u)+\frac{1}{2}\mathcal{E}(v,v).$

In virtue of ($V^{\perp}$) we get $\mathcal{J}(u)\leq\mathcal{J}(v)$ and thus
$u$ solves ($M^{\perp}$).

Conversely assume that $u$ satisfies ($M^{\perp}$) which means that
$\mathcal{J}(u)\leq\mathcal{J}(v)$ for all $v\in