# Bourgain, Brezis and Mironescu theorem for fractional Sobolev spaces with
variable exponents

Minhyun Kim  Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld,
Germany<EMAIL_ADDRESS>
###### Abstract.

A Bourgain–Brezis–Mironescu-type theorem for fractional Sobolev spaces with
variable exponents is established for sufficiently regular functions. We
prove, however, that a limiting embedding theorem for these spaces fails to
hold in general.

###### Key words and phrases:

fractional Sobolev space, variable exponent, limiting embedding

###### 2020 Mathematics Subject Classification:

46E35

Minhyun Kim gratefully acknowledges financial support by the German Research
Foundation (GRK 2235 - 282638148).

##  1\. Introduction

In a celebrated work [15], Bourgain, Brezis and Mironescu study the asymptotic
behavior of the fractional Sobolev seminorms when the order of
differentiability approaches one. Their results are concerned with smooth
bounded domains, but the same arguments work for $W^{1,p}$-extension domain.
More precisely, if $\Omega\subset\mathbb{R}^{n}$ is a $W^{1,p}$-extension
domain, then an equality

(1.1) $\lim_{s\nearrow
1}s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,\mathrm{d}y\,\mathrm{d}x=K_{n,p}\int_{\Omega}|\nabla
u(x)|^{p}\,\mathrm{d}x$

holds for any $u\in W^{1,p}(\Omega)$, where $n\in\mathbb{N}$,
$p\in(1,+\infty)$ and

(1.2)
$K_{n,p}=\frac{1}{p}\int_{\mathbb{S}^{n-1}}|\omega_{n}|^{p}\,\mathrm{d}\mathcal{H}^{n-1}(\omega)=2\pi^{\frac{n-1}{2}}\frac{\Gamma(\frac{p+1}{2})}{\Gamma(\frac{n+p}{2})}\frac{1}{p}.$

The equality (1.1) suggests a nonlocal approximation of the Sobolev norm and
yields a limiting embedding $W^{s,p}(\Omega)\to W^{1,p}(\Omega)$ as $s\nearrow
1$. Note that $\Omega$ is allowed to be unbounded. For the second equality in
(1.2), see, for instance, [34, Lemma 2.1] or [30, Proposition 3.9].

In this paper, we consider the Sobolev and fractional Sobolev spaces with
variable exponents. The Sobolev spaces with variable exponents
$W^{1,p(\cdot)}$ appear naturally from variational problems [2, 5, 19, 26,
46], modeling of the electrorheological fluids [43, 44, 45], thermorheological
fluids [6] and image processing [1, 14, 18, 38]. We refer the reader to the
books [21, 24] for a comprehensive study. The fractional Sobolev spaces with
variable exponents $W^{s,p(\cdot,\cdot)}$ are introduced recently in [35], and
studied extensively in various contexts: [7, 8, 9, 10, 11, 12, 13, 17, 22, 32,
33, 42].

This paper aims to answer a question whether a Bourgain–Brezis–Mironescu-type
result is true for the fractional Sobolev spaces with variable exponents. On
the one hand, the answer is affirmative in the sense that an equality similar
to (1.1), which takes the variable exponents into account, holds for functions
that are sufficiently regular. However, on the other hand, the answer is
negative in the sense that this equality fails to holds in general for
functions in Lebesgue spaces—even in Sobolev spaces—with variable exponents.
It implies that a limiting embedding for the fractional Sobolev spaces with
variable exponents, corresponding to $W^{s,p}\to W^{1,p}$, is not true in
general.

Let $\Omega\subset\mathbb{R}^{n}$ be any open set (not necessarily bounded).
Let $p:\Omega\times\Omega\to\mathbb{R}$ be a measurable function satisfying

(1.3)
$1<p_{-}:=\operatorname*{ess\,inf}_{x,y\in\Omega}p(x,y)\leq\operatorname*{ess\,sup}_{x,y\in\Omega}p(x,y)=:p_{+}<+\infty,$

and define $\bar{p}(x):=p(x,x)$. Let $L^{\bar{p}(\cdot)}(\Omega)$,
$W^{1,\bar{p}(\cdot)}(\Omega)$ and $W^{s,p(\cdot,\cdot)}(\Omega)$ denote the
Lebesgue, Sobolev and fractional Sobolev spaces with variable exponents,
respectively. See Section 2 for the precise definitions. We begin with an
extension of (1.1) to an equality which takes a variable exponent $p$ into
account.

######  Theorem 1.1.

Assume that $p$ satisfies (1.3) and

(1.4) $r^{-\operatorname{osc}_{B_{r}(x)}p(x,\cdot)}\leq L\quad\text{for all
}x\in\Omega\text{ and }0<r<\min\\{\mathrm{dist}(x,\partial\Omega),1\\}$

for some constant $L\geq 1$. Then for any $u\in C^{2}_{c}(\mathbb{R}^{n})$

(1.5) $\lim_{s\nearrow
1}s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\,\mathrm{d}x=\int_{\Omega}K_{n,\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}\,\mathrm{d}x,$

where

$K_{n,\bar{p}(x)}=\frac{1}{\bar{p}(x)}\int_{\mathbb{S}^{n-1}}|\omega_{n}|^{p(x)}\,\mathrm{d}\mathcal{H}^{n-1}(\omega)=2\pi^{\frac{n-1}{2}}\frac{\Gamma(\frac{\bar{p}(x)+1}{2})}{\Gamma(\frac{n+\bar{p}(x)}{2})}\frac{1}{\bar{p}(x)}.$

The condition (1.4) says that for a fixed point $x\in\Omega$ the function
$p(x,\cdot)$ is log-Hölder continuous at $x$. We emphasize that it does not
impose any regularity on $x$-variable.

Theorem 1.1 is not a full extension of (1.1) because (1.5) holds for functions
that are sufficiently regular. As a natural extension of
Bourgain–Brezis–Mironescu theorem, one may expect that the equality (1.5)
holds for all functions $u\in W^{1,\bar{p}(\cdot)}(\Omega)$. However, this is
not true in general even when the variable exponent $p$ is smooth, which is in
sharp contrast to the case when $p$ is constant.

######  Theorem 1.2.

There exists a smooth variable exponent $p(\cdot,\cdot)$ such that

$W^{1,\bar{p}(\cdot)}(\Omega)\not\subset W^{s,p(\cdot,\cdot)}(\Omega)$

for all $s\in(0,1)$.

The following result is an immediate consequence of Theorem 1.2, which shows
that a limiting embedding $W^{s,p(\cdot,\cdot)}\to W^{1,\bar{p}(\cdot)}$ as
$s\nearrow 1$ is not true in general.

######  Corollary 1.3.

There exists a smooth variable exponent $p$ such that the equality (1.5) does
not hold for some $u\in W^{1,\bar{p}(\cdot)}(\Omega)$.

Corollary 1.3 tells us that the target space $W^{1,\bar{p}(\cdot)}$ is too
large for (1.5) to be true in general. The failure of $W^{1,\bar{p}(\cdot)}$
can be explained by the following observation: let us forget about
$p(\cdot,\cdot)$ and suppose that a variable exponent $\bar{p}(\cdot)$ is
given. What is the natural fractional Sobolev space as an interpolation of
$L^{\bar{p}(\cdot)}$ and $W^{1,\bar{p}(\cdot)}$? Since there are too many
choices of $p(\cdot,\cdot)$ satisfying (1.3) and $p(x,x)=\bar{p}(x)$, there is
no way to find the natural interpolation space. This is because the spaces
$L^{\bar{p}(\cdot)}$ and $W^{1,\bar{p}(\cdot)}$ do not contain any information
about $p(\cdot,\cdot)$.

The following result nevertheless shows that some Sobolev functions, in a
smaller space than $W^{1,\bar{p}(\cdot)}$, satisfy the equality (1.5).

######  Corollary 1.4.

Let $\Omega$ be a $W^{1,p_{+}}$\- and $W^{1,p_{-}}$-extension domain. Assume
that $p$ satisfies (1.3) and (1.4). Then the equality (1.5) holds for all
$u\in W^{1,p_{+}}(\Omega)\cap W^{1,p_{-}}(\Omega)$.

It is worth mentioning the results in [20, 29], where the authors provide
nonlocal approximations to the Sobolev norms with variable exponents $p(x)$.
Their approximations involve a suitable class of nonlocal nonconvex energy
functionals, generalizing the constant case [16, 40, 41], and their proofs
make use of the Hardy–Littlewood maximal function with variable exponent
$p(x)$. Since the maximal inequality with variable exponents in a modular form
fails in general [37], these approximations require an additional assumption
$u\in W^{1,p_{+}}\cap W^{1,p_{-}}$ as well.

Let us also mention other related results in the literature. The
Bourgain–Brezis–Mironescu theorem is extended to fractional Sobolev norms with
Lévy-type measures [30, 31], Orlicz growth [3, 28] and anisotropic structure
[25]. The asymptotic behavior of the fractional Sobolev seminorms as
$s\searrow 0$ is first established in [39], and extended those with Orlicz
growth [4] and anisotropic structure [25].

The paper is organized as follows. In Section 2, we recall definitions of the
Lebesgue, Sobolev and fractional Sobolev spaces with variable exponents. The
asymptotic behavior of the fractional Sobolev seminorms with variable
exponents (in a modular form), for regular functions, is proved in Section 3.
Section 4 is devoted to the proof of Theorem 1.2 and Corollary 1.3.

##  2\. Preliminaries

Let us briefly recall definitions of the Lebesgue, Sobolev and fractional
Sobolev spaces with variable exponents. Let $\Omega\subset\mathbb{R}^{n}$ be
an open set and let $q:\Omega\to\mathbb{R}$ be a measurable function
satisfying

$1<\operatorname*{ess\,inf}_{x\in\Omega}q(x)\leq\operatorname*{ess\,sup}_{x\in\Omega}q(x)<+\infty.$

The space

$L^{q(\cdot)}(\Omega)=\left\\{u:\Omega\to\mathbb{R}\text{
measurable}:\varrho_{q(\cdot)}(u/\lambda)<+\infty\text{ for some
}\lambda>0\right\\}$

equipped with the norm

$\|u\|_{q(\cdot)}:=\|u\|_{L^{q(\cdot)}(\Omega)}:=\inf\left\\{\lambda>0:\varrho_{q(\cdot)}(u/\lambda)\leq
1\right\\}$

is called the variable exponent Lebesgue space, where

$\varrho_{q(\cdot)}(u):=\varrho_{L^{q(\cdot)}(\Omega)}(u):=\int_{\Omega}|u(x)|^{q(x)}\,\mathrm{d}x$

is a modular. The Sobolev space with variable exponent is defined by

$W^{1,q(\cdot)}(\Omega)=\left\\{u\in L^{q(\cdot)}(\Omega):D_{i}u\in
L^{q(\cdot)}(\Omega)\text{ for all }i=1,\dots,n\right\\}$

with the norm

$\|u\|_{1,q(\cdot)}:=\|u\|_{W^{1,q(\cdot)}(\Omega)}:=\|u\|_{q(\cdot)}+\sum_{1\leq
i\leq n}\|D_{i}u\|_{q(\cdot)}.$

It is well known that $L^{q(\cdot)}(\Omega)$ and $W^{1,q(\cdot)}(\Omega)$ are
Banach spaces, see [21, 24, 27, 36] for instance.

The fractional Sobolev spaces with variable exponents are first introduced in
[35] and studied in various contexts. Let $p:\Omega\times\Omega\to\mathbb{R}$
be a measurable function satisfying (1.3) and define $\bar{p}(x)=p(x,x)$. For
$s\in(0,1)$, the fractional Sobolev space with variable exponent is defined by

$W^{s,p(\cdot,\cdot)}(\Omega)=\left\\{u\in
L^{\bar{p}(\cdot)}(\Omega):\varrho_{s,p(\cdot,\cdot)}(u/\lambda)<+\infty\text{
for some }\lambda>0\right\\}$

with the norm

$\|u\|_{s,p(\cdot,\cdot)}:=\|u\|_{W^{s,p(\cdot,\cdot)}(\Omega)}:=\|u\|_{p(\cdot)}+[u]_{s,p(\cdot,\cdot)},$

where

$[u]_{s,p(\cdot,\cdot)}:=[u]_{W^{s,p(\cdot,\cdot)}(\Omega)}:=\inf\left\\{\lambda>0:\varrho_{s,p(\cdot,\cdot)}(u/\lambda)\leq
1\right\\}$

is a seminorm and

$\varrho_{s,p(\cdot,\cdot)}(u):=\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u):=s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\,\mathrm{d}x.$

It is easy to check that $W^{s,p(\cdot,\cdot)}(\Omega)$ is a Banach space by
following the arguments in [23, Proposition 4.24]. Obviously, we have
$W^{1,p_{+}}(\Omega)\cap W^{1,p_{-}}(\Omega)\subset W^{s,p_{+}}(\Omega)\cap
W^{s,p_{-}}(\Omega)\subset W^{s,p(\cdot,\cdot)}(\Omega)$. Let us collect some
useful properties for the the fractional Sobolev spaces with variables
exponents. They follow immediately from the definition.

######  Lemma 2.1.

Let $u\in W^{s,p(\cdot,\cdot)}(\Omega)$, then

$\min\left\\{[u]_{s,p(\cdot,\cdot)}^{p_{+}},[u]_{s,p(\cdot,\cdot)}^{p_{-}}\right\\}\leq\varrho_{s,p(\cdot,\cdot)}(u)\leq\max\left\\{[u]_{s,p(\cdot,\cdot)}^{p_{+}},[u]_{s,p(\cdot,\cdot)}^{p_{-}}\right\\}.$

######  Lemma 2.2.

Let $u,u_{k}\in W^{s,p(\cdot,\cdot)}(\Omega)$ for $k\in\mathbb{N}$. Then the
following are equivalent:

  1. (i)

$[u_{k}-u]_{s,p(\cdot,\cdot)}\to 0$ as $k\to\infty$,

  2. (ii)

$\varrho_{s,p(\cdot,\cdot)}(u_{k}-u)\to 0$ as $k\to\infty$.

In particular, if one of the assertions is satisfied, then
$\varrho_{s,p(\cdot,\cdot)}(u_{k})\to\varrho_{s,p(\cdot,\cdot)}(u)$ as
$k\to\infty$.

##  3\. Limiting behavior of norms for regular functions

In this section, we study the limiting behavior of fractional Sobolev
seminorms with variable exponents (in a modular form) for sufficiently regular
functions. Let us first prove Theorem 1.1, which follows from the following
two lemmas: Lemma 3.1 and Lemma 3.2. Throughout the paper, we denote by $C$
strictly positive constants whose exact values are not important. These
constants might change from line to line.

######  Lemma 3.1.

Let $\Omega\subset\mathbb{R}^{n}$ be open and assume that $p$ satisfies (1.3)
and (1.4). Then

(3.1) $\int_{\Omega}K_{n,\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}\,\mathrm{d}x\leq\liminf_{s\nearrow
1}s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\,\mathrm{d}x$

for any $u\in C^{2}(\overline{\Omega})$.

###### Proof.

Let us fix $\varepsilon>0$, $\theta_{0}\in(0,1)$, $R>0$ and $x\in\Omega\cap
B_{R}$. Since $u\in C^{2}(\overline{\Omega})$, there exists a constant $C>0$,
depending on $R$ and $\|u\|_{C^{2}(\overline{\Omega})}$, such that

(3.2) $|u(x+h)-u(x)-\nabla u(x)\cdot h|\leq C|h|^{2}$

for all $|h|<\mathrm{dist}(x,\partial\Omega)$. Since $p(x,\cdot)$ is
continuous at $x$, we find $r_{1}\in(0,\mathrm{dist}(x,\partial\Omega))$,
depending on $\varepsilon$, $x$, $|\nabla u(x)|$ and $p$, such that

(3.3) $(1-\varepsilon)|\nabla u(x)|^{p(x,x)}\leq|\nabla
u(x)|^{p(x,y)}\leq(1+\varepsilon)|\nabla u(x)|^{p(x,x)}$

holds for all $y\in B_{r_{1}}(x)$. Moreover, by continuity of $p(x,\cdot)$ at
$x$ again, there exists $r_{2}\in(0,\mathrm{dist}(x,\partial\Omega))$,
depending on $\varepsilon$, $x$ and $\theta_{0}$, such that

(3.4)
$(1-\varepsilon)\theta^{p(x,x)}\leq\theta^{p(x,y)}\leq(1+\varepsilon)\theta^{p(x,x)}$

holds for all $y\in B_{r_{2}}(x)$ and $\theta\in[\theta_{0},1]$. We set
$r_{0}=\min\\{r_{1},r_{2},1\\}$ and let $r\in(0,r_{0})$.

For $x\in\Omega\cap B_{R}$ and $h\in B_{r}$, we have from (3.2)

$|\nabla u(x)\cdot h|\leq|u(x+h)-u(x)|+C|h|^{2}.$

Using an inequality

(3.5) $(a+b)^{q}\leq(1+\varepsilon)a^{q}+C(q,\varepsilon)b^{q},\quad a,b\geq
0,q>1,$

we obtain

$|\nabla u(x)\cdot
h|^{p(x,x+h)}\leq(1+\varepsilon)|u(x+h)-u(x)|^{p(x,x+h)}+C|h|^{2p(x,x+h)}$

for some $C=C(R,\|u\|_{C^{2}(\overline{\Omega})},p_{+},p_{-},\varepsilon)>0$.
Multiplying by $|h|^{-n-sp(x,x+h)}$ and integrating over $B_{r}$ with respect
to $h$, we have

(3.6) $\begin{split}I&:=\int_{B_{r}}\frac{|\nabla u(x)\cdot
h|^{p(x,x+h)}}{|h|^{n+sp(x,x+h)}}\,\mathrm{d}h\\\
&\leq(1+\varepsilon)\int_{B_{r}}\frac{|u(x+h)-u(x)|^{p(x,x+h)}}{|h|^{n+sp(x,x+h)}}\,\mathrm{d}h+C\int_{B_{r}}\frac{|h|^{2p(x,x+h)}}{|h|^{n+sp(x,x+h)}}\,\mathrm{d}h\end{split}$

for each $x\in\Omega\cap B_{R}$.

Let us estimate $I$ from below. By using the spherical coordinates, we write

$I=\int_{0}^{r}\int_{\mathbb{S}^{n-1}}|\nabla
u(x)\cdot\omega|^{p(x,x+\rho\omega)}\rho^{-1+(1-s)p(x,x+\rho\omega)}\,\mathrm{d}\mathcal{H}^{n-1}(\omega)\,\mathrm{d}\rho.$

We use the change of variables $\omega=A\xi$, where $A\in O(n)$ is a rotation
satisfying $\nabla u(x)\cdot\omega=|\nabla u(x)|\xi_{n}$, so that we have

(3.7) $I=\int_{0}^{r}\int_{\mathbb{S}^{n-1}}|\nabla u(x)|^{p(x,x+\rho
A\xi)}|\xi_{n}|^{p(x,x+\rho A\xi)}\rho^{-1+(1-s)p(x,x+\rho
A\xi)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\,\mathrm{d}\rho.$

Using (3.3) and (3.4), we obtain

$I\geq(1-\varepsilon)^{2}\left(\int_{0}^{r}\int_{\\{|\xi_{n}|\geq\theta_{0}\\}}|\xi_{n}|^{\bar{p}(x)}\rho^{-1+(1-s)p(x,x+\rho
A\xi)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\,\mathrm{d}\rho\right)|\nabla
u(x)|^{\bar{p}(x)}.$

Moreover, since the assumption (1.4) yields

(3.8)
$L^{-1}\leq\min\\{1,\rho^{\operatorname{osc}_{B_{\rho}(x)}p(x,\cdot)}\\}\leq\rho^{p(x,x+\rho
A\xi)-\bar{p}(x)}\leq\max\\{1,\rho^{-\operatorname{osc}_{B_{\rho}(x)}p(x,\cdot)}\\}\leq
L,$

$I$ can be further estimated as

$\begin{split}I&\geq\frac{(1-\varepsilon)^{2}}{L^{1-s}}\left(\int_{0}^{r}\int_{\\{|\xi_{n}|\geq\theta_{0}\\}}|\xi_{n}|^{\bar{p}(x)}\rho^{-1+(1-s)\bar{p}(x)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\,\mathrm{d}\rho\right)|\nabla
u(x)|^{\bar{p}(x)}\\\
&\geq\frac{(1-\varepsilon)^{2}}{(1-s)L^{1-s}}\left(\int_{\\{|\xi_{n}|\geq\theta_{0}\\}}|\xi_{n}|^{\bar{p}(x)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\right)\frac{r^{(1-s)\bar{p}(x)}}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}.\end{split}$

Since

$\begin{split}\int_{\\{|\xi_{n}|\geq\theta_{0}\\}}|\xi_{n}|^{\bar{p}(x)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)&\geq\int_{\mathbb{S}^{n-1}}|\xi_{n}|^{\bar{p}(x)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)-\int_{\\{|\xi_{n}|<\theta_{0}\\}}\theta_{0}^{p_{-}}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\\\
&\geq\bar{p}(x)K_{n,\bar{p}(x)}-\theta_{0}|\mathbb{S}^{n-1}|,\end{split}$

we arrive at

(3.9)
$I\geq\frac{(1-\varepsilon)^{2}}{(1-s)L^{1-s}}\left(\bar{p}(x)K_{n,\bar{p}(x)}-\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{r^{(1-s)\bar{p}(x)}}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}.$

For the last term on the right-hand side of (3.6), we use the assumption (1.3)
to have

(3.10)
$\int_{B_{r}}|h|^{-n+(2-s)p(x,x+h)}\,\mathrm{d}h\leq\int_{B_{r}}|h|^{-n+(2-s)p_{-}}\,\mathrm{d}h\leq|\mathbb{S}^{n-1}|\frac{r^{(2-s)p_{-}}}{(2-s)p_{-}}.$

Therefore, combining (3.6), (3.9), (3.10) and integrating over $\Omega\cap
B_{R}$ yield

$\begin{split}&(1-\varepsilon)^{2}\frac{s}{L^{1-s}}\int_{\Omega\cap
B_{R}}\left(\bar{p}(x)K_{n,\bar{p}(x)}-\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{r^{(1-s)\bar{p}(x)}}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}\,\mathrm{d}x\\\
&\leq(1+\varepsilon)s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\,\mathrm{d}x+C\frac{s(1-s)}{2-s}r^{(2-s)p_{-}}\end{split}$

for some $C>0$ independent of $s$. We take $\liminf_{s\nearrow 1}$ on both
sides to deduce

$\begin{split}&(1-\varepsilon)^{2}\int_{\Omega\cap
B_{R}}\left(\bar{p}(x)K_{n,\bar{p}(x)}-\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{1}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}\,\mathrm{d}x\\\ &\leq(1+\varepsilon)\liminf_{s\nearrow
1}s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\,\mathrm{d}x.\end{split}$

Since $\varepsilon$, $\theta_{0}$ and $R$ are arbitrarily chosen, we conclude
(3.1). ∎

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ be open and assume that $p$ satisfies (1.3)
and (1.4). Then

(3.11) $\limsup_{s\nearrow
1}s(1-s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\,\mathrm{d}x\leq\int_{\Omega}K_{n,\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}\,\mathrm{d}x$

for any $u\in C^{2}_{c}(\mathbb{R}^{n})$.

To prove Lemma 3.2, we use the dominated convergence theorem. For this
purpose, we first prove the following lemma.

######  Lemma 3.3.

Assume that $p$ satisfies (1.3). Let $s_{0}\in(0,1)$ and $u\in
C^{1}_{c}(\mathbb{R}^{n})$. Then, there exists a function $F\in L^{1}(\Omega)$
such that

(3.12)
$F_{s}(x):=s(1-s)\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\leq
F(x),\quad x\in\Omega,$

for all $s\in[s_{0},1)$.

###### Proof.

Let us write $F_{s}(x)=I_{1}+I_{2}$, where

$\begin{split}I_{1}&=s(1-s)\int_{\Omega\cap
B_{1}(x)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\quad\text{and}\\\
I_{2}&=s(1-s)\int_{\Omega\setminus
B_{1}(x)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y.\end{split}$

Using

$|u(y)-u(x)|\leq\int_{0}^{1}|\nabla
u(x+t(y-x))\cdot(y-x)|\,\mathrm{d}t\leq\|\nabla
u\|_{L^{\infty}(\mathbb{R}^{n})}|y-x|$

and (1.3), we have

(3.13) $\begin{split}I_{1}&\leq(1-s)\int_{\Omega\cap B_{1}(x)}\|\nabla
u\|_{L^{\infty}(\mathbb{R}^{n})}^{p(x,y)}|x-y|^{-n+(1-s)p(x,y)}\,\mathrm{d}y\\\
&\leq(1-s)\left(\|\nabla u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|\nabla
u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right)\int_{B_{1}}|h|^{-n+(1-s)p_{-}}\,\mathrm{d}h\\\
&=\frac{|\mathbb{S}^{n-1}|}{p_{-}}\left(\|\nabla
u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|\nabla
u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right).\end{split}$

Moreover, using (1.3) again, we obtain

(3.14) $\begin{split}I_{2}&\leq 2^{p_{+}}s\int_{\Omega\setminus
B_{1}(x)}\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p(x,x+h)}|x-y|^{-n-sp(x,y)}\,\mathrm{d}y\\\
&\leq
2^{p_{+}}s\left(\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right)\int_{\mathbb{R}^{n}\setminus
B_{1}}|h|^{-n-sp_{-}}\,\mathrm{d}h\\\
&=2^{p_{+}}\frac{|\mathbb{S}^{n-1}|}{p_{-}}\left(\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right).\end{split}$

Thus, it follows from (3.13) and (3.14)

(3.15) $F_{s}=I_{1}+I_{2}\leq C\quad\text{in }\Omega$

for all $s\in(0,1)$, where $C$ is a constant depending on $n$, $p_{+}$,
$p_{-}$ and $\|u\|_{C^{1}(\mathbb{R}^{n})}$. This finishes the proof when
$\Omega$ is bounded.

Let us next consider the case when $\Omega$ is bounded. We prove that $F_{s}$
is bounded by an integrable function for all $s\in[s_{0},1)$. To this end, let
us consider a large ball $B_{R/2}$ containing $\mathrm{supp}\,u$. We may
assume that $R>2$. Then, for $x\in\Omega\setminus B_{R}$, we have

$F_{s}(x)=s(1-s)\int_{\Omega\cap
B_{R/2}}\frac{|u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y.$

Since $|x-y|\geq|x|-R/2\geq|x|/2\geq 1$, using (1.3) we obtain

(3.16)
$F_{s}(x)\leq|B_{R/2}|\left(\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right)\left(\frac{2}{|x|}\right)^{n+s_{0}p_{-}}\quad\text{for
}x\in\Omega\setminus B_{R}.$

Combining (3.15) and (3.16), we conclude that

$F_{s}(x)\leq
C\min\left\\{\frac{1}{R^{n+s_{0}p_{-}}},\frac{1}{|x|^{n+s_{0}p_{-}}}\right\\}=:F(x)\in
L^{1}(\Omega),$

where $C$ is a constant depending on $n$, $p_{+}$, $p_{-}$, $s_{0}$ and $u$. ∎

In the next lemma, we find a pointwise limit of $F_{s}$ as $s\nearrow 1$.

######  Lemma 3.4.

Assume that $p$ satisfies (1.3) and (1.4). Let $u\in
C^{2}_{c}(\mathbb{R}^{n})$ and let $F_{s}$ be given as in (3.12). Then

$\lim_{s\nearrow 1}F_{s}(x)=K_{n,\bar{p}(x)}|\nabla u(x)|^{\bar{p}(x)}$

for all $x\in\Omega$.

###### Proof.

Let $u\in C^{2}_{c}(\mathbb{R}^{n})$, then there exists a constant $C>0$,
depending on $\|u\|_{C^{2}(\mathbb{R}^{n})}$, such that

(3.17) $|u(y)-u(x)|\leq|\nabla u(x)\cdot(y-x)|+C|y-x|^{2}$

holds for all $x,y\in\mathbb{R}^{n}$. Let $\varepsilon>0$ and
$\theta_{0}\in(0,1)$. Similarly as in the proof of Lemma 3.1, for a fixed
point $x\in\Omega$ we find $r_{0}\in(0,1)$ such that (3.3) and (3.4) hold for
all $y\in B_{r_{0}}(x)$. Then, using (3.17) and (3.5) we have

(3.18)
$\begin{split}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y&\leq(1+\varepsilon)\int_{B_{r}(x)}\frac{|\nabla
u(x)\cdot(y-x)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\\\
&\quad+C\int_{B_{r}(x)}\frac{|x-y|^{2p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\\\
&\quad+\int_{\Omega\setminus
B_{r}(x)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{n+sp(x,y)}}\,\mathrm{d}y\\\
&=:I_{1}+I_{2}+I_{3}\end{split}$

for $r\in(0,r_{0})$.

Let us estimate $I_{1}$, $I_{2}$ and $I_{3}$. We first recall from (3.7) and
(3.10) that

$I_{1}=(1+\varepsilon)\int_{0}^{r}\int_{\mathbb{S}^{n-1}}|\nabla
u(x)|^{p(x,x+\rho A\xi)}|\xi_{n}|^{p(x,x+\rho A\xi)}\rho^{-1+(1-s)p(x,x+\rho
A\xi)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\,\mathrm{d}\rho$

and

(3.19) $I_{2}\leq C\frac{r^{(2-s)p_{-}}}{2-s}$

hold. For $I_{1}$, we use (3.3) and (3.8) to have

$I_{1}\leq(1+\varepsilon)^{2}L^{1-s}\left(\int_{0}^{r}\int_{\mathbb{S}^{n-1}}|\xi_{n}|^{p(x,x+\rho
A\xi)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)\,\rho^{-1+(1-s)\bar{p}(x)}\,\mathrm{d}\rho\right)|\nabla
u(x)|^{\bar{p}(x)}.$

Moreover, by using (3.4) we obtain

$\begin{split}\int_{\mathbb{S}^{n-1}}|\xi_{n}|^{p(x,x+\rho
A\xi)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)&\leq\int_{\\{|\xi_{n}|\geq\theta_{0}\\}}|\xi_{n}|^{p(x,x+\rho
A\xi)}\,\mathrm{d}\mathcal{H}^{n-1}(\xi)+\theta_{0}|\mathbb{S}^{n-1}|\\\
&\leq(1+\varepsilon)\bar{p}(x)K_{n,\bar{p}(x)}+\theta_{0}|\mathbb{S}^{n-1}|.\end{split}$

Thus, we deduce

(3.20)
$I_{1}\leq(1+\varepsilon)^{2}\frac{L^{1-s}}{1-s}\left((1+\varepsilon)\bar{p}(x)K_{n,\bar{p}(x)}+\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{r^{(1-s)\bar{p}(x)}}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}.$

For $I_{3}$, it follows from $|x-y|^{-sp(x,y)}\leq
r^{sp_{-}-sp(x,y)}|x-y|^{-sp_{-}}\leq r^{s(p_{-}-p_{+})}|x-y|^{-sp_{-}}$

(3.21)
$I_{3}\leq\frac{2^{p_{+}}|\mathbb{S}^{n-1}|}{sp_{-}}\left(\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right)r^{-sp_{+}}.$

Combining (3.18)–(3.21), we arrive at

$\begin{split}F_{s}(x)&=s(1-s)(I_{1}+I_{2}+I_{3})\\\
&\leq(1+\varepsilon)^{2}sL^{1-s}\left((1+\varepsilon)\bar{p}(x)K_{n,\bar{p}(x)}+\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{r^{(1-s)\bar{p}(x)}}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}\\\
&\quad+C\frac{s(1-s)}{2-s}r^{(2-s)p_{-}}+\frac{2^{p_{+}}|\mathbb{S}^{n-1}|}{p_{-}}(1-s)\left(\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{+}}+\|u\|_{L^{\infty}(\mathbb{R}^{n})}^{p_{-}}\right)r^{-sp_{+}},\end{split}$

and hence

$\limsup_{s\nearrow
1}F_{s}(x)\leq(1+\varepsilon)^{2}\left((1+\varepsilon)\bar{p}(x)K_{n,\bar{p}(x)}+\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{1}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}.$

Since $\varepsilon$ and $\theta_{0}$ are arbitrarily chosen, we deduce
$\limsup_{s\nearrow 1}F_{s}(x)\leq K_{n,\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}$. Recall from (3.9)

$F_{s}(x)\geq(1+\varepsilon)(1-\varepsilon)^{2}\frac{s}{L^{1-s}}\left(\bar{p}(x)K_{n,\bar{p}(x)}-\theta_{0}|\mathbb{S}^{n-1}|\right)\frac{r^{(1-s)\bar{p}(x)}}{\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)},$

which results in $\liminf_{s\nearrow 1}F_{s}(x)\geq K_{n,\bar{p}(x)}|\nabla
u(x)|^{\bar{p}(x)}$, finishing the proof. ∎

Lemma 3.2 follows from Lemma 3.3 and Lemma 3.4. Moreover, Theorem 1.1 is an
immediate consequence of Lemma 3.1, Lemma 3.2 and the dominated convergence
theorem. Let us finish this section by providing the proof of Corollary 1.4.

###### Proof of Corollary 1.4.

Since $\Omega$ is a $W^{1,p_{+}}$\- and $W^{1,p_{-}}$-extension domain, we may
assume that $u\in W^{1,p_{+}}(\mathbb{R}^{n})\cap
W^{1,p_{-}}(\mathbb{R}^{n})$. There exists a sequence $\\{u_{k}\\}\subset
C^{2}_{c}(\mathbb{R}^{n})$ such that $u_{k}\to u$ in
$W^{1,p_{+}}(\mathbb{R}^{n})$ and $W^{1,p_{-}}(\mathbb{R}^{n})$ as
$k\to\infty$. By the triangle inequality, we have

$\begin{split}\left|\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u)-\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(|\nabla
u|)\right|&\leq\left|\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u)-\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u_{k})\right|\\\
&\quad+\left|\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u_{k})-\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(|\nabla
u_{k}|)\right|\\\
&\quad+\left|\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(|\nabla
u_{k}|)-\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(|\nabla u|)\right|\\\
&=:I_{1}+I_{2}+I_{3},\end{split}$

where

$\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(v)=\int_{\Omega}K_{n,\bar{p}}(x)|v(x)|^{\bar{p}(x)}\,\mathrm{d}x.$

Since $W^{1,p_{+}}(\mathbb{R}^{n})\cap W^{1,p_{-}}(\mathbb{R}^{n})\subset
W^{s,p(\cdot,\cdot)}(\mathbb{R}^{n})\subset W^{s,p(\cdot,\cdot)}(\Omega)$,
$u_{k}\to u$ in $W^{s,p(\cdot,\cdot)}(\Omega)$ as $k\to\infty$. Thus, by Lemma
2.2, we have $I_{1}\to 0$ as $k\to\infty$. Obviously, we have $I_{3}\to 0$ as
$k\to\infty$. Therefore, for $\varepsilon>0$, we obtain

$\left|\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u)-\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(|\nabla
u|)\right|\leq\varepsilon+I_{2}$

for sufficiently large $k$. By Theorem 1.1 we arrive at

$\limsup_{s\nearrow
1}\left|\varrho_{W^{s,p(\cdot,\cdot)}(\Omega)}(u)-\tilde{\varrho}_{L^{\bar{p}(\cdot)}(\Omega)}(|\nabla
u|)\right|\leq\varepsilon,$

which conclude the theorem. ∎

##  4\. Failure of the limiting embedding

Having Theorem 1.1 at hand, one may expect the limiting embedding
$W^{s,p(\cdot,\cdot)}\to W^{1,\bar{p}(\cdot)}$ as a natural extension of the
limiting embedding $W^{s,p}\to W^{1,p}$. However, in this section, we provide
an example of a variable exponent $p(\cdot,\cdot)$ such that the limiting
embedding $W^{s,p(\cdot,\cdot)}\to W^{1,\bar{p}(\cdot)}$ fails. As explained
in the introduction, the idea is to construct an unnatural fractional Sobolev
space as one of the interpolations of the given Lebesgue and Sobolev spaces.
Such a strange space can be constructed even from $L^{p}$ and $W^{1,p}$ with a
constant $p$. In the following proof, we find a variable exponent
$p(\cdot,\cdot)$ such that $\bar{p}(x)=p(x,x)\equiv p$ is a constant and that
$W^{1,p}\not\subset W^{s,p(\cdot,\cdot)}$ for all $s\in(0,1)$.

###### Proof of Theorem 1.2.

We may assume that $\Omega$ contains $\overline{B}_{2}$. The idea of proof is
to construct a variable exponent $p$ so that $W^{s,p(\cdot,\cdot)}(\Omega)$
requires more integrability than $W^{1,\bar{p}(\cdot)}(\Omega)$. To this end,
let $p:\Omega\times\Omega\to\mathbb{R}$ be a smooth function satisfying (1.3).
Slightly abusing notation, we assume that $p(x,y)=p(|x-y|)$ for all
$x,y\in\Omega$. Then, $\bar{p}=\bar{p}(x)=p(x,x)=p(0)$ is a constant. For
$q\in(1,n)$, we assume

(4.1) $1<\bar{p}<n/q\quad\text{and}\quad p(r)\geq n/(q-1)\quad\text{for }r\geq
1.$

The assumption (4.1) shows how $W^{s,p(\cdot,\cdot)}(\Omega)$ requires more
integrability than $W^{1,\bar{p}}(\Omega)$.

We consider a smooth decreasing function $\phi:(0,+\infty)\to[0,+\infty)$
satisfying $\phi(t)=|t|^{1-q}$ for $t\in(0,1]$, $0\leq\phi(t)\leq 1$ for
$t\in[1,2]$ and $\phi(t)=0$ for $t\in[2,+\infty)$. We claim that a function
$u:\Omega\to[0,+\infty]$ defined by $u(x)=\phi(|x|)$ belongs to
$W^{1,\bar{p}}(\Omega)\setminus W^{s,p(\cdot,\cdot)}(\Omega)$ for all
$s\in(0,1)$. Indeed, a straightforward computation shows

$\int_{\Omega}|u(x)|^{\bar{p}}\,\mathrm{d}x\leq\int_{B_{1}}|x|^{-(q-1)\bar{p}}\,\mathrm{d}x+|B_{2}\setminus
B_{1}|<+\infty$

and

$\int_{\Omega}|\nabla
u(x)|^{\bar{p}}\,\mathrm{d}x\leq(q-1)^{\bar{p}}\int_{B_{1}}|x|^{-q\bar{p}}\,\mathrm{d}x+C|B_{2}\setminus
B_{1}|<+\infty.$

Thus, $u\in W^{1,\bar{p}}(\Omega)$.

It remains to prove $u\not\in W^{s,p(\cdot,\cdot)}(\Omega)$ for all
$s\in(0,1)$. Since $u=0$ outside $B_{2}$, we have

$\varrho_{s,p(\cdot,\cdot)}(u)\geq s(1-s)\int_{\Omega\setminus
B_{2}}\int_{B_{1}}\frac{|y|^{-(q-1)p(|x-y|)}}{|x-y|^{n+sp(|x-y|)}}\,\mathrm{d}y\,\mathrm{d}x.$

If $x\in\Omega\setminus B_{2}$ and $y\in B_{1}$, then
$1\leq|x-y|\leq|x|+1\leq\frac{3}{2}|x|$. Thus, by using (1.3) and (4.1), we
obtain

$\varrho_{s,p(\cdot,\cdot)}(u)\geq s(1-s)\int_{\Omega\setminus
B_{2}}\int_{B_{1}}\frac{|y|^{-n}}{(\frac{3}{2}|x|)^{n+sp_{+}}}\,\mathrm{d}y\,\mathrm{d}x=+\infty.$

Therefore, it follows from Lemma 2.1 $u\notin W^{s,p(\cdot,\cdot)}(\Omega)$
for all $s\in(0,1)$. ∎

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