Compact embeddings of some weighted fractional Sobolev spaces on $\Rn$

TL;DR

This paper investigates compact embeddings of fractional Sobolev spaces on Rn and bounded domains, providing new results on the conditions under which these spaces embed compactly into Lebesgue spaces, with implications for analysis.

Contribution

It introduces new compact embedding results for fractional Sobolev spaces on Rn and bounded Lipschitz domains, extending existing theory.

Findings

Established compact embeddings VsqpRn  LqRn for suitable potentials

Proved embeddings VsqpRn  LlRn under certain conditions

Extended classical Sobolev embedding results to fractional and weighted spaces

Abstract

In this paper, we study a family of general fractional Sobolev spaces $\MsqpOm$ when $\Om=\Rn$ or $\Om$ is a bounded domain, having a compact, Lipschitz boundary $\Bdy$, in $\Rn$ for $n\geq2$. Among other results, some compact embedding results of $\MVsqpRn\hookrightarrow\LqRn$ and $\MVsqpRn\hookrightarrow\LlRn$ for suitable potential functions $V(x)$ are described.

Equations

$$\Vert u{\Vert }_{M^{s;q,p}(\mathbf{\Omega })}:=\Vert u{\Vert }_{q,\mathbf{\Omega }}+[u{]}_{s,p,\mathbf{\Omega }},$$

$$\left\{\begin{array}[]{ll}M^{s;q,p}(\bm{\Omega})=W^{s,p}(\bm{\Omega})&\mathrm{for}\hskip 5.69054pt1\leq q\leq p^{*}_{s},\\ \\ M^{s;q_{1},p}(\bm{\Omega})\to M^{s;q_{2},p}(\bm{\Omega})&\mathrm{for}\hskip 5.69054ptp^{*}_{s}\leq q_{2}\leq q_{1}\leq\infty.\end{array}\right.$$

$$\Vert u{\Vert }_{W^{s,p}(\mathbf{\Omega })}\le C_{p,q}\Vert u{\Vert }_{M^{s;q,p}(\mathbf{\Omega })},\forall u\in M^{s;q,p}(\mathbf{\Omega })\mathrm{and}q\in [1,\infty ].$$

$$M^{s;\infty ,p}(\mathbf{\Omega })\to M^{s;q,p}(\mathbf{\Omega })=W^{s,p}(\mathbf{\Omega })\mathrm{for}1\le q<\infty ,$$

$$\underset{[u{]}_{s,p,{\mathbf{R}}^n}\le 1}{\sup }{\int }_{\mathbf{\Omega }}{e}^{\alpha |u{|}^{\frac{n}{n-s}}}\mathbf{d}x\le C_{\mathbf{\Omega }},\forall u\in {\tilde{W}}_0^{s,p}(\mathbf{\Omega })\mathrm{and}\alpha \in [0,{\alpha }_{\ast }).$$

$$M^{s;q,p}(\mathbf{\Omega })=W^{s,p}(\mathbf{\Omega })\to C^{0,\gamma }(\mathbf{\Omega })\mathrm{for}0\le \gamma \le {\gamma }_s^{\ast }\mathrm{and}1\le q\le \infty .$$

$$\Vert \tilde{u}{\Vert }_{W^{s,p}(\mathbf{Q})}\le \Vert \tilde{u}{\Vert }_{W^{s,p}({\mathbf{R}}^n)}\le C_{\mathbf{\Omega }}^{\prime }\Vert u{\Vert }_{W^{s,p}(\mathbf{\Omega })}\le C_{p,q}^{\prime }\Vert u{\Vert }_{M^{s;q,p}(\mathbf{\Omega })}.$$

$$M^{s;q,p}(\mathbf{\Omega })\hookrightarrow C^{0,\gamma }(\mathbf{\Omega })\mathrm{for}0\le \gamma <{\gamma }_s^{\ast }\mathrm{and}1\le q\le \infty .$$

$$\Bigl{(}\int_{\mathbf{R}^{n}}|u|^{p^{*}_{s}}\,\mathbf{d}x\Bigr{)}^{\frac{n-sp}{n}}\leq C_{1}\int_{\mathbf{R}^{n}}\int_{\mathbf{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,\mathbf{d}x\mathbf{d}y,\hskip 17.07164pt\forall\hskip 5.69054ptu\in D^{s,p}(\mathbf{R}^{n}).$$

$$M^{s;q_{1},p}(\mathbf{R}^{n})\to M^{s;q_{2},p}(\mathbf{R}^{n})\hskip 5.69054pt\mathrm{for}\hskip 5.69054pt\begin{array}[]{ll}\mathrm{either}&1\leq q_{1}\leq q_{2}\leq p^{*}_{s},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \mathrm{or}&p^{*}_{s}\leq q_{2}\leq q_{1}<\infty\end{array}$$

$$\Vert u{\Vert }_{r,{\mathbf{R}}^n}\le C_2\Vert u{\Vert }_{q,{\mathbf{R}}^n}^{\theta }[u{]}_{s,p,{\mathbf{R}}^n}^{1-\theta },\forall u\in M^{s;q,p}({\mathbf{R}}^n).$$

$$M^{s;q_1,p}({\mathbf{R}}^n)\to M^{s;q_2,p}({\mathbf{R}}^n)\mathrm{for}1\le q_1\le q_2<\infty .$$

$$M^{s;q_1,p}({\mathbf{R}}^n)\to M^{s;q_2,p}({\mathbf{R}}^n)\mathrm{for}1\le q_1\le q_2\le \infty .$$

$${\int }_{{\mathbf{R}}^n}\Phi (f^{\ast })\mathbf{d}x={\int }_{{\mathbf{R}}^n}\Phi (|f|)\mathbf{d}x$$

$${\int }_{{\mathbf{R}}^n}{\int }_{{\mathbf{R}}^n}\frac{|f^{\ast }(x)-f^{\ast }(y){|}^p}{|x-y{|}^{n+sp}}\mathbf{d}x\mathbf{d}y\le {\int }_{{\mathbf{R}}^n}{\int }_{{\mathbf{R}}^n}\frac{|f(x)-f(y){|}^p}{|x-y{|}^{n+sp}}\mathbf{d}x\mathbf{d}y.$$

$$f^{*}(x)\leq|x|^{-\frac{n}{q}}\Bigl{(}\frac{n}{\bm{\omega}_{n-1}}\Bigr{)}^{\frac{1}{q}}\|f^{*}\|_{q,\mathbf{R}^{n}}$$

$$\underset{\Vert u{\Vert }_{M^{s;q,p}({\mathbf{R}}^n)}\le 1}{\sup }{\int }_{{\mathbf{R}}^n}{\Psi }_{\alpha ,q,s}(|u|)\mathbf{d}x\le C(\alpha ,q,s).$$

$${\int }_{{\mathbf{R}}^n}{\Psi }_{\alpha ,q,s}(u)\mathbf{d}x={\int }_{{\mathbf{B}}_{R_0}}{\Psi }_{\alpha ,q,s}(u)\mathbf{d}x+{\int }_{{\mathbf{B}}_{R_0}^c}{\Psi }_{\alpha ,q,s}(u)\mathbf{d}x.$$

$$\begin{array}{rl}& \frac{{\mathbf{V}}_n}{(36R_0{)}^n}{\int }_{{\mathbf{B}}_{R_0}}{v}^p(x)\mathbf{d}x\le {\int }_{{\mathbf{B}}_{R_0}}{\int }_{{\mathbf{B}}_{R_0}(x_0)}\frac{|v(x)-v(y){|}^p}{|x-y{|}^{2n}}\mathbf{d}x\mathbf{d}y\\ \le & {\int }_{{\mathbf{B}}_{R_0}}{\int }_{{\mathbf{B}}_{R_0}}\frac{|v(x)-v(y){|}^p}{|x-y{|}^{2n}}\mathbf{d}x\mathbf{d}y+2{\int }_{{\mathbf{B}}_{R_0}}{\int }_{{\mathbf{B}}_{R_0}^c}\frac{|v(x)-v(y){|}^p}{|x-y{|}^{2n}}\mathbf{d}x\mathbf{d}y\\ =& {\int }_{{\mathbf{B}}_{R_0}}{\int }_{{\mathbf{B}}_{R_0}}\frac{|u(|x|)-u(|y|){|}^p}{|x-y{|}^{2n}}\mathbf{d}x\mathbf{d}y+2{\int }_{{\mathbf{B}}_{R_0}}{\int }_{{\mathbf{B}}_{R_0}^c}\frac{|u(|x|)-u_0{|}^p}{|x-y{|}^{2n}}\mathbf{d}x\mathbf{d}y\end{array}$$

$$u^{\frac{n}{n-s}}\le v^{\frac{n}{n-s}}(1+u_0^{\frac{n}{s}})+K_2+u_0^{\frac{n}{n-s}}.$$

$$\int_{\mathbf{B}_{R_{0}}}\Psi_{\alpha,q,s}(u)\,\mathbf{d}x\leq\int_{\mathbf{B}_{R_{0}}}e^{\alpha u^{\frac{n}{n-s}}}\leq e^{\alpha\bigl{(}K_{2}+u_{0}^{\frac{n}{n-s}}\bigr{)}}\int_{\mathbf{B}_{R_{0}}}e^{\alpha\tilde{u}^{\frac{n}{n-s}}}\,\mathbf{d}x\leq C_{1}(\alpha,q,s).$$

$$\int_{\mathbf{B}^{c}_{R_{0}}}u^{\frac{\ell n}{n-s}}\,\mathbf{d}x\leq\bm{\omega}_{n-1}\Bigl{(}\frac{n}{\bm{\omega}_{n-1}}\Bigr{)}^{\frac{\ell n}{q(n-s)}}\int^{\infty}_{R_{0}}t^{-\frac{\ell n^{2}}{q(n-s)}+n-1}\,\mathbf{d}t.$$

$$\sum _{\ell ={\beta }_0}^{\infty }\frac{{\alpha }^{\ell }}{\ell !}{\int }_{{\mathbf{B}}_{R_0}^c}{u}^{\frac{\ell n}{n-s}}\mathbf{d}x\le \frac{{\alpha }^{{\beta }_0}}{{\beta }_0!}{\int }_{{\mathbf{B}}_{R_0}^c}{u}^{\frac{{\beta }_{0}n}{n-s}}\mathbf{d}x+\frac{q(n-s)}{n^2}{R}_0^n{\mathbf{\omega }}_{n-1}{e}^{\alpha {\sigma }^{\frac{s}{n-s}}}\le C_2(\alpha ,q,s).$$

$${\int }_{{\mathbf{R}}^n}|u_l{|}^{r}K\mathbf{d}x={\int }_{{\mathbf{W}}_{ϵ}}|u_l{|}^{r}K\mathbf{d}x+{\int }_{{\mathbf{W}}_{ϵ}^c}|u_l{|}^{r}K\mathbf{d}x.$$

$$\begin{split}&\int_{\mathbf{W}_{\epsilon}^{c}}|u_{l}|^{r}\frac{K}{V^{\tau}}V^{\tau}\,\mathbf{d}x\leq\epsilon\int_{\mathbf{R}^{n}}|u_{l}|^{r}V^{\tau}\,\mathbf{d}x\leq\epsilon\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{q}V\,\mathbf{d}x\Bigr{)}^{\tau}\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{p^{*}_{s}}\,\mathbf{d}x\Bigr{)}^{\frac{r-q\tau}{p^{*}_{s}}}\\ \leq\,&\epsilon C^{\prime}_{1}\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{q}V\,\mathbf{d}x\Bigr{)}^{\tau}\Bigl{(}\int_{\mathbf{R}^{n}}\int_{\mathbf{R}^{n}}\frac{|u_{l}(x)-u_{l}(y)|^{p}}{|x-y|^{n+sp}}\,\mathbf{d}x\mathbf{d}y\Bigr{)}^{\frac{r-q\tau}{p}}\leq\epsilon C^{\prime}_{1}\|u_{l}\|^{r}_{M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})}\end{split}$$

$$\begin{split}&\int_{\mathbf{W}_{\epsilon}}|u_{l}|^{r}K\,\mathbf{d}x=\int_{\mathbf{W}_{\epsilon}\cap\mathbf{B}_{R}}|u_{l}|^{r}K\,\mathbf{d}x+\int_{\mathbf{W}_{\epsilon}\cap\mathbf{B}_{R}^{c}}|u_{l}|^{r}K\,\mathbf{d}x\\ \leq&\Bigl{(}\int_{\mathbf{W}_{\epsilon}}K^{t}\,\mathbf{d}x\Bigr{)}^{\frac{1}{t}}\Biggl{\{}\Bigl{(}\int_{\mathbf{B}_{R}}|u_{l}|^{\frac{rt}{t-1}}\,\mathbf{d}x\Bigr{)}^{\frac{t-1}{t}}\\ &+\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{p^{*}_{s}}\,\mathbf{d}x\Bigr{)}^{\frac{r}{p^{*}_{s}}}\{\mathfrak{L}(\mathbf{W}_{\epsilon}\cap\mathbf{B}_{R}^{c})\}^{1-\frac{1}{t}-\frac{r}{p^{*}_{s}}}\Biggr{\}}\to 0\end{split}$$

$${\int }_{{\mathbf{R}}^n}|u_l{|}^{r}K\mathbf{d}x={\int }_{{\mathbf{B}}_R}|u_l{|}^{r}K\mathbf{d}x+{\int }_{{\mathbf{B}}_R^c}|u_l{|}^{r}K\mathbf{d}x.$$

$$\int_{\mathbf{B}_{R}}|u_{l}|^{r}K\,\mathbf{d}x\leq\Bigl{(}\int_{\mathbf{B}_{R}}K^{t}\,\mathbf{d}x\Bigr{)}^{\frac{1}{t}}\Bigl{(}\int_{\mathbf{B}_{R}}|u_{l}|^{\frac{rt}{t-1}}\,\mathbf{d}x\Bigr{)}^{\frac{t-1}{t}}\to 0$$

$$\begin{split}&\int_{\mathbf{B}^{c}_{R}}|u_{l}|^{r}K\,\mathbf{d}x\leq\|KV^{-\tau}\|_{\infty,\mathbf{B}^{c}_{R}}\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{q}V\,\mathbf{d}x\Bigr{)}^{\tau}\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{p^{*}_{s}}\,\mathbf{d}x\Bigr{)}^{\frac{r-q\tau}{p^{*}_{s}}}\\ \leq\,&C^{\prime}_{1}\|KV^{-\tau}\|_{\infty,\mathbf{B}^{c}_{R}}\|u_{l}\|^{r}_{M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})}\to 0\end{split}$$

$$\begin{split}&\int_{\bm{\Omega}}|u|^{r}\frac{K}{V^{\mathfrak{x}}}V^{\mathfrak{x}}\,\mathbf{d}x\leq\Bigl{(}\int_{\bm{\Omega}}|u|^{q}V\,\mathbf{d}x\Bigr{)}^{\frac{1}{r_{1}}}\Bigl{(}\int_{\bm{\Omega}}|u|^{p^{*}_{s}}\,\mathbf{d}x\Bigr{)}^{\frac{1}{r_{2}}}\Bigl{(}\int_{\bm{\Omega}}K^{r_{3}}V^{-r_{3}\mathfrak{x}}\,\mathbf{d}x\Bigr{)}^{\frac{1}{r_{3}}}\\ \leq&\,C^{\prime}_{1}\Bigl{(}\int_{\bm{\Omega}}|u|^{q}V\,\mathbf{d}x\Bigr{)}^{\frac{1}{r_{1}}}\Bigl{(}\int_{\mathbf{R}^{n}}\int_{\mathbf{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,\mathbf{d}x\mathbf{d}y\Bigr{)}^{\frac{p^{*}_{s}}{pr_{2}}}\Bigl{(}\int_{\bm{\Omega}}K^{\psi_{p^{*}_{s}}(\tau)}V^{-\tau}\,\mathbf{d}x\Bigr{)}^{\frac{1}{r_{3}}}\end{split}$$

Full text

2010 Mathematics Subject Classification. 46E35, 35S05, 47A75, 49R05.

Keywords. Fractional Sobolev spaces, compact embedding results, fractional Laplacian eigenvalue problems.

Recently, fractional Sobolev function spaces $W^{s,p}(\mathbf{R}^{n})$, and the associated nonlocal equations, attracted great attentions due to their impressive applications to various disciplines, as briefed for example in the introduction of Di Nezza, Palatucci and Valdinoci [9]. See also Chapter 4 of Demengel and Demengel [8] or Chapters 1-2 of Molica Bisci, Radulescu and Servadei [21] for some concise, self-contained discussions of this matter.

In his classical monograph [20, Section 5.1.1], Maz’ya defined the general Sobolev function spaces $W^{1}_{p,q}(\mathbf{R}^{n})$; using the notation $M^{q,p}(\mathbf{R}^{n})$, the author described some relationships among those spaces and their compact embedding results in [12, 13, 14, 15] with the desire of providing a complement to Lions [19, Lemma I.1] in the sense of function space settings.

In the hope of suggesting a little bit more insight regarding the similarity between $W^{1,p}(\mathbf{R}^{n})$ and $W^{s,p}(\mathbf{R}^{n})$ as observed in [8, 9, 21], this paper is devoted to the description of some general fractional Sobolev function spaces $M^{s;q,p}(\mathbf{R}^{n})$ when $n\geq 2$, their relations and certain compact embedding results of $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{q}(\mathbf{R}^{n})$ and $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{1}(\mathbf{R}^{n})$ for suitable potentials $V(x)\geq 0$. As expected, $M^{q,p}(\mathbf{R}^{n})$ and $M^{s;q,p}(\mathbf{R}^{n})$ share several common properties.

In the sequel, we always assume that $n\geq 2$, $1\leq p,q\leq\infty$, and $0<s<1$.

Denote $\bm{\Omega}$ a bounded domain in $\mathbf{R}^{n}$ having a compact, Lipschitz boundary $\partial\bm{\Omega}$, or $\bm{\Omega}=\mathbf{R}^{n}$. Define $M^{s;q,p}(\bm{\Omega})$ to be the Banach space as the completion of the set $C^{1}(\bm{\Omega})$ if $\bm{\Omega}$ is bounded, or as that of the set $C^{1}_{c}(\mathbf{R}^{n})$ for $\bm{\Omega}=\mathbf{R}^{n}$, with respect to the norm

[TABLE]

where $\|u\|^{q}_{q,\bm{\Omega}}:=\int_{\bm{\Omega}}|u|^{q}\,\mathbf{d}x$ and $[u]^{p}_{s,p,\bm{\Omega}}:=\int_{\bm{\Omega}}\int_{\bm{\Omega}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,\mathbf{d}x\mathbf{d}y$.

Note $W^{s,p}(\bm{\Omega})=M^{s;p,p}(\bm{\Omega})$, and if $\bm{\Omega}$ is bounded, $M^{s;q,\infty}(\bm{\Omega})=C^{0,s}(\bm{\Omega})$ (independent of $q$).

From now on, we shall write both continuous embedding of function spaces and convergence of functions by “ $\to$ ”, compact embedding of function spaces by “ $\hookrightarrow$ ”, and weak convergence of functions by “ $\rightharpoonup$ ”. Other notations will be specified when appropriate.

Below, let’s embark on the elaboration of our analyses for the spaces $M^{s;q,p}(\bm{\Omega})$.

(I.) $\bm{\Omega}$ is a bounded domain with a compact, Lipschitz boundary $\partial\bm{\Omega}$.

In this case, it is trivial to notice $M^{s;q,p}(\bm{\Omega})\to L^{1}(\bm{\Omega})$, so that from the fractional Poincaré’s inequality (see for instance Bellido and Mora-Corral [3, Lemma 3.1], or Drelichman and Durán [11, Estimate (1.3)]), one derives $M^{s;q,p}(\bm{\Omega})\to W^{s,p}(\bm{\Omega})$ in view of Minkowski’s inequality. This, together with Corollary 4.53 in [8], yields the conclusions listed as follows.

(1.) When $sp<n$, then we have, with $p^{*}_{s}:=\frac{np}{n-sp}$,

[TABLE]

Also, there exists a constant $C_{p,q}>0$, depending on $n,p,q,s,\bm{\Omega}$, such that

[TABLE]

(2.) When $sp=n$, then we have

[TABLE]

and $M^{s;q,p}(\bm{\Omega})$ is not a subset of $L^{\infty}(\bm{\Omega})$ by [8, Example 4.25] unless $q=\infty$. A recent result of Parini and Ruf [23] provides a fractional Moser-Trudinger inequality that says

[TABLE]

Here, $\alpha_{*},C_{\bm{\Omega}}>0$ are constants depending on $n,p,s,\bm{\Omega}$ and $\tilde{W}^{s,p}_{0}(\bm{\Omega})$, as a proper subspace of $W^{s,p}(\bm{\Omega})$, is the completion of the set $C^{1}_{c}(\bm{\Omega})$ with respect to the semi-norm $[u]_{s,p,\mathbf{R}^{n}}$, which is equivalent to (1) in $\mathbf{R}^{n}$ on this occasion by Brasco, Lindgren and Parini [6, Lemma 2.4].

(3.) When $sp>n$, then we have, with $\gamma^{*}_{s}:=\frac{sp-n}{p}$,

[TABLE]

Further, via [8, Theorem 4.54] and [9, Theorem 7.1], one easily verifies the result below.

[TABLE]

Here, $\tilde{u}$ denotes the extension of $u$ to $W^{s,p}(\mathbf{R}^{n})$, and $C^{\prime}_{\bm{\Omega}},C^{\prime}_{p,q}>0$ are constants depending on $n,p,q,s,\bm{\Omega}$. Therefore, we can follow [9, Theorem 7.1] to complete the proof. ∎

Recall for $sp>n$, one actually has

[TABLE]

(II.) $\bm{\Omega}=\mathbf{R}^{n}$.

First, note that $M^{s;q,p}(\mathbf{R}^{n})$ represents the Banach space as the completion of the set $C^{1}_{c}(\mathbf{R}^{n})$ with respect to the norm (1). Now, following Lieb and Loss [18, Sections 3.2 and 4.3], a function $u\in L^{1}_{loc}(\mathbf{R}^{n})$ is said to vanish at infinity provided that $\mathfrak{L}(\{x\in\mathbf{R}^{n}:|u(x)|\geq c\})<\infty$ for each positive constant $c>0$, with $\mathfrak{L}$ being the Lebesgue measure.

(1.) When $sp<n$, denote by $D^{s,p}(\mathbf{R}^{n})$ the space of functions $u\in L^{1}_{loc}(\mathbf{R}^{n})$, where $u$ vanish at infinity and $[u]_{s,p,\mathbf{R}^{n}}<\infty$. Then, a careful check of Lemma 6.3 and Theorem 6.5 in [9] implies $u\in L^{p^{*}_{s}}(\mathbf{R}^{n})$, and one has the fractional Sobolev inequality, saying that

[TABLE]

Here, $C_{1}>0$ is an absolute constant depending on $n,p,s$ with $1\leq p<\infty$.

This leads to $D^{s,p}(\mathbf{R}^{n})=M^{s;p^{*}_{s},p}(\mathbf{R}^{n})$111Equivalently, $D^{s,p}(\mathbf{R}^{n})$ is the completion of the set $C^{1}_{c}(\mathbf{R}^{n})$ with respect to $[\cdot]_{s,p,\mathbf{R}^{n}}$., and one furthermore has

[TABLE]

using Chebyshev’s inequality and the standard interpolation inequality. Notice if one would like to have $q=\infty$ included, then $u$ needs to be compactly supported.

(2.) When $sp=n$, then one has the fractional Gagliardo-Nirenberg inequality (see Nguyen and Squassina [22, Lemma 2.1]) that says

[TABLE]

Here, we only cited a tailored version for our purpose with $1\leq q\leq r<\infty$ and $\theta:=\frac{q}{r}\in(0,1]$, and $C_{2}>0$ is an absolute constant depending on $n,p,q,r,s$. Therefore, we have

[TABLE]

Notice here $q=\infty$ doesn’t contribute to the embedding result (11).

(3.) When $sp>n$, a careful reading of Theorem 8.2 in [9], especially their Estimates (8.3) and (8.9), suggests $M^{s;q,p}(\mathbf{R}^{n})\to C^{0,\gamma^{*}_{s}}(\mathbf{R}^{n})$ so that $M^{s;q,p}(\mathbf{R}^{n})\to L^{\infty}(\mathbf{R}^{n})$. Thus, one has

[TABLE]

Notice $M^{s;q,\infty}(\mathbf{R}^{n})\to C^{0,s}(\mathbf{R}^{n})$ that is consistent with the observation shown at [9, Page 565], while $M^{s;\infty,p}(\mathbf{R}^{n})$ is the largest possible space for fixed $p,s$ in (12).

(III.) A fractional Moser-Trudinger type inequality on $M^{s;q,p}(\mathbf{R}^{n})$ when $sp=n$.

In this section, we employ the preceding result in [23] to characterize a possible extension of (5) to $\mathbf{R}^{n}$ concerning functions $u\in M^{s;q,p}(\mathbf{R}^{n})$ when $sp=n$. The proof presented here follows basically from Ruf [24] and do Ó [10] (see also Li and Ruf [17], and Iula [16] among many other important contributions on $\mathbf{R}^{n}$, which cannot be exhaustively listed here).

We now recall some essential facts about Schwarz symmetrization by Berestycki and Lions [4, Appendix III], Lieb and Loss [18, Section 3.3], and Beckner [2, Theorem 3]. Let $f(x)$ be a Borel measurable function vanishing at infinity in the sense of Lieb and Loss, and let $f^{*}(x)(\geq 0)$ be the Schwarz symmetrization or spherical rearrangement of $f(x)$ such that $\mathfrak{L}(\{x\in\mathbf{R}^{n}:|f(x)|\geq c\})=\mathfrak{L}(\{x\in\mathbf{R}^{n}:f^{*}(x)\geq c\})$ for all positive constants $c>0$. Notice $f^{*}(x)$ is unique, radial, decreasing in $|x|$, and lower semi-continuous (so measurable). Also, we have

[TABLE]

for all continuous functions $\Phi$ with $\Phi(|f|)$ integrable; so, $\|f^{*}\|_{q,\mathbf{R}^{n}}=\|f\|_{q,\mathbf{R}^{n}}$ when $f\in L^{q}(\mathbf{R}^{n})$ for $1\leq q\leq\infty$. Besides, one has

[TABLE]

For $f^{*}(x)\in L^{q}(\mathbf{R}^{n})$, Berestycki-Lions’ radial lemma [4, Lemma A.IV] leads to

[TABLE]

for all $x\neq 0$, where $\bm{\omega}_{n-1}$ presents the surface area of the unit sphere in $\mathbf{R}^{n}$.

We next prove a result following [10, Lemma 1] and [24, Proposition 2.1] via the $M^{s;q,p}(\mathbf{R}^{n})$-norm, which may be viewed as a fractional Moser-Trudinger inequality on $M^{s;q,p}(\mathbf{R}^{n})$.

[TABLE]

Here, for $v\geq 0$ and the least positive integer $\beta_{0}$ with $\frac{\beta_{0}n}{n-s}\geq q\geq 1$, $\Psi_{\alpha,q,s}(v):=\sum\limits_{\ell=\beta_{0}}^{\infty}\frac{\alpha^{\ell}}{\ell!}v^{\frac{\ell n}{n-s}}$.

[TABLE]

Here and hereafter, $\mathbf{B}_{R}$ is the ball of radius $R$ in $\mathbf{R}^{n}$ centered at the origin and $\mathbf{B}^{c}_{R}:=\mathbf{R}^{n}\setminus\mathbf{B}_{R}$, while $R_{0}>0$ is a sufficiently large absolute constant to be determined later.

To estimate the integral over $\mathbf{B}_{R_{0}}$, write $v(x):=\max\{u(|x|)-u_{0},0\}$ for $u_{0}:=u(R_{0})>0$, a constant. Then, $v\equiv 0$ on $\mathbf{B}^{c}_{R_{0}}$ and $v\in\tilde{W}^{s,p}_{0}(\mathbf{B}_{R_{0}})$, since by Lemma 2.4 of [6]

[TABLE]

follows; seeing $u(|y|)\leq u_{0}$ when $|y|\geq R_{0}$, one has $\frac{\mathbf{V}_{n}}{(36R_{0})^{n}}\|v\|^{p}_{p,\mathbf{B}_{R_{0}}}\leq[v]^{p}_{s,p,\mathbf{R}^{n}}\leq[u]^{p}_{s,p,\mathbf{R}^{n}}\leq 1$. Here, $\mathbf{V}_{n}$ presents the volume of the unit ball in $\mathbf{R}^{n}$, and $\mathbf{B}_{R_{0}}(x_{0})$ denotes a ball of radius $R_{0}$ in $\mathbf{B}^{c}_{R_{0}}$ centered at some point $x_{0}$ with $|x_{0}|=4R_{0}$.

Notice [10, Estimate (5)] yields $(1+t)^{\frac{n}{n-s}}\leq t^{\frac{n}{n-s}}+K_{1}t^{\frac{s}{n-s}}+1$ for all $t\in(0,\infty)$ with $K_{1}>0$ an absolute constant. Actually, for $f(t):=\frac{(1+t)^{\frac{n}{n-s}}-t^{\frac{n}{n-s}}-1}{t^{\frac{s}{n-s}}}$ that is continuous on $(0,\infty)$, one sees $\lim\limits_{t\to 0}f(t)=0$ as $n\geq 2>2s$, while using $\tilde{t}:=\frac{1}{t}$ one sees $\lim\limits_{t\to\infty}f(t)=\frac{n}{n-s}$.

Next, for $t=\frac{v(x)}{u_{0}}$, it follows that $u^{\frac{n}{n-s}}\leq\{v+u_{0}\}^{\frac{n}{n-s}}\leq v^{\frac{n}{n-s}}+K_{1}v^{\frac{s}{n-s}}u_{0}+u_{0}^{\frac{n}{n-s}}$. Apply Young’s inequality to $K_{1}v^{\frac{s}{n-s}}u_{0}$ to observe, for an absolute constant $K_{2}>0$,

[TABLE]

Define $\tilde{u}:=v(1+u^{\frac{n}{s}}_{0})^{\frac{n-s}{n}}\in\tilde{W}^{s,p}_{0}(\mathbf{B}_{R_{0}})$ and notice $[\tilde{u}]_{s,p,\mathbf{R}^{n}}\leq(1+u^{\frac{n}{s}}_{0})^{\frac{n-s}{n}}[v]_{s,p,\mathbf{R}^{n}}$, so that by the discussions after (17) it leads to $[\tilde{u}]^{p}_{s,p,\mathbf{R}^{n}}\leq(1+u^{\frac{n}{s}}_{0})^{\frac{n-s}{s}}(1-\|u\|_{q,\mathbf{R}^{n}})^{\frac{n}{s}}$.

From (15), $u^{\frac{n}{s}}_{0}\leq\sigma\|u\|^{\frac{n}{s}}_{q,\mathbf{R}^{n}}$ for \sigma:=\bigl{(}\frac{n}{\bm{\omega}_{n-1}R^{n}_{0}}\bigr{)}^{\frac{n}{qs}}. Write $g(t):=(1+\sigma t^{\frac{n}{s}})^{\frac{n-s}{s}}(1-t)^{\frac{n}{s}}-1$ and take g^{\prime}(t)=\frac{n}{s}(1+\sigma t^{\frac{n}{s}})^{\frac{n-2s}{s}}(1-t)^{\frac{n-s}{s}}\bigl{(}\frac{n-s}{s}\sigma t^{\frac{n-s}{s}}-\frac{n}{s}\sigma t^{\frac{n}{s}}-1\bigr{)}, with $g(0)=0$ and $g(1)=-1$. Define $h(t):=\frac{n-s}{s}\sigma t^{\frac{n-s}{s}}-\frac{n}{s}\sigma t^{\frac{n}{s}}-1$ with $h^{\prime}(t_{0})=0$ for t_{0}:=\bigl{(}\frac{n-s}{n}\bigr{)}^{2}. So, when $R_{0}$ is so large that \sigma\leq\bigl{(}\frac{n}{n-s}\bigr{)}^{\frac{2n-s}{s}}, then one sees $h(t)\leq 0$ and hence $g^{\prime}(t)\leq 0$ on $[0,1]$. That is, $g(t)\leq 0$ for $0\leq t\leq 1$, which clearly implies $[\tilde{u}]_{s,p,\mathbf{R}^{n}}\leq 1$ provided R^{n}_{0}\geq\frac{n}{\bm{\omega}_{n-1}}\bigl{(}\frac{n-s}{n}\bigr{)}^{\frac{q(2n-s)}{n}}.

As a result, one can combine this with (5) and (18) to deduce

[TABLE]

Here, $C_{1}(\alpha,q,s)>0$ is an absolute constant depending on $\alpha,n,q,s$.

To estimate the integral over $\mathbf{B}^{c}_{R_{0}}$, one applies (15) with $\|u\|_{q,\mathbf{R}^{n}}\leq 1$ to observe

[TABLE]

For $\frac{\beta_{0}n}{n-s}\geq q$, one employs Proposition 2 to the starting term in $\Psi_{\alpha,q,s}(u)$ and thus

[TABLE]

Here, $C_{2}(\alpha,q,s)>0$ is an absolute constant depending on $\alpha,n,q,s$.

Accordingly, (16) follows for $C(\alpha,q,s):=\max\{C_{1}(\alpha,q,s),C_{2}(\alpha,q,s)\}>0$. ∎

(IV.) Some compact embedding results regarding $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$.

Let $V(x)>0$ be a Lebesgue measurable function in $\mathbf{R}^{n}$ such that $\inf\limits_{\mathbf{D}}V(x)\geq V_{\mathbf{D}}>0$ for all compact subsets $\mathbf{D}\Subset\mathbf{R}^{n}$. Take $n\geq 2$, $p\in[1,\infty]$, $q\in[1,\infty)$, and $s\in(0,1)$. When $sp<n$, we designate $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})$ to be the Banach space as the completion of the set $C^{1}_{c}(\mathbf{R}^{n})$ with respect to the norm $\|u\|_{M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})}:=\|u\|_{L^{q}_{\mathrm{V}}(\mathbf{R}^{n})}+[u]_{s,p,\mathbf{R}^{n}}$ where $\|u\|^{q}_{L^{q}_{\mathrm{V}}(\mathbf{R}^{n})}:=\int_{\mathbf{R}^{n}}|u|^{q}V\,\mathbf{d}x$, so that $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})$ is a subspace of $D^{s,p}(\mathbf{R}^{n})$. When $sp\geq n$, we further require $\inf\limits_{\mathbf{R}^{n}}V(x)\geq V_{0}>0$ and define $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})$ similarly, so that $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})$ is a subspace of $M^{s;q,p}(\mathbf{R}^{n})$.

One recalls that some closely related results on $W^{1,p}(\mathbf{R}^{n})$ may be found in Chiappinelli [7, Theorem 1], Schneider [25, Theorem 2.3], and Bonheure and Van Schaftingen [5, Theorem 4]. Yet, it seems that the results presented here are not available even on $W^{s,p}(\mathbf{R}^{n})$.

(1.) When $sp<n$, we can prove some compact embedding results listed below.

[TABLE]

For the integral over $\mathbf{W}_{\epsilon}^{c}$, as $\frac{r-q\tau}{p^{*}_{s}}=1-\tau=\frac{r-q}{p^{*}_{s}-q}\in[0,1)$, it is easy to derive

[TABLE]

in view of (8), with $C^{\prime}_{1}>0$ an absolute constant independent of $u_{l}$ for any $l\geq 1$.

For the integral over $\mathbf{W}_{\epsilon}$, noticing $\mathfrak{L}(\mathbf{W}_{\epsilon})<\infty$ and the fact that $\inf\limits_{\mathbf{B}_{R}}V(x)\geq V_{\mathbf{B}_{R}}>0$ yields the embedding $M^{s;q,p}_{\mathrm{V}}(\mathbf{B}_{R})\to M^{s;q,p}(\mathbf{B}_{R})\hookrightarrow L^{\frac{rt}{t-1}}(\mathbf{B}_{R})$ with $\frac{rt}{t-1}\in[r,p^{*}_{s})$, we deduce

[TABLE]

as $R\to\infty$ and $l\to\infty$ for a subsequence of $\{u_{l}:l\geq 1\}$ using the same notation.

Notice the embedding $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\to L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$ is continuous provided $\frac{p^{*}_{s}}{p^{*}_{s}-r}\leq t\leq\infty$. Hence, plugging (20) and (21) altogether back to (19) finishes our proof completely. ∎

[TABLE]

For the integral over $\mathbf{B}_{R}$, seeing that $\inf\limits_{\mathbf{B}_{R}}V(x)\geq V_{\mathbf{B}_{R}}>0$ yields the embedding $M^{s;q,p}_{\mathrm{V}}(\mathbf{B}_{R})\to M^{s;q,p}(\mathbf{B}_{R})\hookrightarrow L^{\frac{rt}{t-1}}(\mathbf{B}_{R})$ with $\frac{rt}{t-1}\in[r,q)$, one deduces

[TABLE]

as $l\to\infty$ for a subsequence of $\{u_{l}:l\geq 1\}$ using the same notation. For the integral over $\mathbf{B}^{c}_{R}$, one deduces, similar to (20),

[TABLE]

as $R\to\infty$. So, $u_{l}\to 0$ in $L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$ for a subsequence relabeled with the same index $l$.

Notice the embedding $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\to L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$ is continuous provided $K(x)\in L^{t}_{loc}(\mathbf{R}^{n})$ for $\frac{q}{q-r}\leq t\leq\infty$ and $K(x)V^{-\tau}(x)$ is eventually bounded at infinity for $\tau=\frac{r-p^{*}_{s}}{q-p^{*}_{s}}$. ∎

Finally, let’s consider the case where $1\leq r<\min\{p^{*}_{s},q\}$ uncovered in the preceding results.

[TABLE]

by Hölder’s inequality and (8) with $p^{*}_{s}=r_{2}(r-q\mathfrak{x})$, provided $\mathfrak{x},\mathfrak{y},\mathfrak{z}\in(0,1)$.

To have $\mathfrak{x}<1$, one sees $0<\tau<\frac{p^{*}_{s}}{q-r}$ when $q\leq p^{*}_{s}$, and either $0<\tau<\frac{p^{*}_{s}}{q-r}$ or $\frac{p^{*}_{s}}{q-p^{*}_{s}}<\tau<\infty$ when $q>p^{*}_{s}$. To have $\mathfrak{x}>0$, one sees $0<\tau<\infty$ when $q\leq p^{*}_{s}$, and $0<\tau<\frac{p^{*}_{s}}{q-p^{*}_{s}}$ when $q>p^{*}_{s}$.

To have $\mathfrak{y}>0$, one sees $0<\tau<\frac{r}{q-r}$ when $q\leq p^{*}_{s}$, and either $0<\tau<\frac{r}{q-r}$ or $\frac{p^{*}_{s}}{q-p^{*}_{s}}<\tau<\infty$ when $q>p^{*}_{s}$. To have $\mathfrak{y}<1$, one sees $0<\tau<\infty$ when $q\leq p^{*}_{s}$, and $0<\tau<\frac{p^{*}_{s}}{q-p^{*}_{s}}$ when $q>p^{*}_{s}$.

Notice $\mathfrak{x}=\tau\mathfrak{z}$. To have $\mathfrak{z}<1$, one sees $0<\tau<\infty$ when $q\leq p^{*}_{s}$, and either $0<\tau<\frac{r}{q-p^{*}_{s}}$ or $\frac{p^{*}_{s}}{q-p^{*}_{s}}<\tau<\infty$ when $q>p^{*}_{s}$.

Summarize the above analyses to conclude that \tau\in\bigl{(}0,\frac{r}{q-r}\bigr{)}.

We certainly can take $\mathfrak{y}=0$ and see $\tau=\frac{r}{q-r}$ without having $p^{*}_{s}$ involved, and may also take $\mathfrak{x}=\tau=0$ corresponding to $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})=D^{s,p}(\mathbf{R}^{n})$ with $V(x)\equiv 0$.

Next, one has $\frac{q}{r_{1}}+\frac{p^{*}_{s}}{r_{2}}=q\mathfrak{x}+p^{*}_{s}\mathfrak{y}=r$ and thus the embedding $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\to L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$, when replacing $\bm{\Omega}$ in (24) by $\mathbf{R}^{n}$, is continuous (even for $\frac{\max\{p^{*}_{s},q\}}{\max\{p^{*}_{s},q\}-r}\leq t\leq\infty$).

Finally, using the same $\{u_{l}:l\geq 1\}$, one sees the decomposition (22) and the estimate (23), because $M^{s;q,p}_{\mathrm{V}}(\mathbf{B}_{R})\to M^{s;q,p}(\mathbf{B}_{R})\hookrightarrow L^{\frac{rt}{t-1}}(\mathbf{B}_{R})$ with $\frac{rt}{t-1}\in[r,\max\{p^{*}_{s},q\})$. For the integral over $\mathbf{B}^{c}_{R}$, we apply (24) on $\bm{\Omega}=\mathbf{B}^{c}_{R}$ to derive, as $R\to\infty$,

[TABLE]

Thus, $u_{l}\to 0$ in $L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$ for a subsequence relabeled with the same index $l$. ∎

Notice in Theorems 4 and 6, we do not need $\inf\limits_{\mathbf{R}^{n}}V(x)\geq V_{0}>0$, which appeared in Theorem 5. This condition can be replaced by an integrability condition as in Theorem 7.

(2.) When $sp\geq n$, we similarly have the compact embedding results as follows.

Recall the fractional Gagliardo-Nirenberg inequality in [22, Lemma 2.1] says that

[TABLE]

Here, $n\geq 2$, $1<p\leq\infty$, $1\leq q<\infty$, $0<s<1$, $\frac{1}{r}=(1-\theta)\frac{n-sp}{np}+\theta\frac{1}{q}$ with $\theta\in[0,1]$, and $r$ lies in between $q$ and $p^{*}_{s}$ when $sp<n$ while $q\leq r<\infty$ when $sp\geq n$, with $C_{2}>0$ an absolute constant depending on $n,p,q,r,s$.

[TABLE]

by virtue of (25), with $C^{\prime}_{2}>0$ an absolute constant independent of $u_{l}$ for any $l\geq 1$. Moreover, seeing $M^{s;q,p}_{\mathrm{V}}(\mathbf{B}_{R})\to M^{s;q,p}(\mathbf{B}_{R})\hookrightarrow L^{\frac{rt}{t-1}}(\mathbf{B}_{R})$ with $\frac{rt}{t-1}\in[r,\infty)$, one observes that, for some \tilde{t}\in\bigl{(}\frac{t}{t-1},\infty\bigr{]} with \lim\limits_{\tilde{t}\to\infty}\Bigl{(}\int_{\mathbf{R}^{n}}|u_{l}|^{r\tilde{t}}\,\mathbf{d}x\Bigr{)}^{\frac{1}{\tilde{t}}}=\int_{\mathbf{R}^{n}}|u_{l}|^{r}\,\mathbf{d}x, like (21),

[TABLE]

as $R\to\infty$ and $l\to\infty$ for a subsequence of $\{u_{l}:l\geq 1\}$ using the same notation.

Note the embedding $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\to L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$ is continuous if $\frac{t}{t-1}\leq\tilde{t}\leq\infty$. ∎

In addition, similar calculations lead to \tau\in\bigl{(}0,\frac{r}{q-r}\bigr{]}.

Finally, using the same $\{u_{l}:l\geq 1\}$, one sees the decomposition (22) and the estimate (23), because $M^{s;q,p}_{\mathrm{V}}(\mathbf{B}_{R})\to M^{s;q,p}(\mathbf{B}_{R})\hookrightarrow L^{\frac{rt}{t-1}}(\mathbf{B}_{R})$ with $\frac{rt}{t-1}\in[r,\infty)$. For the integral over $\mathbf{B}^{c}_{R}$, one analogously has, as $R\to\infty$,

[TABLE]

Thus, $u_{l}\to 0$ in $L^{r}_{\mathrm{K}}(\mathbf{R}^{n})$ for a subsequence relabeled with the same index $l$. ∎

(3.) Some applications.

One feels free to set either $V(x)\equiv 1$ or $K(x)\equiv 1$ to see some compact embedding results, in particular, $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{q}(\mathbf{R}^{n})$ and $M^{s;q,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{1}(\mathbf{R}^{n})$ for appropriate weight functions $V(x)$; when $p=q$, one has $W^{s,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{p}(\mathbf{R}^{n})$ and $W^{s,p}_{\mathrm{V}}(\mathbf{R}^{n})\hookrightarrow L^{1}(\mathbf{R}^{n})$.

Finally, following Servadei and Valdinoci [26, Section 3] (see also [21, Section 3.1]) or applying Auchmuty [1], for the standard fractional Laplacian $(-\Delta)^{s}$, the eigenvalue problems

[TABLE]

as well as the eigenvalue problems

[TABLE]

if $sp<n$, possess families of eigenvalues $\{\lambda_{k}>0:k\geq 1\}$ and $\{\delta_{l}>0:l\geq 1\}$, and sequences of associated eigenfunctions $\{f_{k}\in W^{s,2}_{\mathrm{V}}(\mathbf{R}^{n}):k\geq 1\}$ and $\{g_{l}\in D^{s,2}(\mathbf{R}^{n}):l\geq 1\}$, respectively, by virtue of Theorem 4 (when taking $p=q=r=2$ to work in Hilbert spaces) and Theorem 7 (when taking $V(x)\equiv 0$ and $p=r=2$ to work in Hilbert spaces).

$\lambda_{1},\delta_{1}$ are simple, $\lambda_{k},\delta_{l}$ have finite multiplicities, and $\{\lambda_{k}>0:k\geq 1\}$ and $\{\delta_{l}>0:l\geq 1\}$ are increasing with $\lim\limits_{k\to\infty}\lambda_{k}=\infty$ and $\lim\limits_{l\to\infty}\delta_{l}=\infty$. Besides, $f_{1},g_{1}$ are nonnegative in $\mathbf{R}^{n}$, and the sequences $\{f_{k}\in W^{s,2}_{\mathrm{V}}(\mathbf{R}^{n}):k\geq 1\}$ and $\{g_{l}\in D^{s,2}(\mathbf{R}^{n}):l\geq 1\}$ provide orthogonal bases to the spaces $W^{s,2}_{\mathrm{V}}(\mathbf{R}^{n}),L^{2}_{\mathrm{K}}(\mathbf{R}^{n})$ and $D^{s,2}(\mathbf{R}^{n}),L^{2}_{\tilde{\mathrm{K}}}(\mathbf{R}^{n})$, respectively.

The procedure is standard now, with details left for the interested reader.