Some remarks on rearrangement for nonlocal functionals

TL;DR

This paper investigates the properties of a nonlocal functional related to the Dirichlet p-norm, showing it does not decrease under rearrangement and establishing related properties like decay, compactness, and a fractional Polya-Szeg"o inequality.

Contribution

It demonstrates the failure of rearrangement decrease for a specific nonlocal functional and establishes new properties including a fractional Polya-Szeg"o inequality.

Findings

Nonlocal functional does not decrease under two-point rearrangement.

Established decay and compactness properties for the functional.

Proved a Polya-Szeg"o inequality for Riesz fractional gradients.

Abstract

We prove that a nonlocal functional approximating the standard Dirichlet $p$-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness, and the Polya-Szeg\"o inequality for Riesz fractional gradients, a notion recently introduced in the literature.

Equations

$${\int }_{{\mathbb{R}}^N}|\nabla u^{\ast }{|}^{p}dx\le {\int }_{{\mathbb{R}}^N}|\nabla u{|}^{p}dx.$$

$${\iint }_{{\mathbb{R}}^{2N}}\frac{|u^{\ast }(x)-u^{\ast }(y){|}^p}{|x-y{|}^{N+ps}}dxdy\le {\iint }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+ps}}dxdy,$$

$$\underset{s\nearrow 1}{\lim }(1-s){\iint }_{{\mathbb{R}}^{2N}}\frac{|v(x)-v(y){|}^p}{|x-y{|}^{N+ps}}dxdy=K_{N,p}{\int }_{{\mathbb{R}}^N}|\nabla v{|}^p,\text{mbox}forv\in W^{1,p}({\mathbb{R}}^N),$$

$$K_{N,p}:=\int_{{\mathbb{S}}^{N-1}}|\boldsymbol{e}\cdot\sigma|^{p}\,d\sigma,\quad\mbox{ for some }\boldsymbol{e}\in{\mathbb{S}}^{N-1},\mbox{ the unit sphere in ${\mathbb{R}}^{N}$}.$$

$${\int }_{{\mathbb{R}}^N}|\nabla u^H{|}^{p}dx={\int }_{{\mathbb{R}}^N}|\nabla u{|}^{p}dx,$$

$${\iint }_{{\mathbb{R}}^{2N}}\frac{|u^H(x)-u^H(y){|}^p}{|x-y{|}^{N+ps}}dxdy\le {\iint }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+ps}}dxdy,$$

$$I_{\delta }(u):={\iint }_{\{|u(y)-u(x)|>\delta \}}\frac{{\delta }^p}{|x-y{|}^{N+p}}dxdy.$$

$$\underset{\delta \searrow 0}{\lim }{I}_{\delta }(u)=\frac{1}{p}{K}_{N,p}{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{p}dx,$$

$$\left(\int_{\{|u|>\lambda\delta\}}|u|^{\frac{Np}{N-p}}dx\right)^{\frac{N-p}{Np}}\leq C\big{(}I_{\delta}(u)\big{)}^{1/p},\quad\forall u\in L^{p}({\mathbb{R}}^{N}),$$

$$\{u_{n}\}_{n\in{\mathbb{N}}}\mbox{ is bounded in }L^{p}({\mathbb{R}}^{N})\mbox{ and }\{I_{\delta_{n}}(u_{n})\}_{n\in{\mathbb{N}}}\mbox{ is bounded in ${\mathbb{R}}$}.$$

$${\int }_0^{\infty }\frac{I_{\delta }(u^{\ast }{)}^{\vartheta /p}}{\delta }d\delta <\infty ,$$

$$u^H(x):=\{\begin{array}{ll}\max \{u(x),u({\sigma }_H(x))\},& \text{for x∈H},\\ \min \{u(x),u({\sigma }_H(x))\},& \text{for x∈Hc}.\end{array}$$

$$A:=\big{\{}x\in\mbox{int}(H):u(x)\leq u(x^{H})\big{\}},\quad B:=\big{\{}x\in\mbox{int}(H):u(x)>u(x^{H})\big{\}},$$

$$C:=\big{\{}y\in H^{c}:u(y)\geq u(y^{H})\big{\}},\quad\mbox{ and }\quad D:=\big{\{}y\in H^{c}:u(y)<u(y^{H})\big{\}}.$$

$$C=\sigma (A)\text{mbox}andD=\sigma (B).$$

$$I_{\delta }=I_{\delta }^{(A\cup C)(A\cup C)}+I_{\delta }^{(B\cup D)(B\cup D)}+I_{\delta }^{(A\cup C)(B\cup D)}+I_{\delta }^{(B\cup D)(A\cup C)},$$

$$I_{\delta }^{OP}(u):={{\int }_O{\int }_P}_{\{|u(y)-u(x)|>\delta \}}\frac{{\delta }^p}{|x-y{|}^{N+p}}dxdy.$$

$$I_{\delta }^{(A\cup C)(A\cup C)}(u^H)=I_{\delta }^{(A\cup C)(A\cup C)}(u),$$

$$I_{\delta }^{(B\cup D)(B\cup D)}(u^H)=I_{\delta }^{(B\cup D)(B\cup D)}(u).$$

$$u^H(x)=u(x^H)\text{mbox}ifx\in A\cup C\text{mbox}and|x-y|=|x^H-y^H|,$$

$$I_{\delta }^{(A\cup C)(A\cup C)}(u^H)=I_{\delta }^{(A\cup C)(A\cup C)}(u),$$

$$u^{H}(x)=u(x),\quad\mbox{if $x\in B\cup D$}.$$

$$I_{\delta }^{(A\cup C)(B\cup D)}(u^H)\le I_{\delta }^{(A\cup C)(B\cup D)}(u).$$

$$I_{\delta }^{(A\cup C)(B\cup D)}(v)={\int }_A{\int }_B\left(\frac{\mathcal{L}(v,x,y)}{|x-y{|}^{N+p}}+\frac{\mathcal{L}(v,x,y^H)}{|x-y^H{|}^{N+p}}+\frac{\mathcal{L}(v,x^H,y)}{|x^H-y{|}^{N+p}}+\frac{\mathcal{L}(v,x^H,y^H)}{|x^H-y^H{|}^{N+p}}\right)dxdy,$$

$$\mathcal{L}(v,x,y):={\delta }^p\text{mathds}{1}_{\{|v(x)-v(y)|>\delta \}}(x,y),x,y\in {\mathbb{R}}^N.$$

$$u^H(x)=u(x^H),u^H(x^H)=u(x),u^H(y)=u(y),\text{mbox}and{u}^H(y^H)=u(y^H).$$

$$\frac{\mathcal{L}(u^H,x,y)}{|x-y{|}^{N+p}}=\frac{\mathcal{L}(u,x^H,y)}{|x-y{|}^{N+p}},\frac{\mathcal{L}(u^H,x,y^H)}{|x-y^H{|}^{N+p}}=\frac{\mathcal{L}(u,x^H,y^H)}{|x-y^H{|}^{N+p}}$$

$$\frac{\mathcal{L}(u^H,x^H,y)}{|x^H-y{|}^{N+p}}=\frac{\mathcal{L}(u,x,y)}{|x^H-y{|}^{N+p}},\frac{\mathcal{L}(u^H,x^H,y^H)}{|x^H-y^H{|}^{N+p}}=\frac{\mathcal{L}(u,x,y^H)}{|x-y{|}^{N+p}}.$$

$${\mathcal{D}}_{\delta }^H(u,x,y):=\mathcal{L}(u,x^H,y)+\mathcal{L}(u,x,y^H)-\mathcal{L}(u,x,y)-\mathcal{L}(u,x^H,y^H),$$

$$\mathfrak{d}_{\delta}^{H}(u):=\int_{A}\int_{B}{\mathscr{D}}_{\delta}^{H}(u,x,y)\Big{(}\frac{1}{|x-y|^{N+p}}-\frac{1}{|x^{H}-y|^{N+p}}\Big{)}\,dx\,dy\leq 0.$$

Full text

The second author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

1. Introduction

The well-known Polya-Szegö inequality [29, 30, 15] states that if $u\in W^{1,p}({\mathbb{R}}^{N},{\mathbb{R}}^{+})$ then

[TABLE]

where $u^{*}$ is the Schwarz symmetric rearrangement of $u$. This inequality has relevant applications in the study of isoperimetric inequalities, in the Faber-Krahn inequality and in the determination of optimal constants in the Sobolev inequality [2, 38]. This kind of inequalities still holds in the nonlocal case, e.g. for the standard fractional norm, namely for $u\in W^{s,p}({\mathbb{R}}^{N},{\mathbb{R}}^{+})$

[TABLE]

for $p\geq 1$ and $s\in(0,1)$, see e.g. [1, 3]. Actually this inequality implies (1.1) by a straightforward application of a result by Bourgain, Brezis and Mironescu [4, 5] which confirms that

[TABLE]

where

[TABLE]

Polarization by closed half spaces $H\subset{\mathbb{R}}^{N}$ containing the origin is an elementary form of symmetrization and it is a key tool in order to investigate various rearrangements inequalities. The polarization $u^{H}$ with respect to $H$, see the definition (2.1), also called two-point rearrangement, essentially compares the values of $u$ on the two sides of $\partial H$ and keeps the largest values inside $H$ and the smallest values outside $H$, cf. [16, 17, 18, 14]. Since the first achievements obtained in [14], the approximation in $L^{p}({\mathbb{R}}^{N})$ of $u^{*}$ via iterated polarizations of $u$ has been refined in various ways. It is now known that there exists an explicit and universal (i.e. independent of $u$) sequence of closed half spaces $\{H_{n}\}_{n\in{\mathbb{N}}}$ of ${\mathbb{R}}^{N}$ containing the origin such that a suitable sequence of iterated polarizations of $u$ with respect to $H_{n}$ strongly converges to $u^{*}$ in $L^{p}({\mathbb{R}}^{N})$, see [37] and the references therein. It is thus natural to derive the rearrangement inequalities (1.1)-(1.2) from the (possible) corresponding inequalities for the polarizations using general weak lower semi-continuity properties. In fact, for any closed half space $H$ with $0\in H$ and any $u\in W^{1,p}({\mathbb{R}}^{N})$,

[TABLE]

as well as, for any $W^{s,p}({\mathbb{R}}^{N}),$

[TABLE]

see [3, 14] and references included. For applications of polarization techniques, see [34, 33, 19, 37, 36, 35].

More recently, a new class of nonlocal functionals has been involved in the study of topological degree of a map [7, 13, 27], namely, for $\delta>0$ and $1<p<N$,

[TABLE]

It turns out that this energy also provides a pointwise approximation of $\|\nabla u\|_{p}^{p}$ for $p>1$, precisely

[TABLE]

where $K_{N,p}$ is given by (1.4), see [23, 6] (and also [9, 10, 11, 12, 24]). Various properties of Sobolev spaces in terms of $I_{\delta}$ were investigated in [26], for example, it was shown in [26, Theorem 3] that, for $1<p<N$ and for $\delta>0$,

[TABLE]

for some positive constants $C,\lambda$ depending only on $N$ and $p$.

The first goal of this note is to prove that, however, the nonlocal energy $I_{\delta}$ fails to be decreasing upon polarization. This supports yet again the idea (cf. [25]) that $I_{\delta}$ is a much more delicate approximation of local norm with respect to the other mentioned above. More precisely, we have

Theorem 1.1**.**

Let $N\geq 1$ and $p\geq 1$. Then there exist $\delta_{0}>0$ and a closed half space $H\subset{\mathbb{R}}^{N}$ with $0\in H$ such that for any $0<\delta<\delta_{0}$ there exists a measurable function $u:{\mathbb{R}}^{N}\to{\mathbb{R}}$ such that $I_{\delta}(u^{H})>I_{\delta}(u)$.

The proof of Theorem 1.1 is given in Section 2.2. In Section 2.3, we present a proof of (1.2).

The second goal of this note is to prove some facts related to symmetric functions. Precisely:

$\bullet$ (Global compactness and $I_{\delta}$). Let $1<p<N$, $(\delta_{n})\to 0_{+}$, and let $\{u_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of radially symmetric decreasing functions. Assume that

[TABLE]

Then $\{u_{n}\}_{n\in{\mathbb{N}}}$ is pre-compact in $L^{r}({\mathbb{R}}^{N})$ for every $p<r<Np/(N-p)$. $\bullet$ (Decay and integrability of $I_{\delta}$). If $u:{\mathbb{R}}^{N}\to{\mathbb{R}}^{+}$ and there exists $\vartheta>0$ with

[TABLE]

then there exists $C>0$ depending on $u,N,p,\vartheta$ such that $0\leq u^{*}(x)\leq C|x|^{-(N-p)/p}.$

The proof of these two facts is given in Section 3 (see Theorems 3.2 and 3.4).

2. Symmetrization inequalities

2.1. The defect of decreasingness of $I_{\delta}$

In the following $H$ will denote a closed half-space of ${\mathbb{R}}^{N}$ containing the origin. We denote by ${\mathcal{H}}$ the set of these closed half-spaces. A reflection $\sigma_{H}:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ with respect to $H$ is an isometry such that $\sigma_{H}^{2}={\rm Id}$ and $|x-y|<|x-\sigma_{H}(y)|$ for all $x,y\in H$. We also set $H^{c}:={\mathbb{R}}^{N}\setminus H$. Given $x\in{\mathbb{R}}^{N}$, $\sigma_{H}(x)$ will also be denoted by $x^{H}$. The polarization (or two-point rearrangement) of a nonnegative real valued function $u:{\mathbb{R}}^{N}\to{\mathbb{R}}^{+}$ with respect to a given $H$ is defined as

[TABLE]

Let us set

[TABLE]

[TABLE]

It is clear that

[TABLE]

We have

[TABLE]

where, for two measurable subsets $O,\,P$ of ${\mathbb{R}}^{N}$, we denote

[TABLE]

We claim that

[TABLE]

[TABLE]

We begin with (2.3). Since

[TABLE]

by a change of variable $x=x^{H}$ and $y=y^{H}$, we obtain

[TABLE]

which is (2.3). We next establish (2.4). This is a direct consequence of the fact

[TABLE]

We next concern about the validity of the inequality:

[TABLE]

By a change of variables, we obtain

[TABLE]

where

[TABLE]

We have, for $x\in A$ and $y\in B$,

[TABLE]

It follows that, for $x\in A$ and $y\in B$,

[TABLE]

and

[TABLE]

Setting

[TABLE]

we derive that inequality (2.5) is equivalent to

[TABLE]

On the other hand, in general, concrete examples show that the inequality ${\mathscr{D}}_{u}(x,y)$ fails, so that it is expected that $I_{\delta}(u^{H})-I_{\delta}(u)$ can be positive for some $H,u$ and $\delta>0$, in which case the quantity $\mathfrak{d}_{\delta}^{H}(u)$ provides a measure of the defect of decreasingness.

Remark 2.1** (Vanishing defect).**

For any closed half space $H\subset{\mathbb{R}}^{N}$ with $0\in H$ there holds, for $p>1$,

[TABLE]

In fact, we know that $u^{H}\in W^{1,p}({\mathbb{R}}^{N})$ also and $\|\nabla u^{H}\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}$ from formula (1.5). Furthermore, from formulas (2.2)-(2.5), we infer

[TABLE]

Then, from equality (1.7), we conclude

[TABLE]

proving the assertion.

2.2. Proof of Theorem 1.1

We first deal with the case $N=1$. Here is a counterexample to (2.6) with $p\in[1,+\infty)$. Fix $\varepsilon\in(0,1/8)$ and let $u:{\mathbb{R}}\to{\mathbb{R}}$ be defined by

[TABLE]

Let $H=[0,\infty)$ and $\sigma$ be the standard reflection. It is clear that $u^{H}$ satisfies

[TABLE]

We derive from (2.7) and (2.8) that

[TABLE]

A straightforward computation yields, for small $\delta$ and $\varepsilon$,

[TABLE]

[TABLE]

and

[TABLE]

We obtain that, for small positive $\delta$ and $\varepsilon$, we have $I_{\delta}(u^{H})>I_{\delta}(u)$.

This example can be modified to obtain similar conclusion in the case $\{0\}\in H$ by considering the function $u(\cdot+c)$ for some $c>0$ where $u$ is given above. In the above example, the function $u$ is not non-negative. However, this point can be handled by considering the function given by $u(x)+2\varepsilon\delta$ if $|x|<3$ and 0 otherwise.

We next consider the case $N\geq 2$. Set

[TABLE]

and define $U:{\mathbb{R}}^{N}\to{\mathbb{R}}$ as follows, for $(x_{1},x^{\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{N-1}$,

[TABLE]

where $u$ is given in (2.7). One can check that

[TABLE]

In the above example, the function $U$ is not non-negative and $H$ does not contain the origin. However, this point can be handled similarly as in the case $N=1$.

The following question remains open:

Open problem 2.2**.**

Let $N\geq 1$. It is true that $I_{\delta}(u^{*})\leq I_{\delta}(u)$ for any measurable $u:{\mathbb{R}}^{N}\to{\mathbb{R}}^{+}$ and $\delta>0$?

2.3. Riesz two-point inequality

Let $u\in L^{1}_{{\rm loc}}({\mathbb{R}}^{N})$, let $H$ be a closed half-space of ${\mathbb{R}}^{N}$ and let $G$ be a Young function, i.e. $G:[0,+\infty)\to{\mathbb{R}}$, $G(0)=0$, $G$ is non-decreasing, and $G$ is convex, and let $w$ be a non-negative, non-increasing radial function. The above notations allow to prove the classical inequality

[TABLE]

In fact, set

[TABLE]

and define

[TABLE]

As in (2.6), we have

[TABLE]

We claim that

[TABLE]

for each $x\in A$ and $y\in B$. Assuming this, we then immediately get inequality (1.6) since $w$ is non-decreasing. We now prove (2.10). Observe that if $a\leq b$ and $c\leq d$ then

[TABLE]

and

[TABLE]

It follows that, for $x\in A$ and $y\in B$,

[TABLE]

and

[TABLE]

Assertion (2.10) follows from the properties of Young’s functions of $G$.

As a consequence of (2.9), one has

[TABLE]

for $s\in(0,1)$ and $p>1$. It follows that, for $u\in W^{s,p}({\mathbb{R}}^{N})$ with $u>0$,

[TABLE]

where $u^{*}$ denotes the spherical symmetric rearrangement of $u$. By the BBM formula (1.3), one reaches the Polya-Szegö inequality.

Remark 2.3**.**

Recall that ${\mathcal{H}}$ is the set of half-spaces which contain the origin. One can endow ${\mathcal{H}}$ with a metric that ensures that $H_{n}\to H$ if there exists a sequence of isometries $i_{n}:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ such that $H_{n}=i_{n}(H)$ and $i_{n}$ converges to the identity as $n\to+\infty$, moreover, ${\mathcal{H}}$ is separable with respect to this metric. Let $\{H_{n}\}_{n\in{\mathbb{N}}}$ be a dense set in ${\mathcal{H}}$. For any $u\in L^{p}({\mathbb{R}}^{N})$, let $\{u_{n}\}_{n\in{\mathbb{N}}}$ be the sequence defined by

[TABLE]

Assume that

[TABLE]

Then $u^{*}\in W^{1,p}({\mathbb{R}}^{N})$ ($p>1$), where $u^{*}$ denotes the Schwarz symmetrization of $u$. In fact, by [37, Theorem 1], we have $u_{n}\to u^{*}$ strongly in $L^{p}({\mathbb{R}}^{N})$ as $n\to\infty$. Then, the assertion follows by the Gamma-convergence result in [25, Theorem 2]. If in addition, for all $\delta>0$,

[TABLE]

then $u$ is radially symmetric about some point $x_{0}\in{\mathbb{R}}^{N}$ provided that ${\mathcal{L}}^{N}(\{\nabla u^{*}=0\})=0$. This follows by the Brothers-Ziemer result [15, Theorem 1.1] jointly with [23, Theorem 2].

3. Radially decreasing functions and $I_{\delta}$

For every measurable function $u:\mathbb{R}^{N}\to\mathbb{R}$ we define its distribution function

[TABLE]

Let $0<q<\infty$ and $0<\vartheta<\infty$, the Lorentz space $L^{q,\vartheta}(\mathbb{R}^{N})$ (cf. [20, 22, 28]) is defined by

[TABLE]

In the limit case $\vartheta=\infty$, this is defined by

[TABLE]

We recall from [8, Lemma 2.9] the following

Lemma 3.1**.**

Let $0<\vartheta\leq\infty$ and $0<q<\infty$. Let $u\in L^{q,\vartheta}(\mathbb{R}^{N})$ be a non-negative and radially symmetric decreasing function. Then

[TABLE]

The next proposition shows that the measure of the superlevels of a nonnegative function $u$ is controlled by a quantity involving $I_{\delta}(u^{*})$. Set $p^{*}:=Np/(N-p)$ for $1\leq p<N$.

Theorem 3.2**.**

Let $1<p<N$ and $u:{\mathbb{R}}^{N}\to{\mathbb{R}}$ be a non-negative function. Then there exists two positive constants $C$ and $\lambda$ depending only on $N$ and $p$ such that

[TABLE]

where $u^{*}$ is the Schwarz symmetric rearrangement of $u$. Assume that there exists $\vartheta>0$ such that

[TABLE]

Then $u,u^{*}\in L^{p^{*},\vartheta}(\mathbb{R}^{N})$ and there exists a positive constant $C$ depending on $u,N,p,\vartheta$ such that

[TABLE]

Proof.

By the definition of $u^{*}$ we have that $\mu_{u}(t)=\mu_{u^{*}}(t)$ for $t>0$. Then, by applying inequality (1.8) to $u^{*}$, we have, for all $\delta>0$,

[TABLE]

and

[TABLE]

which implies (3.1). By virtue of (3.2), it follows from (3.1) that

[TABLE]

which yields $u,u^{*}\in L^{p^{*},\vartheta}(\mathbb{R}^{N})$ and the final assertions follows from Lemma 3.1. ∎

Remark 3.3**.**

Since $L^{q,q}({\mathbb{R}}^{N})=L^{q}({\mathbb{R}}^{N})$, it follows that $u,u^{*}\in L^{p^{*}}(\mathbb{R}^{N})$ if

[TABLE]

Concerning the compactness related to $I_{\delta}$, the following result was shown in [26, Theorem 2]. Let $p>1$, $(\delta_{n})\to 0_{+}$ and $(u_{n})\subset L^{p}({\mathbb{R}}^{N})$. Assume that

[TABLE]

Then

[TABLE]

In this paper, we prove the following global compactness result:

Theorem 3.4**.**

Let $1<p<N$, $(\delta_{n})\to 0_{+}$, and let $\{u_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of radially symmetric decreasing functions. Assume that

[TABLE]

Then \big{\{}u_{n}\big{\}}_{n\in{\mathbb{N}}} is pre-compact in $L^{r}({\mathbb{R}}^{N})$ for every $p<r<Np/(N-p)$.

Proof.

From (1.8), (3.3), and (3.4), we derive that

[TABLE]

Since $(u_{n})$ is decreasing and $(u_{n})$ is bounded in $L^{p}({\mathbb{R}}^{N})$, for any $\delta>0$ there exists $R_{\delta}$ such that, for all $n$,

[TABLE]

Fix $\varepsilon>0$. By (3.5), there exists a finite subset $J$ of ${\mathbb{N}}$ such that

[TABLE]

where $B_{R}$ denotes the open ball centered at the origin and of radius $R$ in ${\mathbb{R}}^{N}$ for $R>0$. On the other hand, by (3.6), we have

[TABLE]

since $(u_{n})$ is bounded in $L^{p}({\mathbb{R}}^{N})$. A combination of (3.7) and (3.8) yields, for $\varepsilon$ small enough such that $\varepsilon<C\varepsilon^{1-p/r}$,

[TABLE]

where $C$ is the constant in (3.8). Since (3.9) holds for small $\varepsilon$ and $1-r/p>0$, it follows that $(u_{n})$ is pre-compact in $L^{r}({\mathbb{R}}^{N})$. ∎

Remark 3.5**.**

Let $N\geq 1$ and $p\geq N$. It was shown in [26, Theorem 1] that if $u\in L^{p}({\mathbb{R}}^{N})$ and $I_{\delta}(u)<+\infty$ for some $\delta>0$ then $u\in BMO_{{\rm loc}}({\mathbb{R}}^{N})$ where $BMO$ denotes the space of functions of bounded mean oscillation. By the same proof, under the same assumptions on $\{u_{n}\}$, one obtains the compactness result for $L^{r}({\mathbb{R}}^{N})$ for $p<r<p^{*}$ with $p^{*}=\infty$.

As a consequence of Theorem 3.4, we have the following

Corollary 3.6**.**

Let $1<p<N$ and let $\{u_{n}\}_{n\in{\mathbb{N}}}$ be a nonnegative sequence of functions bounded in $L^{p}({\mathbb{R}}^{N})$ such that

[TABLE]

for some sequence $\{\delta_{n}\}_{n\in{\mathbb{N}}}\to 0_{+}$. Then, up to a subsequence, $\{u_{n}^{*}\}_{n\in{\mathbb{N}}}$ converges strongly in $L^{r}({\mathbb{R}}^{N})$ to a radial decreasing function, for any $p<r<Np/(N-p)$.

Proof.

Since the sequence $\{u_{n}\}_{n\in{\mathbb{N}}}$ is bounded in $L^{p}({\mathbb{R}}^{N})$, it follows that sequence $\{u_{n}^{*}\}_{n\in{\mathbb{N}}}$ is also bounded in $L^{p}({\mathbb{R}}^{N})$ by Cavalieri’s principle. Then, by Theorem 3.4, it follows that $u_{n}^{*}\to v$ in $L^{r}({\mathbb{R}}^{N})$ strongly for any $p<r<Np/(N-p)$. ∎

4. An open problem for Riesz fractional gradients

Recently, a notion of fractional gradients (more precisely distributional Riesz fractional gradients) has been introduced in the literature by Shieh and Spector in the papers [31, 32], where several basic properties of local Sobolev spaces (e.g. Sobolev, Morrey, Hardy, Trudinger inequalities) are proven to extend to fractional spaces defined through this new notion.

More precisely, the fractional gradient $D^{s}u(x)$ at a point $x\in{\mathbb{R}}^{N}$ is defined for locally Lipschitz compactly supported functions $u:{\mathbb{R}}^{N}\to{\mathbb{R}}$, for any $s\in(0,1)$, by

[TABLE]

for a suitable positive constant $c_{N,s}$ depending on $N$ and $s$. This is reminiscent of the classical scalar notion of $\frac{s}{2}$-fractional laplacian

[TABLE]

for a suitable normalization constant $C_{N,s}$ depending on $N$ and $s$. Notice also that [31, 32]

[TABLE]

According to [31, 32], one can define, for $p>1$ and $s\in(0,1)$, the space

[TABLE]

We now formulate a related open problem.

Open problem 4.1**.**

Let $p>1$, $s\in(0,1)$ and $u\in L^{s,p}({\mathbb{R}}^{N})$ with $u\geq 0$. Prove or disprove that $D^{s}u^{*}\in L^{s,p}({\mathbb{R}}^{N})$ and the inequality holds

[TABLE]

In general the inequality

[TABLE]

for all $u\in L^{s,p}({\mathbb{R}}^{N})$ is not expected to hold. In particular, one cannot obtain inequality (4.1) for sign-changing functions. The solution of the above open problem would be very useful in connection with compact injections for radially symmetric functions of $L^{s,p}({\mathbb{R}}^{N})$. In fact, assume $N\geq 2$. Notice that, since for any $\varepsilon\in(0,s)$ the injection

[TABLE]

is continuous (cf.  [31, (g) of Theorem 2.2]) and the injection

[TABLE]

is compact (cf. [21, Theorem II.1]) for all $p<q<p^{*}_{s-\varepsilon}$ it follows that the injection

[TABLE]

is compact for $p<q<Np/(N-sp)$. In particular, nonnegative minimizing sequences for

[TABLE]

could be replaced by new minimizing sequences which are radially symmetric decreasing and strongly converging in $L^{q}({\mathbb{R}}^{N})$ for any $p<q<Np/(N-sp)$.

Remark 4.2**.**

Aiming to prove (4.1), with no loss of generality one may assume $u\in C^{\infty}_{c}({\mathbb{R}}^{N})$. Let $H$ be an arbitrary closed given half-space with $0\in H$. Define the function $\mathfrak{J}(u):{\mathbb{R}}^{N}\to{\mathbb{R}}$,

[TABLE]

Then $\mathfrak{J}(u)\in L^{1}({\mathbb{R}}^{N})$. Setting $v(x):=u(x^{H})$ and $w(x):=u^{H}(x^{H})$ for all $x\in{\mathbb{R}}^{N}$, we have

[TABLE]

Writing $x^{H}=x_{0}+Rx$ where $R$ is a rotation, a change of variable yields

[TABLE]

and, analogously, $\mathfrak{J}(u^{H})(x^{H})=\mathfrak{J}(w)(x)$. In turn, we conclude that

[TABLE]

If one was be able to prove that

[TABLE]

then (4.1) would follow by standard approximations. In the local case (4.3) follows immediately in light of (4.2), while for $\mathfrak{J}$, which is a nonlocal function, the situation is rather unclear.

If instead one finds $u$ and $H$ such that (4.3) holds with opposite inequality, then the Riesz gradients would already fail the basic polarization inequality.

Acknowledgements. The authors would like to warmly thank Daniel Spector for providing some useful remarks about the content of his works [31, 32].