Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds

TL;DR

This paper investigates the existence of extremal functions for Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds using the Concentration-Compactness principle.

Contribution

It extends the analysis of Hardy-Littlewood-Sobolev inequalities to the setting of compact Riemannian manifolds and establishes existence results for extremal functions.

Findings

Existence of extremal functions on compact manifolds.

Application of Concentration-Compactness principle to this problem.

Extension of classical inequalities to geometric settings.

Abstract

This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact manifolds without boundary via Concentration-Compactness principle.

Equations

$$I_{\alpha }f(x)={\int }_{M^n}\frac{f(y)}{|x-y{|}_g^{n-\alpha }}d{V}_y$$

$$\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n},$$

$$||I_{\alpha }f|{|}_{L^q(M^n)}\le C(\alpha ,p,M^n,g)||f|{|}_{L^p(M^n)}$$

$$N_{p,\alpha ,M}:=$$

$$:=$$

$$N_{p,\alpha ,M}:=$$

$$:=$$

$$I_{M^{n},g,\alpha}=\int_{M^{n}}\bigl{[}G_{x}^{g}(y)\bigr{]}^{\frac{\alpha-n}{2-n}}g(y)dV_{y},$$

$$\{\begin{array}{l}{\int }_{{\mathbb{R}}^n}|f{|}^{p}dx={\int }_{{\mathbb{R}}^n}|g{|}^{t}dy=1\\ N_{p,\alpha }={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}f(x)g(y)|x-y{|}^{\alpha -n}dxdy.\end{array}$$

$$\{\begin{array}{l}|x{|}^{\alpha -n}\ast g=N_{p,\alpha }{f}^{p-1}(x),\\ |x{|}^{\alpha -n}\ast f=N_{p,\alpha }{g}^{t-1}(x).\end{array}$$

$$f_{\lambda }(x)={\lambda }^{-p/n}f(x/\lambda ),g_{\lambda }(y)={\lambda }^{-t/n}g(y/\lambda ),\forall \lambda >0$$

$$g\ast h(x)={\int }_{M^n}g(y)h(|y-x{|}_g)d{V}_y.$$

$$||g\ast h|{|}_{L^r}\le C||g|{|}_{L^q}\cdot ||h|{|}_{L^p},$$

$$f_m\rightharpoonup f\text{weakly in}{L}^p(M^n).$$

$$I_{\alpha }{f}_m\to I_{\alpha }f\text{strongly in}{L}^r(M^n).$$

$$I_{\alpha }{f}_m(x)=K^{\rho }\ast f_m(x)+K_{\rho }\ast f_m(x)\triangleq I_{\alpha }^1{f}_m(x)+I_{\alpha }^2{f}_m(x).$$

$$\bigl{|}K^{\rho}*f_{m}(x)\bigr{|}\leq\|K^{\rho}\|_{P^{\prime}}\|f_{m}\|_{p}\leq C(\rho),$$

$$K^{\rho }\ast f_m\to K^{\rho }f\text{strongly in}{L}^r(M^n).$$

$${\int }_{M^n}{K}_{\rho }(|x-y{|}_g{)}^{s}d{V}_x\le C{\rho }^{(\alpha -n)s+n},$$

$$\Vert K_{\rho }\ast (f_m-f){\Vert }_r\le C{\rho }^{(\alpha -n)+n/s}\Vert f_m-f{\Vert }_p\le C{\rho }^{(\alpha -n)+n/s}.$$

$$f_m\rightharpoonup f\text{weakly in}{L}^p(M^n),$$

$$I_{\alpha }{f}_m\rightharpoonup I_{\alpha }f\text{weakly in}{L}^q(M^n),$$

$$I_{\alpha }{f}_m\to I_{\alpha }f\text{strongly in}{L}^r(M^n)$$

$$f_m\rightharpoonup f\text{weakly in}{L}^p(M^n).$$

$$\nu =|I_{\alpha }f{|}^{q}d{V}_x+\sum _{j\in J}{\nu }_j{\delta }_{P_j},$$

$$\mu \ge |f{|}^{p}d{V}_x+\sum _{j\in J}{\mu }_j{\delta }_{P_j}$$

$${\nu }_j^{1/q}\le N_{p,\alpha }{\mu }_j^{1/p}\text{for all}j\in J.$$

$$I_{\alpha }{f}_m\rightharpoonup I_{\alpha }f\text{weakly in}{L}^q(M^n),$$

$$I_{\alpha }{f}_m\to I_{\alpha }f\text{strongly in}{L}^r(M^n)$$

$$I_{\alpha }{f}_m\to I_{\alpha }f\text{pointwisely a.e. in}{M}^n,$$

$$0=$$

$$=$$

$${\left({\int }_{M^n}|\phi {|}^{q}d\nu \right)}^{1/q}\le N_{p,\alpha ,M}{\left({\int }_{M^n}|\phi {|}^{p}d\mu \right)}^{1/p},\forall \phi \in C_0^{\infty }(M^n).$$

$${\left({\int }_{M^n}|\phi (x)I_{\alpha }{f}_m{|}^{q}d{V}_x\right)}^{1/q}$$

Full text

**keywords: ** Hardy-Littlewood-Sobolev inequalities, Existence of extremal, Concentration-compactness principle, Compact manifold.

1 Introduction

It is well known that classical Sobolev inequalities and Hardy-Littlewood- Sobolev(HLS) inequalities are basic tools in analysis and geometry, and their sharp constants play essential role on certain geometric and probabilistic information. In fact, in past decades, these sharp inequalities were applied extensively in the study of curvature equations, see, e.g. [3, 15, 1, 26, 14, 16, 4] and references therein. Recently, there have been some interesting results concerning the globally defined fractional operators such as fractional Yamabe problems, fractional prescribing curvature problems, fractional Paneitz operators, etc. (see, e.g. [11, 12, 13, 20, 21, 22, 23] and references therein), which are closely related to singular integral operator. In particular, the sharp HLS inequality is immediately applied to discuss a class of prescribing integral curvature problems by Zhu [30] and integral equations on bounded domain in [7, 6]. So, HLS inequalities play essential role in the global analysis of some operators of geometric interest.

Motivated by these studies, there are some extensions of classical HLS inequalities, such as HLS inequality on the upper half space, HLS on compact manifolds, reversed HLS inequality, or HLS inequality on the Heisenberg group, see [8, 9, 17, 10, 5, 28, 29] for details. This paper is mainly devoted to discuss the sharp HLS inequality on compact manifolds without boundary.

Let $(M^{n},g)$ be a given compact Riemmanian manifold without boundary, $\alpha\in(0,n)$ be a parameter and $|x-y|_{g}$ represent the distance from $x$ to $y$ on $M^{n}$ under metric $g$. In [17], Han and Zhu have introduced the following integral operator

[TABLE]

and got the following Hardy-Littlewood-Sobolev inequalities:

Proposition 1.1** (Proposition 1.1. in [17]).**

Assume that $\alpha\in(0,n)$, $1<p<\frac{n}{\alpha}$ and $q$ is given by

[TABLE]

then there is a positive constant $C(\alpha,p,M^{n},g)$, such that

[TABLE]

holds for all $f\in L^{p}({M}^{n}).$ Moreover, for $1\leq r<q$, operator $I_{\alpha}:L^{p}(M^{n})\to L^{r}(M^{n})$ is a compact embedding.

As is well known, it is important to study the extremal problems of (1.3), which can be stated as follows:

[TABLE]

Equivalently, we can stated also as

[TABLE]

where $t=\frac{q}{q-1}$. In particular, we denote $N_{p,\alpha,\mathbb{R}^{n}}$ as $N_{p,\alpha}$.

In [17], Han and Zhu have discussed the extremal problems (1) for the conformal case, i.e. the case $p=t$ and $f=g$. Then as an application, they studied a class of integral curvature problems. Particularly, they give a new proof for the Yamabe problem on compact locally conformally flat manifold.

This paper will deal with the remaining cases. Firstly, we will get the following estimate to the sharp constant.

Proposition 1.2** (Estimate).**

$N_{p,\alpha,M}\geq N_{p,\alpha}$.

Then, similar to the existence criteria of classical Yamabe problem, we will give the following the existence criteria of the extremal problems (1) by the Concentration-Compactness principle introduced by Lions (see [24, 25]).

Theorem 1.3** (Criteria of Existence).**

Under the assumption of Proposition 1.1 and if $N_{p,\alpha,M}>N_{p,\alpha}$, then the supremum is attained, i.e., there exists some function $f(x)\in L^{p}(M^{n})$ such that $N_{p,\alpha,M}=\frac{\|I_{\alpha}f\|_{L^{q}(M^{n})}}{\|f\|_{L^{p}(M^{n})}}$.

Remark 1.4**.**

Let $G_{x}^{g}(y)=n(n-2)\omega_{n}\Gamma_{x}^{g}(y)$, where $\Gamma_{x}^{g}(y)$ is the Green’s function with pole at $x$ for the conformal Laplacian operator $-\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}$ and $\omega_{n}$ is the volume of the unit ball. As discussed in [17], for the operator

[TABLE]

we can also get the similar results of estimate (Proposition 1.2) and existence criteria (Theorem 1.3). Since the details of the proof is similar, so we omit it for conciseness.

The plan of the paper is following. In Section 2, we introduce some known facts and give a new proof of the compactness of operator (1.1) for convenience. Then, we present our Concentration-Compactness Lemma in the Section 3. Finally, Section 4 is devoted to get the estimate (Proposition 1.2) and prove the existence of extremal problem (Theorem 1.3).

2 Preliminary

Firstly, we recall the existence of the extremal problem of Classical Hardy-Littlewood-Sobolev inequalities on $\mathbb{R}^{n}$ as follows.

Theorem 2.1** (Theorem 2.3 of [27] & Theorem 2.1 of [25]).**

There exist a pair of nonnegative functions $f\in L^{p}(\mathbb{R}^{n})$ and $g\in L^{t}(\mathbb{R}^{n})$ such that

[TABLE]

Hence, Extremal pair satisfies the Euler-Lagrange equation

[TABLE]

Furthermore, by scaling, we know that function pairs

[TABLE]

also satisfy (2.1) and (2.2).

For convenience, we introduce the following Young’s inequality.

Lemma 2.2** (Young’s inequality, Lemma 2.1 of [17]).**

For a given compact manifold ($M^{n},g)$, define

[TABLE]

Then, there is a constant $C>0$, such that

[TABLE]

where $p,q,r\in(1,\infty)$ and satisfy $1+\frac{1}{r}=\frac{1}{q}+\frac{1}{p}.$

Following, we give a new proof of the compactness about the operator (1.1).

Proposition 2.3** (Compactness).**

For all $r\in[1,q)$, where $q$ is defined as (1.2), operator $I_{\alpha}:L^{p}(M^{n})\rightarrow L^{r}(M^{n})$ is compact.

Proof.

Take any bounded sequence $\{f_{m}\}$ in $L^{p}(M^{n})$. Then, there exists a subsequence (still denoted by $\{f_{m}\}$) and some function $f\in L^{p}(M)$ such that

[TABLE]

It is known that the proof will be completed if it holds that

[TABLE]

Denoted by $K^{\rho}(t)=t^{\alpha-n}\chi_{\{t>\rho\}}$ and $K_{\rho}(t)=t^{\alpha-n}-K^{\rho}(t)$ for $t>0$, where $\rho>0$ is a parameter to be chosen later. Then, we decompose the integral operator as

[TABLE]

Since, for any fixed $x\in M^{n}$, $K^{\rho}(|x-y|_{g})\in L^{p^{\prime}}(M^{n})$ with respect to $y$, then weak convergence implies that $K^{\rho}*f_{m}\rightarrow K^{\rho}*f$ pointwisely. Notice also that

[TABLE]

where $C(\rho)$ is independent of $x$ and $m$. So, by dominated convergence theorem, we have that

[TABLE]

Since

[TABLE]

where $0<s<\frac{n}{n-\alpha}$, then we take parameter $s>1$ satisfying $\frac{1}{r}+1=\frac{1}{p}+\frac{1}{s}$ and get from the Young’s inequality (see Lemma 2.2) that

[TABLE]

By now, through choosing first $\rho$ small and then $m$ large, we deduce the claimed convergence in $L^{r}(M^{n})$. ∎

Based on the Proposition 2.3, we have the following conclusions.

Remark 2.4**.**

For any bounded sequence $\{f_{m}\}\subset L^{p}(M^{n})$, there exists a subsequence (still denoted by $\{f_{m}\}$) and some function $f\in L^{p}(M^{n})$ such that

[TABLE]

for all $r\in[1,q)$. Furthermore, $I_{\alpha}f_{m}\rightarrow I_{\alpha}f$ pointwisely a.e. in $M^{n}$.

3 Concentration-Compactness Lemma

Lemma 3.1**.**

Let $\{f_{m}\}\subset L^{p}(M^{n})$ be a bounded nonnegative sequence and there exists some function $f\in L^{p}(M^{n})$ such that

[TABLE]

After passing to a subsequence, assume that $|I_{\alpha}f_{m}|^{q}dV_{x}$, $|f_{m}|^{p}dV_{x}$ converge weakly in the sense of measure to some bounded nonnegative measures $\nu,\ \mu$ on $M^{n}$. Then we have:

i) There exist some countable set $J$, a family $\{P_{j}:j\in J\}$ of distinct points in $M^{n}$, and a family $\{\nu_{j}:j\in J\}$ of nonnegative numbers such that

[TABLE]

where $\delta_{P_{j}}$ are the Dirac-mass of mass $1$ concentrated at $P_{j}\in M^{n}$;

ii)In addition we have

[TABLE]

for some family $\{\mu_{j}>0:j\in J\}$, where $\mu_{j}$ satisfy

[TABLE]

In particular, $\sum_{j\in J}\nu_{j}^{p/q}<+\infty$.

Proof of i). By the conditions of the sequence $\{f_{m}\}\subset L^{p}(M^{n})$, we know from the Remark 2.4 that

[TABLE]

where $r\in[1,q)$. Then, Brézis-Lieb Lemma leads that

[TABLE]

So, it is sufficient to discuss the case $f\equiv 0$. By the classical argument of Lions (see [24, 25]), it is sufficient to prove

[TABLE]

Since, for any $\varphi(x)\in C_{0}^{\infty}(M^{n})$,

[TABLE]

then we get as $m\rightarrow+\infty$ that

[TABLE]

So, we can obtain (3.4) if

[TABLE]

Notice that

[TABLE]

and

[TABLE]

where $r\leq+\infty$ if $\alpha+1-n\geq 0$ and $r<\frac{n}{n-\alpha-1}$ if $\alpha+1-n<0$. If $\alpha+1-n\geq 0$, we can prove (3.5) by dominated convergence theorem. While for the case $\alpha+1-n<0$, we obtain through the Hardy-Littlewood-Sobolev inequalities (1.3) that

[TABLE]

where $s=(\frac{1}{p}-\frac{\alpha+1}{n})^{-1}>q$. Furthermore, repeating the proof of Proposition 2.3, we have

[TABLE]

So, we get (3.5) and complete the proof of i). $\Box$

Proof of ii). Since

[TABLE]

then, $\mu\geq|f|^{p}dV_{x}$. So, we just have to show that for each fixed $j\in J$,

[TABLE]

For point $P_{j}\in M^{n}$, choose a neighbourhood $\Omega_{P_{j}}\subset M^{n}$ so that for $\delta>0$ small enough, in a normal coordinate, $\exp(B_{\delta})\subset\Omega_{P_{j}}$ and

[TABLE]

Take $\varphi_{\lambda}(x)=\varphi(\frac{x}{\lambda})$, where $\varphi(x)\in C_{0}^{\infty}(\mathbb{R}^{n})$ satisfies $0\leq\varphi(x)\leq 1,\ \varphi(0)=1,\ \text{supp}\ \varphi\subset B_{1}$ and $\lambda\in(0,\delta)$. Then,

[TABLE]

and

[TABLE]

So,

[TABLE]

where

[TABLE]

Repeating the argument of (3.5), we have, as $m\rightarrow+\infty$,

[TABLE]

So, letting $m\rightarrow+\infty$ leads

[TABLE]

Since

[TABLE]

then we can complete the proof by letting $\lambda\rightarrow 0^{+}$ and $\epsilon\rightarrow 0^{+}$. $\Box$

4 Estimate and criteria of existence

Proof of Proposition 1.2. For small positive constant $\lambda>0$, recall that $f_{\lambda}(x)$ and $g_{\lambda}(y)$ are given in (2.3). Take

[TABLE]

where $\delta>0$ is a fixed constant to be determined later. Then, for small enough $\lambda$ and by (2.2),

[TABLE]

where, for fixed $\delta>0$ and as $\lambda\rightarrow 0^{+}$,

[TABLE]

So, for small enough $\lambda$,

[TABLE]

For any given point $P\in M^{n}$, choose a neighbourhood $\Omega_{P}\subset M^{n}$ so that for $\delta>0$ small enough, in a normal coordinate, $\exp(B_{\delta})\subset\Omega_{P}$ and

[TABLE]

Thus,

[TABLE]

In the normal coordinates with respect to the center $P\in M^{n}$, let

[TABLE]

Then

[TABLE]

Thus

[TABLE]

Sending $\epsilon$ and $\lambda$ to [math], we obtain the estimate. $\Box$

Prof of Theorem 1.3. Take a maximizing nonnegative sequence $\{f_{m}(x)\}\subset L^{p}(M^{n})$ satisfying $\int_{M^{n}}f_{m}^{p}dV_{x}=1$ and

[TABLE]

Then, there exist a subsequence of $\{f_{m}\}$ (still denoted by $\{f_{m}\}$) and some function $f\in L^{p}(M^{n})$ such that

[TABLE]

Because of the Hardy-Littlewood-Sobolev inequalities (1.3), we know that

[TABLE]

are two families of bounded measures. So, there exist two nonnegative bounded measures $\mu$ and $\nu$ on $M^{n}$ such that

[TABLE]

weakly in the sense of measures.

Applying the Concentration-Compactness Lemma (see Lemma 3.1), we have

[TABLE]

and $\nu_{j}^{1/q}\leq N_{p,\alpha}\mu_{j}^{1/p}$ for all $j\in J$. Since $\int_{M^{n}}d\mu=\lim_{m\rightarrow+\infty}\int_{M^{n}}|f_{m}|^{p}dV_{x}=1,$ then $\int_{M^{n}}|f|^{p}dV_{x}\leq 1$ and $\mu_{j}\leq 1,\ j\in J$.

We claim that $\mu_{j}=0,\ j\in J$, which deduce that $\nu_{j}=0,\ j\in J$.

In fact, otherwise, combining (4.6) and the fact $\frac{q}{p}>1$, we have

[TABLE]

which is a contradiction.

Repeating the process of (4), we have that

[TABLE]

i.e., $f$ is a maximizer. $\Box$