紧黎曼流形上Hardy-Littlewood-Sobolev不等式的极值问题:次临界逼近法

摘要/Abstract

摘要：

令（Mⁿ，g）为n维无边紧黎曼流形， $0<\alpha<n,\ q>\frac{n}{n-\alpha}$，该文研究了下列Hardy-Littlewood-Sobolev（HLS）不等式

$\|I_\alpha f\|_{L^q(M^n)}\leq C\|f\|_{L^p(M^n)},\quad I_\alpha f(x)= \int_{M^n}\frac{f(y)}{|x-y|_g^{n-\alpha}}{\rm d}V_y,\ p\geq\frac{nq}{n+\alpha q} $

的极值问题.首先,利用算子$I_\alpha: L^p(M^n)\rightarrow L^q(M^n)$在次临界情形(即$p>\frac{nq}{n+\alpha q}$)时的紧致性,证明$p>\frac{nq}{n+\alpha q}$时极值函数$f_p\in L^p(M^n)$的存在性;进而证明函数列$\{f_p\}$为临界情形时HLS不等式的最佳常数的极值列;最后,结合极值列$\{f_p\}$在$L^{\frac{nq}{n+\alpha q}}(M^n)$中的一致有界性,利用文献[32]建立的集中列紧原理证明$\{f_p\}$在$L^{\frac{nq}{n+\alpha q}}(M^n)$中存在收敛子列,从而给出临界情形(即$p=\frac{nq}{n+\alpha q}$)时极值函数的存在性.

关键词: Hardy-Littlewood-Sobolev不等式, 紧黎曼流形, 极值问题

Abstract:

Let $(M^n,g)$ be a $n$-dimensional compact Riemannian manifolds, $0<\alpha<n$ and $q>\frac{n}{n-\alpha}$. This paper is mainly devoted to study the extremal problems of the following HLS inequalities:

$\|I_\alpha f\|_{L^q(M^n)}\leq C\|f\|_{L^p(M^n)},\quad\mbox{with}\quad I_\alpha f(x)=\int_{M^n}\frac{f(y)}{|x-y|_g^{n-\alpha}}{\rm d}V_y,\ p\geq\frac{nq}{n+\alpha q}.$

Firstly, we prove that $I_\alpha: L^p(M^n)\rightarrow L^q(M^n)$ with $p>\frac{nq}{n+\alpha q}$ is compact and then get the existence of extremal functions $f_p, p>\frac{nq}{n+\alpha q}$. Secondly, we find that the function sequence $\{f_p\}$ is a maximizing sequence for the sharp constant of HLS inequality with $p=\frac{nq}{n+\alpha q}$. Finally, by the Concentration-Compactness principle established in [32], we can prove that there exists a convergence subsequence of $\{f_p\}$ and then give the existence of extremal function for critical case.

Key words: Hardy-Littlewood-Sobolev intqualities, Compact Riemannian manifolds, Extremal problems

中图分类号:

O175.2

张书陶,韩亚洲. 紧黎曼流形上Hardy-Littlewood-Sobolev不等式的极值问题:次临界逼近法[J]. 数学物理学报, 2020, 40(1): 63-71.

Shutao Zhang,Yazhou Han. Extremal Problems of Hardy-Littlewood-Sobolev Inequalities on Compact Riemannian Manifolds: the Approximation Method from Subcritical to Critical[J]. Acta mathematica scientia,Series A, 2020, 40(1): 63-71.

参考文献 33

1	Brendle S . Global existence and convergence for a higher order flow in conformal geometry. Ann of Math, 2003, 158 (1): 323- 343
2	Brezis H , Lieb E . A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88, 486- 490 doi: 10.1090/S0002-9939-1983-0699419-3
3	Chang S Y , Gursky M , Yang P C . An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann of Math, 2002, 155 (3): 709- 787 doi: 10.2307/3062131
4	Chen W, Li C. Methods on Nonlinear Elliptic Equations. Missouri: American Institute of Mathematical Sciences, 2010
5	Djadli Z , Malchiodi A . Existence of conformal metrics with constant Q-curvature. Ann of Math, 2008, 168 (3): 813- 858 doi: 10.4007/annals.2008.168.813
6	Dou J , Guo Q , Zhu M . Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space. Advances in Mathematics, 2017, 312, 1- 45 doi: 10.1016/j.aim.2017.03.007
7	Dou J, Guo Q, Zhu M. Negative power nonlinear integral equations on bounded domains. 2019, arxiv: 1904.03878[math.AP]
8	Dou J , Zhu M . Nonlinear integral equations on bounded domains. J Funct Anal, 2019, 277 (1): 111- 134 doi: 10.1016/j.jfa.2018.05.020
9	Dou J , Zhu M . Sharp Hardy-Littlewood-Sobolev inequality on the upper half space. Int Math Res Notices, 2015, 2015, 651- 687 doi: 10.1093/imrn/rnt213
10	Dou J , Zhu M . Reversed Hardy-Littewood-Sobolev inequality. Int Math Res Notices, 2015, 2015, 9696- 9726 doi: 10.1093/imrn/rnu241
11	Frank R L , Lieb E H . Sharp constants in several inequalities on the Heisenberg group. Annals of Mathematics, 2012, 176, 349- 381 doi: 10.4007/annals.2012.176.1.6
12	González M , Mazzeo R , Sire Y . Singular solutions of fractional order conformal Laplacians. J Geom Anal, 2012, 22 (2): 845- 863
13	González M , Qing J . Fractional conformal Laplacians and fractional Yamabe problems. Analysis & PDE, 2013, 6 (7): 1535- 1576
14	Graham C R , Zworski M . Scattering matrix in conformal geometry. Invent Math, 2013, 152 (1): 89- 118
15	Guan P , Lin C S , Wang G . Application of the method of moving planes to conformally invariant equations. Math Z, 2004, 247 (1): 1- 19
16	Gursky M , Viaclovsky J . A fully nonlinear equation on four-manifolds with positive scalar curvature. J Differential Geom, 2003, 63 (1): 131- 154 doi: 10.4310/jdg/1080835660
17	Gursky M , Viaclovsky J . Prescribing symmetric functions of the eigenvalues of the Ricci tensor. Ann of Math, 2007, 166 (2): 475- 531 doi: 10.4007/annals.2007.166.475
18	Han Y , Zhu M . Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications. J Differential Equations, 2016, 260, 1- 25 doi: 10.1016/j.jde.2015.06.032
19	Hang F , Wang X , Yan X . An integral equation in conformal geometry. Ann Inst H Poincaré Analyse Non Linéaire, 2009, 26, 1- 21 doi: 10.1016/j.anihpc.2007.03.006
20	Hang F , Wang X , Yan X . Sharp integral inequalities for Harmonic functions. Comm Pure Appl Math, 2007, 61 (1): 54- 95
21	Jin T , Li Y Y , Xiong J . On a fractional Nirenberg problem, part Ⅰ:blow up analysis and compactness of solutions. J Eur Math Soc, 2014, 16, 1111- 1171 doi: 10.4171/JEMS/456
22	Jin T , Li Y Y , Xiong J . On a fractional Nirenberg problem, part Ⅱ:existence of solutions. Int Math Res Notices, 2015, 2015, 1555- 1589
23	Jin T , Li Y Y , Xiong J . The Nirenberg problem and its generalizations:A unified approach. Math Ann, 2017, 369, 109- 151 doi: 10.1007/s00208-016-1477-z
24	Jin T , Xiong J . A fractional Yamabe flow and some applications. J Reine Angew Math, 2014, 696, 187- 223
25	Lee J M , Parker T H . The Yamabe problem. Bull Amer Math Soc, 1987, 17 (1): 37- 91 doi: 10.1090/S0273-0979-1987-15514-5
26	Lions P L . The concentration-compactness principle in the calculus of variations, the limit case, Part 1. Rev Mat Iberoamericana, 1985, 1 (1): 145- 201
27	Lions P L . The concentration-compactness principle in the calculus of variations, the limit case, Part 2. Rev Mat Iberoamericana, 1985, 1 (2): 45- 121
28	Li A , Li Y Y . On some conformally invariant fully nonlinear equations Ⅱ:Liouville, Harnack and Yamabe. Acta Math, 2005, 195, 117- 154 doi: 10.1007/BF02588052
29	Lieb E . Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann of Math, 1983, 118, 349- 374 doi: 10.2307/2007032
30	Ngô Q A , Nguyen V H . Sharp reversed Hardy-Littlewood-Sobolev inequality on Rⁿ. Israel Journal of Mathematics, 2017, 220, 189- 223 doi: 10.1007/s11856-017-1515-x
31	Ngô Q A , Nguyen V H . Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space R₊ⁿ. International Mathematics Research Notices, 2017, 2017, 6187- 6230
32	Zhang S, Han Y. Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemanniannian manifolds. 2019, arXiv: 1901.02309[math. AP]
33	Zhu M . Prescribing integral curvature equation. Differential and Integral Equations, 2016, 29 (9/10): 889- 904

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