Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Reverse Stein-Weiss inequalities and existence of their extremal functions


Authors: Lu Chen, Zhao Liu, Guozhen Lu and Chunxia Tao
Journal: Trans. Amer. Math. Soc. 370 (2018), 8429-8450
MSC (2010): Primary 42B99, 35B40
DOI: https://doi.org/10.1090/tran/7273
Published electronically: August 21, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in $ \mathbb{R}^n$:

$\displaystyle \int _{\mathbb{R}^n}\int _{\mathbb{R}^n}\vert x\vert^\alpha \vert... ...eta dxdy\geq C_{n,\alpha ,\beta ,p,q'}\Vert f\Vert _{L^{q'}}\Vert g\Vert _{L^p}$    

for any nonnegative functions $ f\in L^{q'}(\mathbb{R}^n)$, $ g\in L^p(\mathbb{R}^n)$, and $ p,\ q'\in (0,1)$, $ \alpha $, $ \beta $, $ \lambda >0$ such that $ \frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta +\lambda }{n}=2$. We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the $ n$-dimensional sphere $ \mathbb{S}^n$ by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42B99, 35B40

Retrieve articles in all journals with MSC (2010): 42B99, 35B40


Additional Information

Lu Chen
Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
Address at time of publication: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
Email: luchen2015@mail.bnu.edu.cn

Zhao Liu
Affiliation: School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, People’s Republic of China
Email: liuzhao2008tj@sina.com

Guozhen Lu
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: guozhen.lu@uconn.edu

Chunxia Tao
Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
Email: taochunxia@mail.bnu.edu.cn

DOI: https://doi.org/10.1090/tran/7273
Keywords: Reverse Stein--Weiss inequality, asymptotic behavior, existence of extremal functions, Pohozaev identity, reverse Hardy--Littlewood--Sobolev inequality
Received by editor(s): November 11, 2016
Received by editor(s) in revised form: March 25, 2017
Published electronically: August 21, 2018
Additional Notes: The first two authors and the fourth author were partly supported by a grant from the NNSF of China (No.11371056).
The third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation.
The third and fourth authors are the corresponding authors.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society