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Volume 19, Issue 3

# Reverse Stein–Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

Lu Chen
/ Guozhen Lu
/ Chunxia Tao
Published Online: 2019-01-22 | DOI: https://doi.org/10.1515/ans-2018-2038

## Abstract

The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space:

${\int }_{{ℝ}_{+}^{n}}{\int }_{\partial {ℝ}_{+}^{n}}|x{|}^{\alpha }|x-y{|}^{\lambda }f\left(x\right)g\left(y\right)|y{|}^{\beta }dydx\ge {C}_{n,\alpha ,\beta ,p,{q}^{\prime }}\parallel f{\parallel }_{{L}^{{q}^{\prime }}\left({ℝ}_{+}^{n}\right)}\parallel g{\parallel }_{{L}^{p}\left(\partial {ℝ}_{+}^{n}\right)}$

for any nonnegative functions $f\in {L}^{{q}^{\prime }}\left({ℝ}_{+}^{n}\right)$, $g\in {L}^{p}\left(\partial {ℝ}_{+}^{n}\right)$, and $p,{q}^{\prime }\in \left(0,1\right)$, $\beta <\frac{1-n}{{p}^{\prime }}$ or $\alpha <-\frac{n}{q}$, $\lambda >0$ satisfying

$\frac{n-1}{n}\frac{1}{p}+\frac{1}{{q}^{\prime }}-\frac{\alpha +\beta +\lambda -1}{n}=2.$

Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space ${ℝ}_{+}^{n}$.

MSC 2010: 42B37; 42B35; 35B40; 45G15

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Revised: 2018-12-04

Accepted: 2018-12-05

Published Online: 2019-01-22

Published in Print: 2019-08-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11371056

The first and third authors were partly supported by grant from the NNSF of China (No.11371056), the second author was partly supported by a US NSF grant and a grant from the Simons foundation.

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 3, Pages 475–494, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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