Israel Journal of Mathematics

, Volume 220, Issue 1, pp 189–223 | Cite as

Sharp reversed Hardy–Littlewood–Sobolev inequality on Rn

Article

Abstract

This is the first in our series of papers that concerns Hardy–Littlewood–Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space Rn, \(\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}\), for any non-negative functions fLp(Rn), gLr(Rn), and p, r ∈ (0, 1), λ > 0 such that 1/p+1/r −λ/n = 2. We will also explore some estimates for ℓn,p,r and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsCollege of Science Viêt Nam National UniversityHà NôiViet Nam
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse cédex 09France

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