Sharp reversed Hardy–Littlewood–Sobolev inequality on R n

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 R n, \(\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}\), for any non-negative functions fL p(R n), gL r(R n), 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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Correspondence to Quốc Anh Ngô.

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Dedicated to Professor Hoàng Quốc Toàn on the occasion of his 70th birthday

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Ngô, Q.A., Nguyen, V.H. Sharp reversed Hardy–Littlewood–Sobolev inequality on R n . Isr. J. Math. 220, 189–223 (2017). https://doi.org/10.1007/s11856-017-1515-x

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