Sharp reversed Hardy–Littlewood–Sobolev inequality on Rn

By Q. Ngô, Van Hoang Nguyen. 2015

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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, $$\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}$$∫Rn∫Rnf(x)|x−y|λg(y)dxdy≥ℓn,p,r‖f‖Lp(Rn)‖g‖Lr(Rn), for any non-negative functions f ∈ Lp(Rn), g ∈ Lr(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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/8a6a9cbfa1e9908f4e724c314aef5eb3c10f7906