Sharp reversed Hardy-Littlewood-Sobolev inequality on $\mathbb R^n$

TL;DR

This paper establishes a sharp reversed Hardy-Littlewood-Sobolev inequality on ^n for nonnegative functions in certain Lebesgue spaces, providing explicit constants, estimates, and existence results for optimal functions.

Contribution

It introduces the first sharp reversed HLS inequality on ^n with explicit constants and investigates the existence of extremal functions, extending classical HLS results.

Findings

Proved the sharp reversed HLS inequality for p,r in (0,1)

Derived estimates for the optimal constant _{n,p,r}

Demonstrated existence of extremal functions for the inequality

Abstract

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbb R^n$ \[\int_{\mathbb R^n} \int_{\mathbb R^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p (\mathbb R^n)}\, \|g\|_{L^r (\mathbb R^n)}\] for any nonnegative functions $f\in L^p(\mathbb R^n)$, $g\in L^r(\mathbb R^n)$, and $p,r\in (0,1)$, $\lambda > 0$ such that $1/p + 1/r -\lambda /n =2$. We will also explore some estimates for $\mathscr C_{n,p,r}$ and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.