Reversed Hardy-Littewood-Sobolev inequality

Abstract

The classical sharp Hardy-Littlewood-Sobolev inequality states that, for $1<p, t<\infty$ and $0<\lambda=n-\alpha <n$ with $ 1/p +1 /t+ \lambda /n=2$, there is a best constant $N(n,\lambda,p)>0$, such that $$ |\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(x)|x-y|^{-\lambda} g(y) dx dy|\le N(n,\lambda,p)||f||_{L^p(\mathbb{R}^n)}||g||_{L^t(\mathbb{R}^n)} $$ holds for all $f\in L^p(\mathbb{R}^n), g\in L^t(\mathbb{R}^n).$ The sharp form is due to Lieb, who proved the existence of the extremal functions to the inequality with sharp constant, and computed the best constant in the case of $p=t$ (or one of them is 2). Except that the case for $p\in ((n-1)/n, n/\alpha)$ (thus $\alpha$ may be greater than $n$) was considered by Stein and Weiss in 1960, there is no other result for $\alpha>n$. In this paper, we prove that the reversed Hardy-Littlewood-Sobolev inequality for $0<p, t<1$, $\lambda<0$ holds for all nonnegative $f\in L^p(\mathbb{R}^n), g\in L^t(\mathbb{R}^n).$ For $p=t$, the existence of extremal functions is proved, all extremal functions are classified via the method of moving sphere, and the best constant is computed.