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 $fin L^p(mathbb{R}^n), gin L^t(mathbb{R}^n).$ 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 $pin ((n-1)/n, n/alpha)$ (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 $fin L^p(mathbb{R}^n), gin L^t(mathbb{R}^n).$ 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.
تحميل البحث