Reverse Stein–Weiss inequalities and existence of their extremal functions
Reverse Stein–Weiss inequalities and existence of their extremal functions
In this paper, we establish the following reverse Stein–Weiss inequality, namely the reversed weighted Hardy–Littlewood–Sobolev inequality, in $\mathbb {R}^n$: \begin{equation*} \int _{\mathbb {R}^n}\int _{\mathbb {R}^n}|x|^\alpha |x-y|^\lambda f(x)g(y)|y|^\beta dxdy\geq C_{n,\alpha ,\beta ,p,q'}\|f\|_{L^{q'}}\|g\|_{L^p} \end{equation*} for any nonnegative functions $f\in L^{q'}(\mathbb {R}^n)$, $g\in L^p(\mathbb {R}^n)$, and $p,\ q'\in (0,1)$, $\alpha$, $\beta$, $\lambda >0$ …