Reverse Stein–Weiss inequalities and existence of their extremal functions

Abstract:

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 $p,\ q’\in (0,1)$, $\alpha$, $\beta$, $\lambda >0$ such that $\frac {1}{p}+\frac {1}{q’}-\frac {\alpha +\beta +\lambda }{n}=2$. We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler–Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein–Weiss and reverse Stein–Weiss inequalities on the $n$-dimensional sphere $\mathbb {S}^n$ by using the stereographic projections. Our proof of the reverse Stein–Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy–Littlewood–Sobolev inequalities.