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 R n : R n R n |x| α |x − y| λ f (x)g(y)|y| β dxdy ≥ C n,α,β,p,q f L q g L p for any nonnegative functions f ∈ L q (R n), g ∈ L p (R n), and p, q ∈ (0, 1), α, β, λ > 0 such that 1 p + 1 q − α+β+λ 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 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.