Some Reverse Hardy-Littlewood-Sobolev Type Inequalities

Abstract

We establish some sharp reverse Hardy-Littlewood-Sobolev (HLS) type inequalities on \(\mathbb{R}^n\) and \(\mathbb{R}_+^n\). Using an operator representation, we overcome the difficulty that the symmetric double-integral structure is unavailable in the half-space setting. On \(\mathbb{R}^n\), for \(1 \le n < α\), \(\frac{n}α < t < 1\), and \(0 < q < 1\), there holds for nonnegative \(f \) that \[ \|E_αf \|_{L^{t^\prime}(\mathbb{R}^n)} \ge \mathscr{C}(n,α,q,t) \|f \|_{L^1(\mathbb{R}^n)}^γ \|f \|_{L^q(\mathbb{R}^n)}^{1-γ}, \quad γ:= \frac{n - qα- \frac{n}{t^\prime}q}{n(1-q)} \] for some $\mathscr{C}(n,α,q,t)>0$ iff \(q>\frac{n}α\), where \(E_α\) is the extension operator with Riesz kernel and \(t^\prime\) is the conjugate of \(t\). The sharp constant is achieved when \(\frac{n t^\prime}{n + αt^\prime} \le q < 1\). On \(\mathbb{R}_+^n\), with \(2 \le n < α\), \(\frac{n}α < t < 1\), and \(0 < q < 1\), we show for nonnegative \(f \) that \[ \|\widetilde{E}_αf \|_{L^{t^\prime}(\mathbb{R}_+^n)} \ge\widetilde{\mathscr{C}}(n,α,q,t) \|f\|_{L^1(\partial \mathbb{R}_+^n)}^{\widetildeγ} \|f\|_{L^q(\partial \mathbb{R}_+^n)}^{1-\widetildeγ}, \quad \widetildeγ := \frac{(n-1) - q(α-1) - \frac{n}{t^\prime}q}{(n-1)(1-q)}, \] for some $\widetilde{\mathscr{C}}(n,α,q,t)>0$ iff \(q > \frac{n-1}{α-1}\), where \(\widetilde{E}_α\) is the extension operator with Poisson-type kernel. The sharp constant is achieved when \(\frac{t^\prime(n-1)}{n + t^\prime(α-1)} \le q < 1\). We further extend results to \(q\ge1\). The proofs use rearrangement inequalities, the sharp Carlson--Levin inequality, and refined pointwise lower bounds for the Riesz and Poisson-type potentials. Our results unify and extend the classical reverse HLS inequalities, especially on \(\mathbb{R}_+^n\).