Some Reverse Hardy-Littlewood-Sobolev Type Inequalities
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 \(1 \le n < \alpha\), \(\frac{n}{\alpha} < t < 1\), and \(0 < q < 1\), there holds for nonnegative \(f \) that \[ \|E_\alpha f \|_{L^{t^\prime}(\mathbb{R}^n)} \ge \mathscr{C}(n,\alpha,q,t) \|f \|_{L^1(\mathbb{R}^n)}^{\gamma} \|f \|_{L^q(\mathbb{R}^n)}^{1-\gamma}, \quad \gamma := \frac{n - q\alpha - \frac{n}{t^\prime}q}{n(1-q)} \] for some $\mathscr{C}(n,\alpha,q,t)>0$ iff \(q>\frac{n}{\alpha}\), where \(E_\alpha\) is the extension operator with Riesz kernel and \(t^\prime\) is the conjugate of \(t\). The sharp constant is achieved when \(\frac{n t^\prime}{n + \alpha t^\prime} \le q < 1\). On \(\mathbb{R}_+^n\), with \(2 \le n < \alpha\), \(\frac{n}{\alpha} < t < 1\), and \(0 < q < 1\), we show for nonnegative \(f \) that \[ \|\widetilde{E}_\alpha f \|_{L^{t^\prime}(\mathbb{R}_+^n)} \ge\widetilde{\mathscr{C}}(n,\alpha,q,t) \|f\|_{L^1(\partial \mathbb{R}_+^n)}^{\widetilde{\gamma}} \|f\|_{L^q(\partial \mathbb{R}_+^n)}^{1-\widetilde{\gamma}}, \quad \widetilde{\gamma} := \frac{(n-1) - q(\alpha-1) - \frac{n}{t^\prime}q}{(n-1)(1-q)}, \] for some $\widetilde{\mathscr{C}}(n,\alpha,q,t)>0$ iff \(q > \frac{n-1}{\alpha-1}\), where \(\widetilde{E}_\alpha\) is the extension operator with Poisson-type kernel. The sharp constant is achieved when \(\frac{t^\prime(n-1)}{n + t^\prime(\alpha-1)} \le q < 1\). 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 \(\mathbb{R}_+^n\).