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 $$\mathrm{\parallel }{E}_{\alpha }f{\mathrm{\parallel }}_{L^{t^{\mathrm{\prime }}}({\mathbb{R}}^n)}\ge \mathcal{C}(n,\alpha ,q,t)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1({\mathbb{R}}^n)}^{\gamma }\mathrm{\parallel }f{\mathrm{\parallel }}_{L^q({\mathbb{R}}^n)}^{1-\gamma },\gamma :=\frac{n-q\alpha -\frac{n}{t^{\mathrm{\prime }}}q}{n(1-q)}$$ for some $\mathcal{C}(n,\alpha ,q,t)>0$ iff $q>\frac{n}{\alpha }$, where $E_{\alpha }$ is the extension operator with Riesz kernel and $t^{\mathrm{\prime }}$ is the conjugate of $t$. The sharp constant is achieved when $\frac{n{t}^{\mathrm{\prime }}}{n+\alpha t^{\mathrm{\prime }}}\le q<1$. On ${\mathbb{R}}_{+}^n$, with $2\le n<\alpha$, $\frac{n}{\alpha }<t<1$, and $0<q<1$, we show for nonnegative $f$ that $$\mathrm{\parallel }{\tilde{E}}_{\alpha }f{\mathrm{\parallel }}_{L^{t^{\mathrm{\prime }}}({\mathbb{R}}_{+}^n)}\ge \tilde{\mathcal{C}}(n,\alpha ,q,t)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(\mathrm{\partial }{\mathbb{R}}_{+}^n)}^{\tilde{\gamma }}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^q(\mathrm{\partial }{\mathbb{R}}_{+}^n)}^{1-\tilde{\gamma }},\tilde{\gamma }:=\frac{(n-1)-q(\alpha -1)-\frac{n}{t^{\mathrm{\prime }}}q}{(n-1)(1-q)},$$ for some $\tilde{\mathcal{C}}(n,\alpha ,q,t)>0$ iff $q>\frac{n-1}{\alpha -1}$, where ${\tilde{E}}_{\alpha }$ is the extension operator with Poisson-type kernel. The sharp constant is achieved when $\frac{t^{\mathrm{\prime }}(n-1)}{n+t^{\mathrm{\prime }}(\alpha -1)}\le q<1$. We further extend results to $q\ge 1$. 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 ${\mathbb{R}}_{+}^n$.