Sharp Poincar\'e--Sobolev Inequalities of Choquet--Lorentz Integrals with Respect to Hausdorff Contents on Bounded John Domains
Let $\Omega$ be a bounded John domain in $\mathbb R^n$ with $n\ge 2$, and let $\mathcal{H}_{\infty }^{\delta}$ denote the Hausdorff content of dimension $\delta\in (0,n]$. In this article, the authors prove the Poincar\'e and the Poincar\'e--Sobolev inequalities, with sharp ranges of indices, on Choquet--Lorentz integrals with respect to $\mathcal{H}_{\infty }^{\delta}$ for all continuously differentiable functions on $\Omega$. These results not only extend the recent Poincar\'e and Poincar\'e--Sobolev inequalities to the Choquet--Lorentz integrals, but also provide some endpoint estimates (weak type) in the critical case. One of the main novelties exists in that, to achieve the goals, the authors develop some new tools associated with Choquet--Lorentz integrals on $\mathcal{H}_{\infty }^{\delta}$, such as the fractional Hardy--Littlewood maximal inequality and the Hedberg-type pointwise estimate on the Riesz potential. As an application, the authors obtain the sharp boundedness of the Riesz potential on Choquet--Lorentz integrals. Moreover, even for classical Lorentz integrals, these Poincar\'e and Poincar\'e--Sobolev inequalities are also new.