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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 thePoincar\'e--Sobolev inequalities, with sharp ranges of indices, onChoquet--Lorentz integrals with respect to $\mathcal{H}_{\infty }^{\delta}$ forall continuously differentiable functions on $\Omega$. These results not onlyextend the recent Poincar\'e and Poincar\'e--Sobolev inequalities to theChoquet--Lorentz integrals, but also provide some endpoint estimates (weaktype) in the critical case. One of the main novelties exists in that, toachieve the goals, the authors develop some new tools associated withChoquet--Lorentz integrals on $\mathcal{H}_{\infty }^{\delta}$, such as thefractional Hardy--Littlewood maximal inequality and the Hedberg-type pointwiseestimate on the Riesz potential. As an application, the authors obtain thesharp boundedness of the Riesz potential on Choquet--Lorentz integrals.Moreover, even for classical Lorentz integrals, these Poincar\'e andPoincar\'e--Sobolev inequalities are also new.

Sharp Poincar\'e--Sobolev Inequalities of Choquet--Lorentz Integrals with Respect to Hausdorff Contents on Bounded John Domains