On Choquet integrals and Poincaré-Sobolev inequalities

2022/03/29 by P. Harjulehto, Harjulehto, P., Ritva Hurri-Syrjänen +1
Mathematics · #26D10) #42B20 #46E35 (31C15 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.15623

openalex publication_date 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content H_∞^δ. In particular, if Ω is a bounded John domain in ℝⁿ, n≥ 2, and 0 <δ≤ n, we prove that the corresponding (δp/(δ-p),p)-Poincaré-Sobolev inequalities hold for all continuously differentiable functions defined on Ω whenever δ/n < p < δ. We prove also that the (p,p)-Poincaré inequality is valid for all p>δ/n.

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