On a Hardy-Morrey inequality

2024/01/11 by Ryan Hynd, Hynd, Ryan, Simon Larson +3
Mathematics · Social Sciences · #26D10 #35P30 #46E35 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #South African History and Culture

paper · pdf · doi:10.48550/arxiv.2401.05781

openalex publication_date 2024/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality λ\biggl‖\fracud_Ω^1-n/p\biggr‖_∞^p≤ ∫_Ω|Du|^p dx for any open set Ω\subsetneq ℝⁿ. This inequality is valid for functions supported in Ω and with λ a positive constant independent of u. The crucial hypothesis is that the exponent p exceeds the dimension n. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of Ω, sharp constants, and the existence of a nontrivial u which saturates the inequality.

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