On a Hardy–Morrey inequality

Journal article, 2025

Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality [Formula presented] for any open set Ω⊊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.