Hardy-type inequalities and principle frequency of the $p-$Laplacian

Abstract

We prove a sharp $L^p$ weighted Hardy inequality involving boundary distance $\delta$ for any domain $\Omega ⊊{\mathbb{R}}^n$. The inequality may be improved substantially under the additional assumption that $-\log \delta$ is subharmonic. Applications of these inequalities to the principle frequency of the $p-$Laplacian are given.

Theorem 1.1 is known, see F. G. Avkhadiev, Hardy type inequalities in high dimensions with explicit estimate of constants, Lobachevskii J. Math. 21 (2006), 3-31