Refined geometric L^p Hardy inequalities

TL;DR

This paper establishes refined Hardy inequalities for convex domains in R^N, incorporating boundary distance, volume, and sharp logarithmic corrections, and discusses their optimal constants.

Contribution

It introduces new refined Hardy inequalities with sharp logarithmic corrections for convex domains, enhancing previous inequalities with additional geometric and correction terms.

Findings

Derived inequalities involving boundary distance and volume

Identified sharp logarithmic correction terms

Analyzed the optimal constants for these inequalities

Abstract

For a bounded convex domain \Omega in R^N we prove refined Hardy inequalities that involve the Hardy potential corresponding to the distance to the boundary of \Omega, the volume of $\Omega$, as well as a finite number of sharp logarithmic corrections. We also discuss the best constant of these inequalities.