$L^1$ Hardy inequalities with weights

TL;DR

This paper establishes sharp weighted Hardy inequalities in L^1 spaces with boundary distance, providing precise estimates for the best constants and improvements in special domains like balls and strips.

Contribution

It introduces sharp homogeneous improvements to L^1 weighted Hardy inequalities involving boundary distance, with exact constants for specific domains.

Findings

Sharp estimates for the best constants in Hardy inequalities.

Exact results for domains like balls and strips.

Additional improvements for the case of a ball.

Abstract

We prove sharp homogeneous improvements to $L^1$ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These estimates are sharp in the sense that they coincide when the domain is a ball or an infinite strip. In the case of a ball, we also obtain further improvements.