On $L^p$-Hardy inequalities for magnetic $p$-Laplacians

TL;DR

This paper investigates the sharp constants in $L^p$-Hardy inequalities involving magnetic $p$-Laplacians, providing integral representations and revisiting remainder terms to deepen understanding of these inequalities.

Contribution

It offers a new integral representation of the sharp constant in $L^p$-Hardy inequalities for magnetic $p$-Laplacians, enhancing previous algebraic results.

Findings

Derived an integral representation of the sharp constant.

Revisited and refined the remainder terms in the inequalities.

Connected algebraic inequalities to integral formulations.

Abstract

In this paper we revisit the remainder terms of $L^p$-Hardy inequalities for magnetic $p$-Laplacians. In particular, we will give an integral representation of the sharp constant for a crucial algebraic inequality established by C. Cazacu, D. Krej\v{c}i\v{r}\'ik, N. Lam, and A. Laptev.