First order Hardy inequalities revisited

TL;DR

This paper revisits first order Hardy inequalities using simple equalities, enabling quick derivation of known inequalities and providing improved or new estimates across various mathematical contexts.

Contribution

It introduces a unified approach with simple equalities to derive and improve Hardy inequalities in diverse settings.

Findings

Derived many well-known Hardy inequalities with optimal constants

Provided improved estimates for multipolar potential and exponential weights

Extended Hardy inequalities to hyperbolic space, Heisenberg group, and other operators

Abstract

In this paper, we consider the first order Hardy inequalities using simple equalities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new estimates in miscellaneous situations, such as multipolar potential, the exponential weight, hyperbolic space, Heisenberg group, the edge Laplacian, or the Grushin type operator.