A bipolar Hardy inequality on Finsler manifolds

TL;DR

This paper proves a new bipolar Hardy inequality on Finsler manifolds, revealing how geometric properties like reversibility and uniformity constants influence the inequality, extending Euclidean and Riemannian results to the Finsler setting.

Contribution

It introduces a bipolar Hardy inequality on Finsler manifolds, highlighting the dependence on geometric constants and generalizing previous Euclidean and Riemannian inequalities.

Findings

The inequality depends on the reversibility constant $r_F$.

The inequality depends on the uniformity constant $l_F$.

It extends classical Hardy inequalities to Finsler geometry.

Abstract

We establish a bipolar Hardy inequality on complete, not necessarily reversible Finsler manifolds. We show that our result strongly depends on the geometry of the Finsler structure, namely on the reversibility constant $r_F$ and the uniformity constant $l_F$. Our result represents a Finslerian counterpart of the Euclidean multipolar Hardy inequality due to Cazacu and Zuazua (2013) and the Riemannian case considered by Faraci, Farkas and Krist\'aly (2018).