# Series expansion of weighted Finsler-Kato-Hardy inequalities

## TL;DR

This paper develops a series expansion method for weighted Finsler-Kato-Hardy inequalities, providing sharp improvements and extensions of classical inequalities in the Finsler geometric setting.

## Contribution

It introduces a unifying approach to interpolate and improve weighted Hardy inequalities using series expansions in the Finsler context, extending Euclidean results.

## Key findings

- Established a sharp interpolation between weighted Hardy inequalities.
- Derived successive sharp improvements with remainder terms.
- Extended results to Finsler cones and bounded domains.

## Abstract

In this work, we consider weighted anisotropic Hardy inequalities and trace Hardy inequalities involving a general Finsler metric. We follow a unifying approach, by establishing first a sharp interpolation between them, extending the corresponding nonweighted version, being established recently by a different approach. Then, passing to bounded domains, we obtain successive sharp improvements by adding remainder terms involving sharp weights and optimal constants, resulting in an infinite series-type improvement. The results extend, into the Finsler context, the earlier known ones within the Euclidean setting. The generalization of our results to cones is also discussed.