# Some functional inequalities on non-reversible Finsler manifolds

## TL;DR

This paper extends fundamental functional inequalities like Poincare, Sobolev, and logarithmic Sobolev inequalities to non-reversible Finsler manifolds using the Bochner inequality and $\Gamma$-calculus, demonstrating sharp estimates in a broader setting.

## Contribution

It establishes the dimensional versions of key inequalities on non-reversible Finsler manifolds, generalizing previous reversible case results using new analytical techniques.

## Key findings

- Dimensional Poincare--Lichnerowicz inequality proved for non-reversible Finsler manifolds.
- Logarithmic Sobolev and Sobolev inequalities established with sharp estimates.
- Results extend known inequalities from reversible to non-reversible Finsler geometries.

## Abstract

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $\Gamma$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti--Mondino in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.