Li--Yau Inequalities for Dunkl Heat Equations

TL;DR

This paper establishes Li--Yau inequalities for Dunkl heat equations, extending classical results to non-local operators associated with reflection groups, with sharp results in specific symmetric cases.

Contribution

It introduces Li--Yau inequalities for Dunkl heat equations, a novel extension of classical inequalities to non-local Dunkl operators linked to reflection groups.

Findings

Derived sharp Li--Yau inequalities for Dunkl heat equations.

Extended classical heat inequality results to non-local Dunkl operators.

Identified cases where inequalities are optimal, such as reflection group isometric to Z_2^d.

Abstract

Motivated by recent works due to Yu--Zhao [J. Geom. Anal. 2020] and Weber--Zacher [arXiv:2012.12974], we study Li--Yau inequalities for the heat equation corresponding to the Dunkl Laplacian, which is a non-local operator parameterized by reflection groups and multiplicity functions. The results are sharp in the particular case when the reflection group is isometric to $\mathbb{Z}_2^d$.