Relative Nash-type and $L^2$-Sobolev inequalities for Dunkl operators and applications

TL;DR

This paper develops local Nash inequalities for Dunkl operators, establishing Pseudo-Poincaré inequalities via heat kernel gradient estimates, leading to mean value inequalities for Dunkl heat equation subsolutions.

Contribution

It introduces local Nash inequalities for Dunkl operators and derives mean value inequalities, advancing analysis in Dunkl operator theory.

Findings

Established Pseudo-Poincaré inequalities using Dunkl heat kernel estimates

Derived relative Nash-type inequalities for Dunkl operators

Obtained mean value inequalities for Dunkl heat equation subsolutions

Abstract

We investigate local variants of Nash inequalities in the context of Dunkl operators. Pseudo-Poincar\'e inequalities are first established using pointwise gradient estimates of the Dunkl heat kernel. These inequalities allow to obtain relative Nash-type inequalities which are used to derive mean value inequalities for subsolutions of the heat equation on orbits of balls not necessarily centered on the origin.