Sharp estimates for $W$-invariant Dunkl and heat kernels in the $A_n$ case

TL;DR

This paper provides precise estimates for the $W$-invariant Dunkl and heat kernels associated with the root system of type $A$, and applies these to analyze related potential kernels and semigroups.

Contribution

It delivers exact bounds for Dunkl and heat kernels in the $A_n$ case with arbitrary multiplicities, advancing understanding of their behavior and applications.

Findings

Exact estimates for $W$-invariant Dunkl kernel

Sharp bounds for Dunkl heat kernel

Applications to Newton kernel and fractional Dunkl Laplacian semigroups

Abstract

In this article, we prove exact estimates for the $W$-invariant Dunkl kernel and heat kernel, for the root system of type $A$ with arbitrary positive multiplicities. We apply the estimates of the $W$-invariant Dunkl heat kernel to compute sharp estimates for the Newton kernel and for the $s$-stable semigroups generated by a fractional power of the $W$-invariant Dunkl Laplacian.