On sharp heat and subordinated kernel estimates in the Fourier-Bessel setting

TL;DR

This paper establishes sharp heat kernel bounds and subordinated kernel estimates in Fourier-Bessel settings for half-integer parameters, with conjectures extending results to general parameters and implications for heat semigroup maximal operators.

Contribution

It provides the first sharp heat kernel bounds for Fourier-Bessel expansions at half-integer parameters and extends estimates to subordinated kernels, proposing conjectures for all parameters.

Findings

Sharp heat kernel bounds for half-integer f u

Sharp estimates of subordinated kernels for half-integer f u

Implications for heat semigroup maximal operators

Abstract

We prove qualitatively sharp heat kernel bounds in the setting of Fourier-Bessel expansions when the associated type parameter $\nu$ is half-integer. Moreover, still for half-integer $\nu$, we also obtain sharp estimates of all kernels subordinated to the heat kernel. Analogous estimates for general $\nu > -1$ are conjectured. Some consequences concerning the related heat semigroup maximal operator are discussed.