Sharp heat kernel estimates in the Fourier-Bessel setting for a continuous range of the type parameter

TL;DR

This paper derives sharp estimates for the heat kernel in Fourier-Bessel expansions by connecting it to Jacobi polynomial expansions and transferring known bounds, addressing the challenge of explicit computation.

Contribution

It introduces a novel approach to estimate the Fourier-Bessel heat kernel by leveraging Jacobi polynomial bounds, providing qualitative sharp estimates.

Findings

Established sharp heat kernel estimates in Fourier-Bessel setting

Connected Fourier-Bessel expansions with Jacobi polynomial expansions

Transferred known bounds to obtain new heat kernel estimates

Abstract

The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.