On sharp heat kernel estimates in the context of Fourier-Dini expansions

TL;DR

This paper establishes precise heat kernel estimates for Fourier-Dini expansions on (0,1) with Neumann boundary conditions, leading to sharp bounds and convergence results for related operators.

Contribution

It provides the first sharp heat kernel estimates in the Fourier-Dini context and applies these to various operators and boundary convergence analysis.

Findings

Sharp heat kernel bounds for Fourier-Dini expansions

Optimal estimates for Poisson and potential kernels

Enhanced understanding of boundary convergence of Fourier-Dini semigroup

Abstract

We prove sharp estimates of the heat kernel associated with Fourier-Dini expansions on $(0,1)$ equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier-Dini semigroup.