Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators

TL;DR

This paper derives upper and lower bounds for the heat kernel associated with elliptic pseudodifferential operators, extending known results for fractional Laplacians and applying to Dirichlet-to-Neumann operators on closed manifolds.

Contribution

It provides new heat kernel estimates for a broad class of pseudodifferential operators, including fractional powers of Laplacians and Dirichlet-to-Neumann operators, on closed manifolds.

Findings

Established upper and lower bounds for the heat kernel of pseudodifferential operators.

Extended Poissonian bounds to perturbations of fractional Laplacians.

Applied results to the Dirichlet-to-Neumann semigroup.

Abstract

The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.