Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results

TL;DR

This paper derives small-time asymptotic expansions for hypoelliptic heat kernels near the diagonal, generalizing previous results and providing tools for analyzing subelliptic Laplacians and related operators.

Contribution

It introduces a purely analytic approach to obtain asymptotics valid in neighborhoods of the diagonal, linking coefficients to nilpotentization of sub-Riemannian structures.

Findings

Asymptotic expansions valid near the diagonal.

Identification of coefficients via nilpotentization.

Results applicable to Weyl laws and boundary behavior.

Abstract

We establish small-time asymptotic expansions for heat kernels of hypoelliptic H\"ormander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by M\'etivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac's principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.