Heat kernel asymptotics for real powers of Laplacians

TL;DR

This paper investigates the small-time asymptotic behavior of heat kernels for fractional powers of Laplacians on manifolds, revealing new coefficients and the structure of the heat kernel in special cases.

Contribution

It provides detailed asymptotic expansions for real powers of Laplacians, including the appearance of logarithmic terms and the structure of the heat kernel for half-integer powers.

Findings

Logarithmic terms appear only for odd dimensions and rational powers with even denominators.

Coefficients in the asymptotics are non-trivial and some are non-local.

The heat kernel for the half-power case is a polyhomogeneous conormal section on a blow-up space.

Abstract

We describe the small-time heat kernel asymptotics of real powers $\Delta^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $\Delta$ acting on the sections of a hermitian vector bundle $\mathcal E$ over a closed oriented manifold $M$. First we treat separately the asymptotic on the diagonal of $M \times M$ and in a compact set away from it. Logarithmic terms appear only if $n$ is odd and $r$ is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case $r=1/2$, we give a simultaneous formula by proving that the heat kernel of $\Delta^{1/2}$ is a polyhomogeneous conormal section in $\mathcal E \boxtimes {\mathcal E}^* $ on the standard blow-up space $M_{heat}$ of the diagonal at time $t=0$ inside $[0,\infty)\times M \times M$.