Heat kernel asymptotics for real powers of Laplacians

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 $\mathcal{E}⊠{\mathcal{E}}^{\ast }$ on the standard blow-up space $M_{heat}$ of the diagonal at time $t=0$ inside $[0,\infty )\times M\times M$.

27 pages