Heat kernel asymptotics for real powers of Laplacians

Abstract

A BSTRACT . We describe the small-time heat kernel asymptotics of real powers ∆ r , r ∈ (0 , 1) of a non-negative self-adjoint generalized Laplacian ∆ acting on the sections of a hermitian vector bundle E over a closed oriented manifold M . First we treat separately the asymptotic on the diagonal of M × 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 ∆ 1 / 2 is a polyhomogeneous conormal section in E ⊠ E ∗ on the standard blow-up space M heat of the diagonal at time t = 0 inside [0 , ∞ ) × M × M .