Heat kernel asymptotics for real powers of Laplacians

Abstract

We describe the small-time heat kernel asymptotics of real powers Δ 𝑟 , 𝑟 ∈ (0, 1) of a non-negative self-adjoint generalized Laplacian Δ 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 Δ 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 .