Heat kernel asymptotics for real powers of Laplacians

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