We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of $m$-Laplace type operators $L$ on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns turns out to be given by the local zeta function of $L$. In particular, the constant term in the asymptotic expansion of the Greens function $L^{-1}$ is often called the mass of $L$, which (in case that $L$ is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant $m$-Laplace operators $L$ (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of $M$ is odd and that $ker L = 0$, and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.
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