Trace Asymptotics for Subordinate Semigroups

We address a conjecture of D. Applebaum on small time trace asymptotics for subordinate Brownian motion on compact manifolds. In [1], heat trace asymptotics are computed for the semigroup of the square root of the Laplacian (the generator of the Cauchy process) on the n-dimensional torus, SU(2) and SO(3), and a conjecture is made [1, pp. 2493-2494] that such asymptotics should hold for all α-stable processes, 0 < α < 2, on an arbitrary compact Lie group. The purpose of this note is to point out that this is indeed the case and that such results hold for a wide class of subordinations of the Laplacian on compact manifolds. In fact, as we shall see, such asymptotics follow from Weyl’s law (3). We first record some definitions and set some notations. Let M be a compact Riemannian manifold of dimension n and denote its Riemannian measure by μ. We denote the Laplace-Beltrami operator on M by −∆ and denote its eigenvalues by 0 = λ0 < λ1 ≤ λ2 ≤ · · · → ∞. As is well known, −∆ generates a heat semigroup {Pt} on Lμ(M,R) which possesses a heat kernel denoted here by p(t, x, y). Thus Ptf(x) = e −t∆f(x) = ∫