We study a “div-grad type” sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spectral geometric meaning when Popp's measure is considered. Our proof is probabilistic. In particular, we use Watanabe's distributional Malliavin calculus.
The first-named author is partially supported by JSPS KAKENHI Grant Number 15K04922, and the second-named author is partially supported by JSPS KAKENHI Grant Number 15K04931.
"Heat trace asymptotics on equiregular sub-Riemannian manifolds." J. Math. Soc. Japan 72 (4) 1049 - 1096, October, 2020. https://doi.org/10.2969/jmsj/82348234