*Published Paper*

**Inserted:** 5 may 2020

**Last Updated:** 29 mar 2021

**Journal:** J. Math. Pures Appl.

**Volume:** 148

**Pages:** 267-307

**Year:** 2021

**Doi:** 10.1016/j.matpur.2020.12.004

**Abstract:**

We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series: \[ Q_\Omega(t) = \sum_{k=0}^{\infty} a_k t^{k/2}, \qquad \text{as } t\to 0. \] We compute explicitly the coefficients up to order $k=5$, in terms of sub-Riemannian invariants of the domain and its boundary. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension. As a particular case we recover, using non-probabilistic techniques, the order $2$ formula due to Tyson and Wang in the first Heisenberg group J. Tyson, J. Wang, Comm. PDE, 2018. An intriguing byproduct of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points. In particular, we prove that the higher order coefficients in the expansion can blow-up in their presence. A key tool for this last result is an exact formula for the sub-Riemannian distance from a surface with an isolated characteristic point in the first Heisenberg group, which is of independent interest.