Estimates of the Green function and the initial-Dirichlet problem for the heat equation in sub-Riemannian spaces

Abstract

In a cylinder $D_T = \Omega \times (0,T)$, where $\Omega\subset \mathbb{R}^n$, we examine the relation between the $L$-caloric measure, $d\omega^{(x,t)}$, where $L$ is the heat operator associated with a system of vector fields of H\"ormander type, and the measure $d\sigma_X\times dt$, where $d\sigma_X$ is the intrinsic $X$-perimeter measure. The latter constitutes the appropriate replacement for the standard surface measure on the boundary and plays a central role in sub-Riemannian geometric measure theory. Under suitable assumptions on the domain $\Omega$ we establish the mutual absolute continuity of $d\omega^{(x,t)}$ and $d\sigma_X\times dt$. We also derive the solvability of the initial-Dirichlet problem for $L$ with boundary data in appropriate $ L^p$ spaces, for every $p>1$.