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

In a cylinder $D_T = Ω\times (0,T)$, where $Ω\subset \mathbb{R}^n$, we examine the relation between the $L$-caloric measure, $dω^{(x,t)}$, where $L$ is the heat operator associated with a system of vector fields of Hörmander type, and the measure $dσ_X\times dt$, where $dσ_X$ 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 $Ω$ we establish the mutual absolute continuity of $dω^{(x,t)}$ and $dσ_X\times dt$. We also derive the solvability of the initial-Dirichlet problem for $L$ with boundary data in appropriate $ L^p$ spaces, for every $p>1$.