Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions

Abstract

In a cylinder ${\Omega }_T=\Omega \times (0,T)\subset {\mathbb{R}}_{+}^{n+1}$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form \[ Hu =\sum_{i,j=1}^ma_{ij}(x,t) X_iX_ju - \p_tu = 0, \ (x,t)\in\R^{n+1}_+, \] where $X=\{X_1,...,X_m\}$ is a system of $C^{\infty }$ vector fields in $\Rn$ satisfying Hörmander's finite rank condition \eqref{frc}, and $\Omega$ is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance $d$ induced by $X$. Concerning the matrix-valued function $A=\{a_{ij}\}$, we assume that it be real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries $a_{ij}$ be Hölder continuous with respect to the parabolic distance associated with $d$. Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem \ref{T:back}); 2) the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem \ref{T:quotients}); 3) the doubling property for the parabolic measure associated with the operator $H$ (Theorem \ref{T:doubling}). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [FSY] and [SY]. With one proviso: in those papers the authors assume that the coefficients $a_{ij}$ be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.