Boundary Behavior of Non-Negative Solutions of the Heat Equation in Sub-Riemannian Spaces

TL;DR

This paper establishes boundary behavior theorems for non-negative solutions of the heat equation in sub-Riemannian spaces, including measure doubling, Dahlberg estimates, and Harnack inequalities.

Contribution

It introduces new boundary regularity results and estimates for heat equation solutions in sub-Riemannian geometries, extending classical theories to these complex spaces.

Findings

Fatou type theorems proved for sub-Riemannian heat solutions

Doubling property of L-caloric measure established

Backward Harnack inequality for solutions vanishing at boundary

Abstract

We prove Fatou type theorems for solutions of the heat equation in sub- Riemannian spaces. The doubling property of L-caloric measure, the Dahlberg estimate, the local comparison theorem, among other results, are established here. A backward Harnack inequality is proved for non-negative solutions vanishing in the lateral boundary.