Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators

2008/03/06 by Luca Capogna, Nicola Garofalo, Capogna, Luca +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP

paper · pdf · doi:10.48550/arxiv.0803.0784

arxiv created 2008/03/06 · openalex publication_date 2008/03/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the Sub-Laplacian L which consists of sum of squares of smooth vector fields that satisfy Hormander's finite rank condition. We study the Dirichlet problem for this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1) A reversed Holder inequality for the Poisson kernel 2) Harmonic measure (corresponding to L) and surface measure (as well as the H-Perimeter measure) are mutually absolutely continuous 3) A representation (hence solvability of the Dirichlet problem) for solutions to the Dirichlet problem.

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• Boundary regularity for subelliptic equations in the Heisenberg group
• Estimates of the Green function and the initial-Dirichlet problem for the heat equation in sub-Riemannian spaces

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