Contribution to Book
Perspectives in Partial Differential Equations, Harmonic Analysis and Applications
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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First published in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications in Vol. 79, 2008, published by the American Mathematical Society.
Copyright 2008 by the American Mathematical Society
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
Capogna, L., Garofalo, N., & Nhieu, D.-M. (2008). Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators. In D. Mitrea & M. Mitrea (Eds.), Perspectives in Partial Differential Equations, Harmonic Analysis and Applications (49-100). Providence: American Mathematical Society.