Properties of harmonic measures in the Dirichlet problem for nilpotent lie groups of Heisenberg type

Luca Capogna, Nicola Garofalo, Duy Minh Nhieu

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

In groups of Heisenberg type we introduce a large class of domains, which we call ADP, admissible for the Dirichlet problem, and we prove that on the boundary of such domains, harmonic measure, ordinary surface measure, and the perimeter measure, are mutually absolutely continuous. We also establish the solvability of the Dirichlet problem when the boundary datum belongs to LP, 1 < p ≤ ∞, with respect to the ordinary surface measure. Here, the harmonic measure is that relative to a sub-Laplacian associated with a basis of the first layer of the Lie algebra. A domain is called ADP if it is a nontangentially accessible domain and it satisfies an intrinsic outer ball condition.

Original languageEnglish
Pages (from-to)273-306
Number of pages34
JournalAmerican Journal of Mathematics
Volume124
Issue number2
DOIs
StatePublished - 2002

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