Boundary Value Problems for Harmonic Functions on the Heisenberg Group

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-paramet...

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Bibliographic Details
Published in:Canadian journal of mathematics Vol. 38; no. 2; pp. 478 - 512
Main Author: Dunkl, Charles F.
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.04.1986
University of Toronto
Subjects:
Exact sciences and technology
Function theory, analysis
Mathematical methods in physics
Physics
Heisenberg group
Boundary value problem
Mathematical analysis
Mathematical operator
Dirichlet problem
Differential operator
Harmonic function
ISSN:0008-414X, 1496-4279
Online Access:Get full text
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Summary:Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ , (γ ∊ C), associated to the Laplacian L 0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ -harmonic). This is the topic of this paper. Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L 0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L 0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L 0.
ISSN:0008-414X
1496-4279
DOI:10.4153/CJM-1986-024-9