$$p$$ -Harmonic functions in the Heisenberg group: boundary behaviour in domains well-approximated by non-characteristic hyperplanes

In this paper, we give measure estimates of nodal sets of ℍ-harmonic functions on the Heisenberg group ℍn. We also introduce a definition of horizontal singular sets and show the geometric structure of the horizontal singular sets of ℍ-harmonic functions.

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r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

Annals of Global Analysis and Geometry ◽

10.1007/s10455-020-09736-3 ◽

2020 ◽

Vol 58 (4) ◽

pp. 477-496

Author(s):

Sigmundur Gudmundsson ◽

Marko Sobak

Keyword(s):

Heisenberg Group ◽

Lie Groups ◽

Lie Group ◽

Riemannian Manifolds ◽

Harmonic Functions ◽

Semidirect Products ◽

Simply Connected ◽

Isoparametric Functions ◽

General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

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Boundary Value Problems for Harmonic Functions on the Heisenberg Group

Canadian Journal of Mathematics ◽

10.4153/cjm-1986-024-9 ◽

1986 ◽

Vol 38 (2) ◽

pp. 478-512 ◽

Cited By ~ 3

Author(s):

Charles F. Dunkl

Keyword(s):

Heisenberg Group ◽

Harmonic Functions ◽

Poisson Kernel ◽

Differential Operators ◽

Special Functions ◽

Continuous Functions ◽

Group Representations ◽

Hypoelliptic Operators ◽

Analysis Group ◽

Partial Differential Operators

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 L0 (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 L0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L0.

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A boundary estimate for harmonic functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100059132 ◽

1982 ◽

Vol 91 (1) ◽

pp. 79-90 ◽

Cited By ~ 3

Author(s):

P. J. Rippon

Keyword(s):

Euclidean Space ◽

Harmonic Functions ◽

Conformal Mappings ◽

Dimensional Euclidean Space ◽

Boundary Behaviour ◽

Boundary Estimate ◽

Distortion Theorems

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.

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