The Poisson kernel for Heisenberg polynomials on the disk
type: Item
instance of: scholarly article
title: The Poisson kernel for Heisenberg polynomials on the disk
zbMATH Open document ID: 0538.43005
DOI: 10.1007/BF01174188
author: Charles F. Dunkl
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173510
review text: The study of harmonic polynomials of the Heisenberg groups was begun by \textit{P. Greiner} [Can. Math. Bull. 23, 383-396 (1980; Zbl 0496.22012)] and involves the Heisenberg polynomials \(C_ n^{(\alpha,\beta)}(z)\) defined by \((1-r{\bar z})^{-\alpha} (1-rz)^{- \beta}=\sum^{\infty}_{n=0}C_ n^{(\alpha,\beta)} (z)r^ n\) for \(| rz|<1\), where \(\alpha,\beta>0\) (these are polynomials in z and \({\bar z})\). \textit{B. Gaveau} [Acta Math. 139, 95-153 (1977; Zbl 0366.22010) used existential methods to show that the Dirichlet problem for the subelliptic Laplacian can be solved in the ball \(\{\) (w,t): \(w\in {\mathbb{C}}^ N\), \(t\in {\mathbb{R}}\), \(| w|^ 2+t^ 4\leq 1\}\). This problem is related to that of expanding continuous functions on 0\(\leq \theta \leq \pi\) in series of the form \(\sum^{\infty}_{n=0}a_ nC_ n^{(\alpha,\beta)}(e^{i\theta})\). (An explicit method remains to be found.) In this paper the basis \[ \{C_ n^{(\alpha,\beta)}(e^{i\theta}):\quad n\geq 0\}\quad \cup(2i \sin \theta C_{n-1}^{(\alpha +1,\beta +1)}(e^{i\theta}):\quad n\geq 1\} \] is introduced for functions on the entire circle \(T:=\{e^{i\theta}: -\pi<\theta \leq \pi \}.\) The biorthogonal set in \(L^ 2(T,| \sin \theta |^{\alpha +\beta}d\theta)\) is found and is used to form a Poisson kernel. (\textit{G. Gasper} discovered a related orthogonality structure [Can. J. Math. 33, 1261-1270 (1981; Zbl 0428.42010)]. The kernel is expressed using a single sum (indeed, an \({}_ 2F_ 1\)- series) by means of the author's addition theorem in [Harmonic analysis, Conf. in Honor A. Zygmund, Chicago 1981, Vol. 2, 690-707 (1983; Zbl 0502.43008]. The resulting Poisson integral transforms each function f in \(L^ 1(T,| \sin \theta |^{\alpha +\beta}d\theta)\) to a smooth function on the disk \(\{z\in {\mathbb{C}}:| z|<1\}.\)
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 3855713
zbMATH Keywords: harmonic polynomials
zbMATH Keywords: Heisenberg groups
zbMATH Keywords: subelliptic Laplacian
zbMATH Keywords: Poisson kernel
MaRDI profile type: Publication
cites work: Q4760447
cites work: Q3967952
cites work: The Tangential Cauchy-Riemann Complex on Spheres
cites work: Estimates for the \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \partial \limits^ - _b $\end{document} complex and analysis on the heisenberg group
cites work: Least action principle, heat propagation and subelliptic estimates on certain nilpotent groups
cites work: Spherical Harmonics on the Heisenberg Group
cites work: Q4125200
cites work: The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I
cites work: Kelvin transforms and harmonic polynomials on the Heisenberg group
cites work: Classical Expansions and Their Relation to Conjugate Harmonic Functions
description: scientific article; zbMATH DE number 3855713