The Poisson kernel for Heisenberg polynomials on the disk (Q793267)

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 \(\{z\in {\mathbb{C}}:| z|<1\}.\)