$H^p$ theory of separately $(\alpha, \beta)$-harmonic functions in the unit polydisc

Abstract

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$ - harmonic functions on the unit polydisc $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ - Poisson kernel. A characterization by hypergeometric functions of such functions which are also m - homogeneous is given, this characterization is used to obtain series expansion of these functions. Basic $H^p$ theory of such functions is developed: integral representations by measures and $L^p$ functions on $\mathbb T^n$, norm and weak star convergence at the distinguished boundary $\mathbb T^n$. Weak $(1, 1)$ - type estimate for a restricted non-tangential maximal function is derived. Slice functions $u(z_1, . . . , z_k, \zeta_{k+1}, . . . , \zeta_n)$, where some of the variables are fixed, are shown to belong in the appropriate space of functions of $k$ variables. We prove a Fatou type theorem on a. e. existence of restricted non-tangential limits for these functions and a corresponding result for unrestricted limit at a point in $\mathbb T^n$. Our results extend earlier results for $(\alpha, \beta)$ harmonic functions in the disc and for n - harmonic functions in $\mathbb D^n$.