$H^p$ 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 ${\mathbb{D}}^n$.

28 pages