Hᵖ spaces of separately (α, β)-harmonic functions in the unit polydisc

We prove existence and uniqueness of a solution of the Dirichlet problem for separately (α, β)-harmonic functions on Dⁿ with boundary data in C( Tⁿ) using (α, β)-Poisson kernel Pα, β (z, ζ). A characterization by hypergeometric functions of separately (α, β)-harmonic functions which are also m-homogeneous is given, it is used to obtain series expansion of separately (α, β)-harmonic functions. Basic Hᵖ theory of such functions is developed: integral representations by measures and Lᵖ functions on Tⁿ, norm and weak^ convergence at the distinguished boundary Tⁿ. Weak $(1,1)$-type estimate for a restricted non-tangential maximal function MA, BNT is derived. We show that slice functions u(z₁, …, zₖ, ζₖ₊₁, …, ζₙ), where some of the variables are fixed, belong in the appropriate space of separately (α', β')-harmonic 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 Tⁿ. Our results extend earlier results for (α, β)-harmonic functions in the disc and for n-harmonic functions in Dⁿ. Bibliography: 23 titles.