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

We prove existence and uniqueness of a solution of the Dirichlet problem for separately (, ) -harmonic functions on Dⁿ with boundary data in C (Tⁿ) using (, ) -Poisson kernel P, (z, ). A characterization by hypergeometric functions of separately (, ) -harmonic functions which are also m-homogeneous is given, it is used to obtain series expansion of separately (, ) -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 M₀, ₁^NT is derived. We show that slice functions u (z₁, …, zₖ, ₊+₁, …, ₙ), where some of the variables are fixed, belong in the appropriate space of separately (', ') -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 (, ) -harmonic functions in the disc and for n-harmonic functions in Dⁿ. Bibliography: 23 titles.