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

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$-harmonic functions on $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$-Poisson kernel $P_{\alpha, \beta} (z, \zeta)$. A characterization by hypergeometric functions of separately $(\alpha, \beta)$-harmonic functions which are also $m$-homogeneous is given, it is used to obtain series expansion of separately $(\alpha, \beta)$-harmonic 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$^\ast$ convergence at the distinguished boundary $\mathbb T^n$. Weak $(1,1)$-type estimate for a restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$ is derived. We show that slice functions $u(z_1, …, z_k, \zeta_{k+1}, …, \zeta_n)$, where some of the variables are fixed, belong in the appropriate space of separately $(\alpha', \beta')$-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 $\mathbb T^n$. Our results extend earlier results for $(\alpha, \beta)$-harmonic functions in the disc and for $n$-harmonic functions in $\mathbb D^n$.

separately $(\alpha,\beta)$-harmonic functions, $H^p$ spaces, non-tangential limits, polydisc