Discrete Hardy Spaces for Bounded Domains in $${\mathbb {R}}^{n}$$

Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in $${\mathbb {R}}^{n}$$ R n . On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.

Discrete Hardy Spaces Related to Powers of the Poisson Kernel

Discrete Hardy spaces $H^{1}_{\alpha}(\partial{T})$, related to powers $\alpha \ge 1/2$ of the Poisson kernels on boundaries $\partial{T}$ of regular rooted trees, are studied. The spaces for $\alpha > 1/2$ coincide with the ordinary atomic Hardy space on $\partial{T}$, which in turn is strictly smaller than $H^{1}_{1/2}(\partial{T})$. The Orlicz space $L\log\log L(\partial{T})$ characterizes the positive and increasing functions in $H^{1}_{1/2}(\partial{T})$, but there is no such characterization for general positive functions.