Discrete Hardy Spaces Related to Powers of the Poisson Kernel

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