HARDY–BLOCH TYPE SPACES AND LACUNARY SERIES ON THE POLYDISK

We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of $\C^n$. Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.