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Hardy-Bloch type spaces and lacunary series on the polydisk. (English) Zbl 1123.32004
Summary: We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of \(\mathbb C^n\). Then we apply them to obtain sharp estimates for the mean growth in weighted spaces \(h(p,\alpha), h(p, \log(\alpha))\) 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.

MSC:
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
32A05 Power series, series of functions of several complex variables
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