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Functional properties of Privalov spaces of holomorphic functions in several variables. (English. Russian original) Zbl 0944.32005

Math. Notes 65, No. 2, 230-237 (1999); translation from Mat. Zametki 65, No. 2, 280-288 (1999).
Summary: We consider Privalov classes of degree \(q>1\) in the unit ball and the polydisk in \(\mathbb{C}^n\). They are defined, say, for the ball, as the sets of functions \(f(z)\) such that the average of \(\ln^q_+|f(z)|\) over a sphere centered at the origin remains bounded as the radius increases to 1. These classes, which were introduced (in the one-dimensional case) by Privalov before 1941, were often used in the foreign literature in the last 10-20 years; typically, the notation varied and Privalov was not mentioned. We discuss various equivalent definitions of these classes as well as the most general properties, such as growth estimates, properties of the natural metric, and boundedness or total boundedness of subsets.

MSC:

32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
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References:

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