Functional properties of privalov spaces of holomorphic functions in several variables

Abstract

We consider Privalov classes of degreeq>1 in the unit ball and the polydisk in ℂn. They are defined, say, for the ball, as the sets of functionsf(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.