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.
Similar content being viewed by others
References
W. Rudin,Function Theory in the Unit Ball of ℂ n, Springer-Verlag, New York-Berlin (1980).
W. Rudin,Function Theory in Polydiscs Benjamin, New York-Amsterdam (1969).
I. I. Privalov,Boundary Properties of Single-Valued Analytic Functions [in Russian], Izd. Moskov. Univ., Moscow (1941).
M. Stoll, “Mean growth and Taylor coefficients of some topological algebras of analytic functions,”Ann. Polon. Math.,35, 139–158 (1977).
N. Mochizuki, “Algebras of holomorphic functions betweenH p andN *”Proc. Amer. Math. Soc.,105, 898–902 (1989).
Meštrović R. and Ž. Pavićević, “Remarks on some classes of holomorphic functions,”Math. Montisnigri,6, 27 (1996).
M. Stoll, “Harmonic majorants for plurisubharmonic functions,”J. Reine Angew. Math.,282, 80–87 (1976).
H. O. Kim, “On closed maximal ideals ofM,”Proc. Japan Acad. Ser. A. Math. Sci.,62, No. 9, 343–346 (1986).
H. O. Kim, “On anF-algebra of holomorphic functions,”Canad. J. Math.,40, No. 3, 718–741 (1988).
B. R. Choe and H. O. Kim, “On the boundary behavior of functions holomorphic on the ball”Complex Variables Theory Appl.,20, 53–61 (1992).
H. O. Kim and Y. Y. Park, “Maximal functions of plurisubharmonic functions,”Tsukuba J. Math.,16, No. 1, 11–18 (1992).
I. I. Privalov,Boundary Properties of Analytic Functions [in Russian], 2d ed., Gostekhizdat, Moscow-Leningrad (1950).
J. Shapiro and A. Shields, “Unusual topological properties of the Nevanlinna class,”Amer. J. Math.,97, 915–936 (1975).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki Vol. 65, No. 2, pp. 280–288, February, 1999.
Rights and permissions
About this article
Cite this article
Subbotin, A.V. Functional properties of privalov spaces of holomorphic functions in several variables. Math Notes 65, 230–237 (1999). https://doi.org/10.1007/BF02679821
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02679821