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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 |

### Keywords:

Hardy class; Nevanlinna class; polydisk; Poisson kernel; growth estimate; total boundedness; Privalov classes
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\textit{A. V. Subbotin}, Math. Notes 65, No. 2, 230--237 (1999; Zbl 0944.32005); translation from Mat. Zametki 65, No. 2, 280--288 (1999)

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### References:

[1] | W. Rudin,Function Theory in the Unit Ball of \(\mathbb{C}\) n, Springer-Verlag, New York-Berlin (1980). · Zbl 0495.32001 |

[2] | W. Rudin,Function Theory in Polydiscs Benjamin, New York-Amsterdam (1969). · Zbl 0177.34101 |

[3] | I. I. Privalov,Boundary Properties of Single-Valued Analytic Functions [in Russian], Izd. Moskov. Univ., Moscow (1941). |

[4] | M. Stoll, ”Mean growth and Taylor coefficients of some topological algebras of analytic functions,”Ann. Polon. Math.,35, 139–158 (1977). · Zbl 0377.30036 |

[5] | N. Mochizuki, ”Algebras of holomorphic functions betweenH p andN *”Proc. Amer. Math. Soc.,105, 898–902 (1989). · Zbl 0679.30040 |

[6] | Meštrović R. and Ž. Pavićević, ”Remarks on some classes of holomorphic functions,”Math. Montisnigri,6, 27 (1996). · Zbl 0901.30030 |

[7] | M. Stoll, ”Harmonic majorants for plurisubharmonic functions,”J. Reine Angew. Math.,282, 80–87 (1976). · Zbl 0318.32014 |

[8] | H. O. Kim, ”On closed maximal ideals ofM,”Proc. Japan Acad. Ser. A. Math. Sci.,62, No. 9, 343–346 (1986). · Zbl 0614.30028 |

[9] | H. O. Kim, ”On anF-algebra of holomorphic functions,”Canad. J. Math.,40, No. 3, 718–741 (1988). · Zbl 0646.30033 |

[10] | 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). · Zbl 0728.32004 |

[11] | H. O. Kim and Y. Y. Park, ”Maximal functions of plurisubharmonic functions,”Tsukuba J. Math.,16, No. 1, 11–18 (1992). · Zbl 0770.31007 |

[12] | I. I. Privalov,Boundary Properties of Analytic Functions [in Russian], 2d ed., Gostekhizdat, Moscow-Leningrad (1950). |

[13] | J. Shapiro and A. Shields, ”Unusual topological properties of the Nevanlinna class,”Amer. J. Math.,97, 915–936 (1975). · Zbl 0323.30033 |

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