Real Analysis Exchange

Bloch and gap subharmonic functions.

R. Supper

Full-text: Open access

Abstract

For subharmonic functions \(u\geq 0\) in the unit ball \(B_N\) of \(\mathbb{R}^N \), the paper characterizes this kind of growth: \(\sup_{x\in B_N} (1- \vert x\vert ^2 )^\alpha u(x) <+\infty \) (given \(\alpha >0\)), through criteria involving such integrals as \(\int u(x)\, dx \) or \(\int u(x) ( 1- \vert x\vert ^2 )^{\alpha -N} \, dx \) over balls centered at points \(a\in B_N\). Given \(p \in \mathbb{R}\) and \(\omega\) some non--negative function, this article compares subharmonic functions with the previous kind of growth to subharmonic functions satisfying: \( \sup_{a\in B_N} \int_{ B_N } u(x) ( 1- \vert x\vert ^2 )^p \omega (\vert \varphi _a (x)\vert)\, dx <+\infty \), where \(\varphi _a\) are Möbius transformations. The paper also studies subharmonic functions which are sums of lacunary series and their links with both previous kinds of subharmonic functions.

Article information

Source
Real Anal. Exchange, Volume 28, Number 2 (2002), 395-414.

Dates
First available in Project Euclid: 20 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184963803

Mathematical Reviews number (MathSciNet)
MR2009762

Zentralblatt MATH identifier
1056.31003

Subjects
Primary: 31B05: Harmonic, subharmonic, superharmonic functions 30B10: Power series (including lacunary series) 26B10: Implicit function theorems, Jacobians, transformations with several variables 26D15: Inequalities for sums, series and integrals 30D45: Bloch functions, normal functions, normal families

Keywords
subharmonic function unit ball ellipsoid M\" obius transformation gap series

Citation

Supper, R. Bloch and gap subharmonic functions. Real Anal. Exchange 28 (2002), no. 2, 395--414. https://projecteuclid.org/euclid.rae/1184963803


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References

  • A. B. Aleksandrov, Function Theory in the Ball, Several Complex Variables, Part II (G.M. Khenkin and A.G. Vitushkin Editors) Encyclopedia of Mathematical Sciences, Volume 8, Springer Verlag, 1994.
  • W. K. Hayman and P. B. Kennedy, Subharmonic functions, Vol.I, London Mathematical Society Monographs, No. 9. Academic Press,
  • M. Mateljevic and M. Pavlovic, $L\sp{p}$-behavior of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc., 87 (1983), 309–316.
  • R. Remmert, Classical topics in complex function theory, Graduate Texts in Mathematics, 172. Springer-Verlag, New York, 1998.
  • W. Rudin, Function theory in the unit ball of $\C^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241, Springer-Verlag, New York-Berlin, 1980.
  • R. Supper, Subharmonic functions with a Bloch type growth, submitted.
  • S. Yamashita, Gap series and $\alpha $-Bloch functions, Yokohama Math. J., 28 (1980), 31–36.
  • R. H. Zhao, On a general family of function spaces, Ann. Acad. Sci. Fenn. Math. Diss. No. 105 (1996).
  • R. H. Zhao, On $\alpha$-Bloch functions and VMOA, Acta Math. Sci., 16 (1996), 349–360.
  • K. H. Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, 139, Marcel Dekker, Inc., New York, 1990.
  • C. Zuily, Distributions et équations aux dérivées partielles, Collection Méthodes, Hermann, Paris, 1986.