Volume integral means of holomorphic functions
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- by Jie Xiao and Kehe Zhu PDF
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Abstract:
The classical integral means of a holomorphic function $f$ in the unit disk are defined by \[ \Bigg [\frac 1{2\pi }\int _0^{2\pi }|f(re^{i\theta })|^p d\theta \Bigg ]^{1/p}, \quad 0\le r<1.\] These integral means play an important role in modern complex analysis. In this note we consider integral means of holomorphic functions in the unit ball $\mathbb B_n$ in $\mathbb C^n$ with respect to weighted volume measures, \[ M_{p,\alpha }(f,r)=\left [\frac {1}{v_\alpha (r\mathbb B_n)} \int _{r\mathbb B_n}|f(z)|^p dv_\alpha (z)\right ]^{1/p}, \quad 0\le r<1,\] where $\alpha$ is real, $dv_\alpha (z)=(1-|z|^2)^\alpha dv(z)$, and $dv$ is volume measure on $\mathbb B_n$. We show that $M_{p,\alpha }(f,r)$ increases with $r$ strictly unless $f$ is a constant, but in contrast with the classical case, $\log M_{p,\alpha }(f,r)$ is not always convex in $\log r$. As an application, we show that if $\alpha \le -1$, $M_{p,\alpha }(f,r)$ is bounded in $r$ if and only if $f$ belongs to the Hardy space $H^p$, while if $\alpha >-1$, $M_{p,\alpha }(f,r)$ is bounded in $r$ if and only if $f$ is in the weighted Bergman space $A^p_\alpha$.References
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Additional Information
- Jie Xiao
- Affiliation: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada
- MR Author ID: 247959
- Email: jxiao@mun.ca
- Kehe Zhu
- Affiliation: Department of Mathematics and Statistics, State University of New York, Albany, New York 12222
- MR Author ID: 187055
- Email: kzhu@albany.edu
- Received by editor(s): May 3, 2010
- Published electronically: November 18, 2010
- Additional Notes: The first author was supported in part by NSERC of Canada
- Communicated by: Richard Rochberg
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1455-1465
- MSC (2010): Primary 32A10, 32A36, 32A35, 51M25
- DOI: https://doi.org/10.1090/S0002-9939-2010-10797-9
- MathSciNet review: 2748439