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Weighted integrals of holomorphic functions

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Abstract

Given an admissible weight w and 0<p<∞, the estimate {\(\smallint _D \left| {f(z)} \right|{}^pw(z)dm(z) \sim \left| {f(0)} \right|{}^pw + \smallint _D \left| {f(z)} \right|{}^p{\text{ }}\psi ^p (z)w(z)dm(z)\)} is valid for all holomorphic functions f in the unit disc D. Here, {\(\psi \left( r \right) = \frac{{\smallint _r^{w\left( t \right)dt} }}{{w\left( r \right)}}\)} is the distortion of w. As an application of the above estimate, it is proved that the Cesàro operator C[•] is bounded on the weighted Bergman spaces L pa,w (D).

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Authors and Affiliations

  1. Dept. of Math., Univ. of Sci. and Tech. of China, 230026, Hefei, China

    Hu Zhangjian & Liu Taishun

  2. Dept. of Math., Huzhou Teachers College, 313000, Huzhou, China

    Hu Zhangjian

Authors
  1. Hu Zhangjian
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  2. Liu Taishun
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Additional information

Partially supported by the 151 Projection and the Natural Science Foundation of Zhejiang Province (M103104).

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Zhangjian, H., Taishun, L. Weighted integrals of holomorphic functions. Appl. Math. Chin. Univ. 19, 474–480 (2004). https://doi.org/10.1007/s11766-004-0014-0

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  • DOI: https://doi.org/10.1007/s11766-004-0014-0

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