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The Weighted Super-Bergman Kernels of \(\mathbb {B}^{m|n}\) and Integral Representations of the Invariant Inner Products on \(H^2_{\nu }(\mathbb {B}^m)\)

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Abstract

The purpose of this paper is twofold. Firstly, we give explicit expressions of the super-Szeg\(\ddot{o}\) kernel and the weighted super-Bergman kernel for the super-ball \(\mathbb {B}^{m|n}\). Secondly, using orthogonal decompositions of the weighted super-Bergman spaces \(L_{a}^2(\mathbb {B}^{m|n},\mu ^{\nu }_{m|n})\), we obtain new integral representations for the invariant inner products on \( H^2_{\nu }(\mathbb {B}^m)\) of holomorphic functions on the unit ball \(\mathbb {B}^m\) of \(\mathbb {C}^m\).

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Acknowledgments

The part of the work was completed when the first author visited School of Mathematics and Statistics at Wuhan University during 2013, and he wishes to thank the School for its kind hospitality. In addition, the authors would like to thank the referees for many helpful suggestions. The first author was supported by the Scientific Research Fund of Sichuan Provincial Education Department (No.11ZA156), and the second author was supported by the National Natural Science Foundation of China (No.11271291).

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Authors and Affiliations

  1. School of Mathematical and Information Sciences, Leshan Normal University, Leshan, 614000, Sichuan, People’s Republic of China

    Zhiming Feng

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Zhenhan Tu

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  1. Zhiming Feng
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  2. Zhenhan Tu
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Correspondence to Zhenhan Tu.

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Communicated by Frank Sommen.

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Feng, Z., Tu, Z. The Weighted Super-Bergman Kernels of \(\mathbb {B}^{m|n}\) and Integral Representations of the Invariant Inner Products on \(H^2_{\nu }(\mathbb {B}^m)\) . Complex Anal. Oper. Theory 9, 1037–1063 (2015). https://doi.org/10.1007/s11785-014-0392-0

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  • DOI: https://doi.org/10.1007/s11785-014-0392-0

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