Abstract
In this paper, we introduce the notion of poly-Berezin transforms over the unit ball \(\mathbb {B}^d\) and investigate the \(\mathcal {M}\)-harmonicity of the fixed points of the poly-Berezin transform. We prove that the fixed points of a class of poly-Berezin transforms on some weighted \(L^p\) spaces are \(\mathcal {M}\)-harmonic if and only if the corresponding analytic functions have only zero point 0. In particular, when all the coefficients are nonnegative, we prove that, for sufficient large p, the fixed points of the poly-Berezin transforms on some \(L^p\) spaces are \(\mathcal {M}\)-harmonic. When d is small, we prove that the fixed points of a class of poly-Berezin transforms on the standard \(L^p\) spaces for any \(1\le p\le \infty \) are \(\mathcal {M}\)-harmonic.
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Acknowledgements
The authors would like to express their great gratitude to Professor K. Guo for his valuable guidance and encouragement over the years. The first author would also like to thank Professor K. Wang and Professor G. Zhang for their helpful discussions and support.
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Communicated by H. Turgay Kaptanoglu.
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Wei He was partially supported by NSFC NO. 11671078 and NO. 11571253.
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Ding, L., He, W. Fixed Points of Poly-Berezin Transforms. Complex Anal. Oper. Theory 13, 3695–3716 (2019). https://doi.org/10.1007/s11785-019-00925-y
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DOI: https://doi.org/10.1007/s11785-019-00925-y