Abstract
We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderón–Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.
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Acknowledgements
A. Karapetyants was partially supported by Southern Federal University Project No. 07/2017-31 and partially supported by the Grant 18-51-05009-Apm_a of Russian Foundation of Basic Research. H. Rafeiro was partially supported by Pontificia Universidad Javeriana. S. Samko was partially supported by the RFBR Grant 15-01-02732 and partially supported by the Grant 18-01-00094-a of Russian Foundation of Basic Research.
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Communicated by Ahmed Sebbar.
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Karapetyants, A., Rafeiro, H. & Samko, S. Boundedness of the Bergman Projection and Some Properties of Bergman Type Spaces. Complex Anal. Oper. Theory 13, 275–289 (2019). https://doi.org/10.1007/s11785-018-0780-y
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DOI: https://doi.org/10.1007/s11785-018-0780-y
Keywords
- Bergman space
- Bergman projection
- Variable exponent Lebesgue space
- Orlicz space
- Variable exponent generalized Morrey space