Abstract
We give a proof of the boundedness of the Bergman projection in generalized variable-exponent vanishing Morrey spaces over the unit disc and the upper half-plane. To this end, we prove the boundedness of the Calderón—Zygmund operators on generalized variable-exponent vanishing Morrey spaces. We give the proof of the latter in the general context of real functions on Rn, since it is new in such a setting and is of independent interest. We also study the approximation by mollified dilations and estimate the growth of functions near the boundary.
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Funding
The work of A. N. Karapetyants and S. G. Samko was supported in part by the Russian Foundation for Basic Research under grant 18-01-00094. The work of A. N. Karapetyants was also supported in part by the Russian Foundation for Basic Research under grant 18-51-05009 Arm-a and by Visiting Fulbright Scholar Program. The research of H. Rafeiro was supported by a Research Start-up Grant of United Arab Emirates University, Al Ain, United Arab Emirates, via grant no. G00002994. The research of S. Samko was supported by the Russian Foundation for Basic Research under grant 19-01-00223.
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Karapetyants, A.N., Rafeiro, H. & Samko, S.G. On Singular Operators in Vanishing Generalized Variable-Exponent Morrey Spaces and Applications to Bergman-Type Spaces. Math Notes 106, 727–739 (2019). https://doi.org/10.1134/S0001434619110075
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DOI: https://doi.org/10.1134/S0001434619110075