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Boundedness of the Hilbert Transform in Besov Spaces

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Abstract

Boundedness conditions are found for the Hilbert transform H in Besov spaces with Muckenhoupt weights. The operator H in this situation acts on subclasses of functions from Hardy spaces. The results are obtained by representing the Hilbert transform H via Riemann–Liouville operators of fractional integration on ℝ, on the norms of images and pre-images of which independent estimates are established in the paper. Separately, a boundedness criterion is given for the transform H in weighted Besov and Triebel–Lizorkin spaces restricted to the subclass of Schwartz functions.

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Acknowledgements

The author thanks a reviewer for careful proofreading of the paper and valuable comments, which significantly improved the results of the paper.

For providing the proof of the formula (4.14), the author is grateful to Professor V. I. Chebotarev, Head of Laboratory of Approximate Methods and Functional Analysis of the Computing Center of Far Eastern Branch of Russian Academy of Sciences in Khabarovsk.

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Authors and Affiliations

  1. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya str., Moscow, 117997, Russia

    E. P. Ushakova

  2. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina str., Moscow, 119991, Russia

    E. P. Ushakova

  3. Computing Center of Far Eastern Branch of Russian Academy of Sciences, 65 Kim Yu Chena str., Khabarovsk, 680000, Russia

    E. P. Ushakova

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  1. E. P. Ushakova
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Correspondence to E. P. Ushakova.

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Dedicated to Professor O. V. Besov on the occasion of his 90th birthday

The work of the author presented in Sections 5 and 6 was supported by the Russian Science Foundation (Grant no. 19-11-00087, https://rscf.ru/project/19-11-00087/) and performed at the Steklov Institute of Mathematics of Russian Academy of Sciences. The work presented in the rest part of the paper was carried out within the framework of the state task of the Ministry of Science and Higher Education of the Russian Federation to the Computing Center of the Far Eastern Branch of the Russian Academy of Sciences and V. A. Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences.

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Ushakova, E.P. Boundedness of the Hilbert Transform in Besov Spaces. Anal Math 49, 1137–1174 (2023). https://doi.org/10.1007/s10476-023-0242-2

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  • DOI: https://doi.org/10.1007/s10476-023-0242-2

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