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Improvement of Menshov’s Theorem on Upper Estimate for the Norm of Maximal Operator

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We study the problem of estimating the majorant of partial sums of a series with respect to an orthogonal system. D. E. Menshov established that the norm of the maximal operator does not exceed log2 N +1 and this estimate is order-sharp. We prove that the norm of the maximal operator does not exceed 0.5 log2 N + 1. The estimate obtained provides new tools for constructing orthogonal systems with extremely large norm of the majorant of partial sums.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics, 1, Leninskie Gory, Moscow, 119991, Russia

    A. P. Solodov

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  1. A. P. Solodov
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Correspondence to A. P. Solodov.

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Solodov, A.P. Improvement of Menshov’s Theorem on Upper Estimate for the Norm of Maximal Operator. J Math Sci 274, 544–551 (2023). https://doi.org/10.1007/s10958-023-06619-3

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  • DOI: https://doi.org/10.1007/s10958-023-06619-3

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