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Oscillation of Functions in the Hölder Class

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Abstract

We study the size of the set of points where the α-divided difference of a function in the Hölder class Λα is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 40126, Bologna, Italy

    Pavel Mozolyako

  2. Department of Mathematics and Computer Science, Saint Petersburg State University, 199178, Line 14th (Vasilyevsky Island), 29, Saint Petersburg, Russia

    Pavel Mozolyako

  3. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Barcelona, Spain

    Artur Nicolau

Authors
  1. Pavel Mozolyako
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  2. Artur Nicolau
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Correspondence to Artur Nicolau.

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The first author is supported by the joint grant of Russian Foundation for Basic Research (project 17-51-150005-NCNI-a) and CNRS, France (project PRC CNRS/RFBR 2017-2019 “Noyaux reproduisants dans des espaces de Hilbert de fonctions analytiques”

The second author is supported in part by the Generalitat de Catalunya (grant 2017 SGR 395) and the Spanish Ministerio de Ciencia e Innovación (projects MTM2017-85666-P and Maria de Maeztu Unit of Excellence MDM-2014-0445)

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Mozolyako, P., Nicolau, A. Oscillation of Functions in the Hölder Class. Potential Anal 55, 53–74 (2021). https://doi.org/10.1007/s11118-020-09849-1

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  • DOI: https://doi.org/10.1007/s11118-020-09849-1

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