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Regularity of the local fractional maximal function

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Arkiv för Matematik

Abstract

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply norm estimates in Sobolev spaces. An unexpected feature is that these estimates contain extra terms involving spherical and fractional maximal functions. Moreover, we construct several explicit examples, which show that our results are essentially optimal. Extensions to metric measure spaces are also discussed.

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Author information

Authors and Affiliations

  1. Department of Mathematics, P.O. Box 11100, FI-00076, Aalto University, Finland

    Toni Heikkinen, Juha Kinnunen & Janne Korvenpää

  2. Department of Mathematics and Statistics, P.O. Box 35, FI-40014, University of Jyväskylä, Finland

    Heli Tuominen

Authors
  1. Toni Heikkinen
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  2. Juha Kinnunen
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  3. Janne Korvenpää
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  4. Heli Tuominen
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Corresponding author

Correspondence to Juha Kinnunen.

Additional information

This work was supported by the Academy of Finland.

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Heikkinen, T., Kinnunen, J., Korvenpää, J. et al. Regularity of the local fractional maximal function. Ark Mat 53, 127–154 (2015). https://doi.org/10.1007/s11512-014-0199-2

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  • DOI: https://doi.org/10.1007/s11512-014-0199-2

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