Regularity of the local fractional maximal function

Toni Heikkinen Department of Mathematics, P.O. Box 11100, Aalto University, Finland Juha Kinnunen Department of Mathematics, P.O. Box 11100, Aalto University, Finland Janne Korvenpää Department of Mathematics, P.O. Box 11100, Aalto University, Finland Heli Tuominen Department of Mathematics and Statistics, P.O. Box 35, University of Jyväskylä, Finland

Functional Analysis mathscidoc:1701.12028

Arkiv for Matematik, 53, (1), 127-154, 2013.10
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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@inproceedings{toni2013regularity,
  title={Regularity of the local fractional maximal function},
  author={Toni Heikkinen, Juha Kinnunen, Janne Korvenpää, and Heli Tuominen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203639543140075},
  booktitle={Arkiv for Matematik},
  volume={53},
  number={1},
  pages={127-154},
  year={2013},
}
Toni Heikkinen, Juha Kinnunen, Janne Korvenpää, and Heli Tuominen. Regularity of the local fractional maximal function. 2013. Vol. 53. In Arkiv for Matematik. pp.127-154. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203639543140075.
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