Arkiv för Matematik

  • Ark. Mat.
  • Volume 53, Number 1 (2015), 127-154.

Regularity of the local fractional maximal function

Toni Heikkinen, Juha Kinnunen, Janne Korvenpää, and Heli Tuominen

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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.


This work was supported by the Academy of Finland.

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Ark. Mat., Volume 53, Number 1 (2015), 127-154.

Received: 16 October 2013
First available in Project Euclid: 30 January 2017

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2014 © Institut Mittag-Leffler


Heikkinen, Toni; Kinnunen, Juha; Korvenpää, Janne; Tuominen, Heli. Regularity of the local fractional maximal function. Ark. Mat. 53 (2015), no. 1, 127--154. doi:10.1007/s11512-014-0199-2.

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