Regularity of the local fractional maximal function

Regularity of the local fractional maximal function

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...

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Veröffentlicht in:arXiv.org 2013-10
Hauptverfasser: Heikkinen, Toni, Kinnunen, Juha, Korvenpää, Janne, Tuominen, Heli
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Estimates
Euclidean geometry
Euclidean space
Mathematics - Functional Analysis
Operators (mathematics)
Sobolev space
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Korvenpää, Janne
Tuominen, Heli
description 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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subjects Estimates
Euclidean geometry
Euclidean space
Mathematics - Functional Analysis
Operators (mathematics)
Sobolev space
title Regularity of the local fractional maximal function
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