The hardy-littlewood maximal function of a sobolev function

Kinnunen, Juha

doi:10.1007/bf02773636

Cited by 169 publications

(136 citation statements)

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“…We do not have these tools available. In the classical case we can also use the fact that the Hardy-Littlewood maximal operator is bounded in the Sobolev space, see [16]. However, examples in [1] show that the Hardy-Littlewood maximal operator does not have the required regularity properties in metric spaces.…”

Section: Introductionmentioning

confidence: 99%

Lebesgue points for Sobolev functions on metric spaces

Kinnunen¹,

Latvala²

2002

Rev. Mat. Iberoam.

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Our main objective is to study the pointwise behaviour of Sobolev functions on a metric measure space. We prove that a Sobolev function has Lebesgue points outside a set of capacity zero if the measure is doubling. This result seems to be new even for the weighted Sobolev spaces on Euclidean spaces. The crucial ingredient of our argument is a maximal function related to discrete convolution approximations. In particular, we do not use the Besicovitch covering theorem, extension theorems or representation formulas for Sobolev functions.

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Section: Introductionmentioning

confidence: 99%

Lebesgue points for Sobolev functions on metric spaces

Kinnunen¹,

Latvala²

2002

Rev. Mat. Iberoam.

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“…[7], [35], [41], [52]). It is well known that the Hardy-Littlewood maximal operator is bounded on the Lebesgue space L p (R N ) if p > 1 (see [52]).…”

Section: Introductionmentioning

confidence: 99%

Musielak-Orlicz-Sobolev spaces on metric measure spaces

Ohno

Shimomura

2015

Czech Math J

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“…In the case of the classical Hardy-Littlewood maximal operator M it is well known, by [5], [7] and [4], that it is bounded in the first order Sobolev spaces, when p > 1. Observe that in general bounded nonlinear operators do not need to be continuous, as is the case for some natural maximal operators, see e.g.…”

Section: Introductionmentioning

confidence: 99%