The Lebesgue Set of a Function whose Distribution Derivatives are p-th Power Summable

Federer, Herbert; Ziemer, William P.

doi:10.1512/iumj.1973.22.22013

Cited by 151 publications

(93 citation statements)

References 6 publications

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“…Standard proofs of refinements of Lebesgue's theorem are based on a capacitary weak type estimate for the Hardy-Littlewood maximal function, see [5], [7], [15], [19] or [22]. This estimate is usually proved by using the Besicovitch covering theorem, extension results or representation formulas for Sobolev functions.…”

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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“…for every x ∈ X \ E, whenever 0 < s < np n−p [6]. The proofs of the refinements of Lebesgue differentiation theorem for Sobolev functions on Euclidean spaces are based on weak capacitary estimates for Hardy-Littlewood maximal function and use tools that are not available in metric spaces.…”

Section: Lebesgue Points Of Orlicz-sobolev Functionsmentioning

confidence: 99%

Lebesgue Points for Orlicz-Sobolev Functions on Metric Measure Spaces

Mocanu¹

2011

Annals of the Alexandru Ioan Cuza University - Mathematics

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Abstract. In this paper we deal with a metric measure space equipped with a doubling measure and supporting an Orlicz-Poincaré inequality, namely a weak (1, Φ)-Poincaré inequality, that is more general than the (1, 1)-Poincaré inequality. For a wide class of Orlicz spaces, we prove that the corresponding Orlicz-Sobolev functions have Lebesgue points outside a set of zero Orlicz-Sobolev capacity. This results extends a theorem of Tuominen (2009)

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“…It is well known (see e.g. [14,15,21]) that every u ∈ W 1,p (Ω) has a quasi continuous representative. We shall always identify u ∈ W 1,p (Ω) with this quasi continuous representative.…”

Section: Notations and Definitionsmentioning

confidence: 99%

Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets

Calvo‐Jurado¹,

Casado–Díaz²,

Luna-Laynez³

2009

ESAIM: COCV

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Abstract. For a fixed bounded open set Ω ⊂ R N , a sequence of open sets Ωn ⊂ Ω and a sequence of sets Γn ⊂ ∂Ω ∩ ∂Ωn, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on Ωn, satisfying Neumann boundary conditions on Γn and Dirichlet boundary conditions on ∂Ωn \ Γn. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on Ωn and Γn locally.Mathematics Subject Classification. 35B40.

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The Lebesgue Set of a Function whose Distribution Derivatives are p-th Power Summable

Cited by 151 publications

References 6 publications

Lebesgue points for Sobolev functions on metric spaces

Lebesgue points for Sobolev functions on metric spaces

Lebesgue Points for Orlicz-Sobolev Functions on Metric Measure Spaces

Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets

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