0000000000669976

AUTHOR

Piotr Hajłasz

showing 2 related works from this author

Sobolev embeddings, extensions and measure density condition

2008

AbstractThere are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density condition. The second main result, Theorem 5, provides several characterizations of the Wm,p-extension domains for 1<p<∞. As a corollary we prove that the property of being a W1,p-extension domain, 1<p⩽∞, is invariant under bi-Lipschitz mappings, Theorem 8.

Discrete mathematicsExtension operator010102 general mathematicsEberlein–Šmulian theoremMeasure density condition01 natural sciencesSobolev embeddingSobolev inequality010101 applied mathematicsSobolev spaceCorollarySobolev spaces0101 mathematicsInvariant (mathematics)AnalysisEdge-of-the-wedge theoremSobolev spaces for planar domainsMathematicsTrace operatorJournal of Functional Analysis
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Maximal potentials, maximal singular integrals, and the spherical maximal function

2014

We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

Sobolev spaceMathematics::Functional AnalysisHomogeneousApplied MathematicsGeneral MathematicsBounded functionMathematical analysisMathematics::Analysis of PDEsMaximal operatorMaximal functionSingular integralMathematicsSobolev inequalityProceedings of the American Mathematical Society
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