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Musielak−Orlicz−Sobolev spaces on arbitrary metrique space

100%
Youssef A., Aissaoui N., Cherif Hassib M.
Commentationes Mathematicae
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2016
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tom 56
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nr 2
EN
In this article we define Musielak−Orlicz−Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each Musielak−Orlicz−Sobolev function has a quasi-continuous representative.
2
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Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

88%
Cheeger J., Kleiner B., Schioppa A.
Analysis and Geometry in Metric Spaces
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2016
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tom 4
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nr 1
EN
We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to 1, and new nonembeddability results.
3
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Resistance Conditions and Applications

75%
Kinnunen J., Silvestre P.
Analysis and Geometry in Metric Spaces
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2013
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tom 1
276-294
EN
This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.
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