Displaying similar documents to “The Sobolev capacity on metric spaces.”

Thin and fat sets for doubling measures in metric spaces

Tuomo Ojala, Tapio Rajala, Ville Suomala (2012)

Studia Mathematica

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We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.

Some properties of the Hausdorff distance in metric spaces.

Jozef Banas, Antonio Martinón (1990)

Extracta Mathematicae

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Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness.

The Besov capacity in metric spaces

Juho Nuutinen (2016)

Annales Polonici Mathematici

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We study a capacity theory based on a definition of Hajłasz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are γ-medians, for which we also prove a new version of a Poincaré type inequality.

Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger, Raanan Schul (2017)

Analysis and Geometry in Metric Spaces

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A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical...

Sobolev inequalities for probability measures on the real line

F. Barthe, C. Roberto (2003)

Studia Mathematica

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We give a characterization of those probability measures on the real line which satisfy certain Sobolev inequalities. Our starting point is a simpler approach to the Bobkov-Götze characterization of measures satisfying a logarithmic Sobolev inequality. As an application of the criterion we present a soft proof of the Latała-Oleszkiewicz inequality for exponential measures, and describe the measures on the line which have the same property. New concentration inequalities for product measures...

Effect of choice of dissimilarity measure on classification efficiency with nearest neighbor method

Tomasz Górecki (2005)

Discussiones Mathematicae Probability and Statistics

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In this paper we will precisely analyze the nearest neighbor method for different dissimilarity measures, classical and weighed, for which methods of distinguishing were worked out. We will propose looking for weights in the space of discriminant coordinates. Experimental results based on a number of real data sets are presented and analyzed to illustrate the benefits of the proposed methods. As classical dissimilarity measures we will use the Euclidean metric, Manhattan and post office...

Hölder quasicontinuity of Sobolev functions on metric spaces.

Piotr Hajlasz, Juha Kinnunen (1998)

Revista Matemática Iberoamericana

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We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].