Displaying similar documents to “A note on global integrability of upper gradients of p-superharmonic functions”

Characterizations of p-superharmonic functions on metric spaces

Anders Björn (2005)

Studia Mathematica

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We show the equivalence of some different definitions of p-superharmonic functions given in the literature. We also provide several other characterizations of p-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our...

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.

A note on the Poincaré inequality

Alireza Ranjbar-Motlagh (2003)

Studia Mathematica

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The Poincaré inequality is extended to uniformly doubling metric-measure spaces which satisfy a version of the triangle comparison property. The proof is based on a generalization of the change of variables formula.

An area formula in metric spaces

Valentino Magnani (2011)

Colloquium Mathematicae

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We present an area formula for continuous mappings between metric spaces, under minimal regularity assumptions. In particular, we do not require any notion of differentiability. This is a consequence of a measure-theoretic notion of Jacobian, defined as the density of a suitable "pull-back measure". Finally, we give some applications and examples.

John-Nirenberg lemmas for a doubling measure

Daniel Aalto, Lauri Berkovits, Outi Elina Kansanen, Hong Yue (2011)

Studia Mathematica

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We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderón-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

Heikki Hakkarainen, Juha Kinnunen, Panu Lahti, Pekka Lehtelä (2016)

Analysis and Geometry in Metric Spaces

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This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted...

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

Jeff Cheeger, Bruce Kleiner, Andrea Schioppa (2016)

Analysis and Geometry in Metric Spaces

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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.

Spaces of σ-finite linear measure

Ihor Stasyuk, Edward D. Tymchatyn (2013)

Colloquium Mathematicae

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Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear...

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.

Stability and Continuity of Functions of Least Gradient

H. Hakkarainen, R. Korte, P. Lahti, N. Shanmugalingam (2015)

Analysis and Geometry in Metric Spaces

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In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.

The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

Marcello Lucia, Michael J. Puls (2015)

Analysis and Geometry in Metric Spaces

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Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

Some remarks about metric spaces, spherical mappings, functions and their derivatives.

Stephen Semmes (1996)

Publicacions Matemàtiques

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If p ∈ R, then we have the radial projection map from R {p} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In...