Displaying similar documents to “Singular functions on metric measure spaces.”

Quasiconformal mappings and Sobolev spaces

Pekka Koskela, Paul MacManus (1998)

Studia Mathematica

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We examine how Poincaré change under quasiconformal maps between appropriate metric spaces having the same Hausdorff dimension. We also show that for many metric spaces the Sobolev functions can be identified with functions satisfying Poincaré, and this allows us to extend to the metric space setting the fact that quasiconformal maps from ℝ Q onto ℝ Q preserve the Sobolev space L 1 , Q ( ℝ Q ) .

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 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...

A note on global integrability of upper gradients of p-superharmonic functions

Outi Elina Maasalo, Anna Zatorska-Goldstein (2009)

Colloquium Mathematicae

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We consider a complete metric space equipped with a doubling measure and a weak Poincaré inequality. We prove that for all p-superharmonic functions there exists an upper gradient that is integrable on H-chain sets with a positive exponent.