Displaying similar documents to “Characterizations of p-superharmonic functions on metric spaces”

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.

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...

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.

Fubini’s Theorem on Measure

Noboru Endou (2017)

Formalized Mathematics

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The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini’s theorem, and prove the relevant proposition. Finally we show Fubini’s theorem on measure.

Product Pre-Measure

Noboru Endou (2016)

Formalized Mathematics

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In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.

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.

Construction of Measure from Semialgebra of Sets1

Noboru Endou (2015)

Formalized Mathematics

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In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a σ-field and the measure on it. In general, the existence of the measure on σ-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a σ-field. Therefore, in this article we define a measure (pre-measure) on a semialgebra and extend it to a measure on a σ-field. Furthermore,...