Compact measure spaces

Abstract We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for $\mathbb {Z}^d$ -actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for the natural action of $\mathbb {Z}^d$ on a product of d measure spaces.

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Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0072 ◽

2020 ◽

Vol 23 (5) ◽

pp. 1452-1471

Author(s):

Vakhtang Kokilashvili ◽

Alexander Meskhi

Keyword(s):

Riesz Potential ◽

Fractional Integrals ◽

Mixed Norm ◽

Lebesgue Spaces ◽

Fractional Integral Operators ◽

Grand Lebesgue Spaces ◽

Measure Spaces ◽

Necessary And Sufficient ◽

Bounded Sets ◽

Trace Inequalities

Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

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Jump processes and nonlinear fractional heat equations on metric measure spaces

Mathematische Nachrichten ◽

10.1002/mana.200310352 ◽

2006 ◽

Vol 279 (1-2) ◽

pp. 150-163 ◽

Cited By ~ 5

Author(s):

Jiaxin Hu ◽

Martina Zähle

Keyword(s):

Jump Processes ◽

Heat Equations ◽

Metric Measure Spaces ◽

Measure Spaces

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Geometry and dynamics of admissible metrics in measure spaces

Open Mathematics ◽

10.2478/s11533-012-0149-9 ◽

2013 ◽

Vol 11 (3) ◽

Cited By ~ 9

Author(s):

Anatoly Vershik ◽

Pavel Zatitskiy ◽

Fedor Petrov

Keyword(s):

Invariant Measure ◽

Wide Class ◽

Lebesgue Space ◽

Measure Space ◽

Asymptotic Properties ◽

Ergodic Transformation ◽

Measure Spaces ◽

Admissible Metric ◽

Discreteness Criterion ◽

Uniformly Bounded

AbstractWe study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

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