Orlicz–Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups☆

By Cyril Roberto, B. Zegarlinski. 2006

47 citations recorded by the source corpus.

Abstract We study coercive inequalities in Orlicz spaces associated to the probability measures on finite- and infinite-dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities.

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f713c81dab3b7251ca29e774c4683f7be2fbb003

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In Applied Mathematics

In Geometry and Topology

In Statistics and Probability

Citations in the corpus

On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “These include Beckner-type inequalities [9] (including the Lata la–Oleszkiewicz inequality as in [60]) and additive Φ-Sobolev inequalities [84].”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “ediate observation that for any Borel set A: kχAkN(µ) = kχAkN(µ),∞ = N ∧(µ(A)) . For this reason, the expression 1/N−1(1/t) already appears in the works of Maz’ya [40, p. 112] and Roberto–Zegarlinski [46]. Deﬁnition. An inequality of the form: ∀f∈ F Dkf−MµfkN(µ),∞ ≤ k|∇f|kL q(µ) (3.4) is called a we”
Coercive Inequalities on Metric Measure Spaces (2009, background): “For probability measures on the real line the necessary and sufficient condition for Poincaré inequality characterising the density (of the absolutely continuous part with respect to the Lebesgue measure) were established long time ago by Muckenphout, [36], ([34]).”
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, background): “We refer to [32, 33, 4, 5, 24] for related results on super-Poincaré and general F -Sobolev inequalities, as well as their consequences for the decay of the semi-group in appropriate Orlicz norms.”
Isoperimetry between exponential and Gaussian (2006, result): “Remark 7. Part of the previous Theorem is proved in a slightly difierent form in [ 35 ].”