Functional Inequalities for Empty Essential Spectrum

By Feng-Yu Wang. 2000

167 citations recorded by the source corpus.

Abstract In terms of the equivalence of Poincare inequality and the existence of spectral gap, the super-Poincare inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for super-Poincare inequalities is studied, the resulting estimates extend previous ones for Poincare and log-Sobolev inequalities.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/6ac7e90093d3dab678a65d15f0b0d2bfeea4dfb8

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In Applied Mathematics

Citations in the corpus

On the role of convexity in functional and isoperimetric inequalities (2008, background): “However, this approach relies on another class of functional inequalities (‘super-Poincaré’ inequalities) and a result due to Wang [54], relating between these inequalities and isoperimetric ones.”
DEVIATION BOUNDS FOR ADDITIVE FUNCTIONALS OF MARKOV PROCESSES (2006, background): “A precise discussion is done in [23] (also see [13]).”
Isoperimetry between exponential and Gaussian (2006, methodology): “Wang discovered that super-Poincaré inequalities imply precise isoperimetric estimates, and are related to Beckner-type inequalities via F -Sobolev inequalities.”
Functional inequalities for uniformly integrable semigroups and application to essential spectrums (2002, methodology): “It was shown in [18] that, for Dirichlet forms of symmetric diffusions on a Riemannian manifold, (1.”
FUNCTIONAL INEQUALITIES, SEMIGROUP PROPERTIES AND SPECTRUM ESTIMATES (2000, methodology): “See [59] for the detailed relationship between β in (1.”
Lyapunov conditions for Super Poincaré inequalities (2009, background): “In particular we thus obtain similar results as Wang’s one, but for measures satisfying sub-gaussian concentration phenomenon.”
Elliptic equations for invariant measures on finite and infinite dimensional manifolds (2001, background): “12) implies the Poincaré inequality (see [47]) and furthermore the super-Poincaré inequality (see [48]), since it implies thatLZ →−∞ as shown in the proof of Theorem 3.”
Transport Inequalities. A Survey (2010, methodology): “This class of functional inequalities was introduced by Wang in [106] with applications in spectral theory.”
Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups (2001, methodology): “, using an increasing function F to replace log in the log-Sobolev inequality), and by Wang in [28] for the super-Poincaré inequality (1.”
Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds (2003, methodology): “1 in [35]), we have the following Nash inequality”
Invariances in variance estimates (2011, background): “Indeed, the paper [33] puts forward functional inequalities involving μ which ensure that the operator L has an empty essential spectrum.”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, methodology): “In practice one often proves Sobolev inequalities first and then deduce the isoperimetric inequalities from a method of Ledoux (see [48], [11], [65], [14]) which applies when the curvature is bounded below to certain Sobolev inequalities with energy term ∫ |∇f |dμ.”
Functional Inequalities and Spectrum Estimates: The Infinite Measure Case (2002, methodology): “1 in [13], for a Riemannian manifold of Ricci curvature bounded below, (4.”
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.”
Essential spectral radius for Markov semigroups (I): discrete time case (2004, background): “Wang [41] (2000) found an infinitesimal criterion for exactly evaluating ress ( Pt |L2(μ) ) .”
Poincaré inequalities and dimension free concentration of measure (2010, methodology): “The general case is more intricate (see [39], Theorem 6.”
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, background): “Their motivation was a criterion for the generator to have a non-empty essential spectrum (see [30,58] for connections with super-Poincaré inequalities).”