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49
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. Funct. Anal
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarith ..."
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Cited by 80 (4 self)
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We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarithmic Sobolev inequalities; exponential integrability; concentration of measure; transportation inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields
, 2008
"... Abstract. Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a ..."
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Cited by 24 (3 self)
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Abstract. Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices. 1.
Discrete Isoperimetric And PoincaréType Inequalities
, 1996
"... We study some discrete isoperimetric and Poincar'etype inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discrete cube. More g ..."
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Cited by 18 (0 self)
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We study some discrete isoperimetric and Poincar'etype inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions ¯ on Z which satisfy these inequalities on Z n . The class of these distributions can be described by a certain class of monotone transforms of the twosided exponential measure. A similar characterization of distributions on R which satisfy Poincar'e inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. 1 Introduction For a number p 2 (0; 1), let ¯ p denote the Bernoulli measure on f0; 1g with probabilities q = 1 \Gamma p and p, respectively, and let ¯ n p denote the corresponding product measure on f0; 1g n . Denote by s i (x) the 'neighbour' of x 2 f0; 1g n obtained b...
Entropy jumps in the presence of a spectral gap
 Duke Math. J
, 2003
"... Abstract It is shown that if X is a random variable whose density satisfies a Poincar'e inequality, and Y is an independent copy of X, then the entropy of (X + Y)/p2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a ..."
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Cited by 12 (4 self)
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Abstract It is shown that if X is a random variable whose density satisfies a Poincar'e inequality, and Y is an independent copy of X, then the entropy of (X + Y)/p2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the BrunnMinkowski inequality (in its functional form due to Pr'ekopa and Leindler). 1 Introduction The informationtheoretic entropy of a real random variable X with density f: R! [0, 1) is defined as
Weighted Poincarétype inequalities for Cauchy and other convex measures
 Annals of Probability
, 2007
"... Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the ..."
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Cited by 9 (1 self)
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Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheegertype isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration. 1. Introduction. The
Weierstrass’ theorem in weighted Sobolev spaces with k derivatives
 J. Math
, 2005
"... Abstract. We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm �f � W k, ∞ (w):= k� �f (j) �L ∞ (wj), j=0 for a wide range of (even nonbounded) weights wj’s. We allow a great deal of independence among the weights wj’s. ..."
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Cited by 7 (5 self)
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Abstract. We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm �f � W k, ∞ (w):= k� �f (j) �L ∞ (wj), j=0 for a wide range of (even nonbounded) weights wj’s. We allow a great deal of independence among the weights wj’s.
Weighted exponential inequalities
, 1992
"... Abstract. Necessary and sufficient conditions on weight pairs are found for the validity of a class of weighted exponential inequalities involving certain classical operators. Among the operators considered are the Hardy averaging operator and its variants in one and two dimensions, as well as the L ..."
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Cited by 6 (0 self)
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Abstract. Necessary and sufficient conditions on weight pairs are found for the validity of a class of weighted exponential inequalities involving certain classical operators. Among the operators considered are the Hardy averaging operator and its variants in one and two dimensions, as well as the Laplace transform. Discrete analogues yield characterizations of weighted forms of Carleman’s inequality.
Spectral Analysis Of Darboux Transformations For The Focusing Nls Hierarchy
, 2002
"... We study Darbouxtype transformations associated with the focusing nonlinear Schrodinger equation (NLS ) and their e#ect on spectral properties of the underlying Lax operator. The latter is a formally selfadjoint (but nonselfadjoint) Diractype di#erential expression of the form d satis ..."
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Cited by 6 (2 self)
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We study Darbouxtype transformations associated with the focusing nonlinear Schrodinger equation (NLS ) and their e#ect on spectral properties of the underlying Lax operator. The latter is a formally selfadjoint (but nonselfadjoint) Diractype di#erential expression of the form d satisfying = M(q) # , where C . As one of our principal results we prove that under the most general hypothesis loc (R) on q, the maximally defined operator D(q) generated by M(q) is actually . Moreover, we establish the existence of WeylTitchmarshtype solutions #+ (z, and # (z, R of M(q)# (z) = z# (z) for z in the resolvent set of D.
Optimal Sobolev Imbeddings Involving RearrangementInvariant Quasinorms
, 1999
"... . Let m and n be positive integers with n 2 and 1 m n \Gamma 1. We study rearrangementinvariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain\Omega in R n , with Lebesgue measure j\Omega j, there corresponds C = C(j\Omega j) ? 0 for which one has the S ..."
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Cited by 5 (0 self)
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. Let m and n be positive integers with n 2 and 1 m n \Gamma 1. We study rearrangementinvariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain\Omega in R n , with Lebesgue measure j\Omega j, there corresponds C = C(j\Omega j) ? 0 for which one has the Sobolev imbedding inequality % R \Gamma u (j\Omega jt) \Delta C%D \Gamma jr m uj (j\Omega jt) \Delta ; u 2 C m 0 (\Omega\Gamma ; involving the nonincreasing rearrangements of u and a certain m th order gradient of u. When m = 1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which % D need not be rearrangementinvariant, % R \Gamma u (j\Omega jt) \Delta C%D ` d d t Z fx2R n : ju(x)j?u (j\Omega jt)g j(ru)(x)j dx ' ; u 2 C 1 0 : In both cases we are especially interested in when the quasinorms are optimal, in the sense that % R cannot be replaced by an essentially larger quasinorm and % D cannot be replaced by an essen...