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36
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Risk communication
 Proceedings of the national conference on risk communication, Conservation Foundation,Washington, DC
, 1987
"... We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, OO. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which t ..."
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Cited by 29 (1 self)
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We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x) 1.x‘.:Ta 1991 Academic Press, Inc. 1.
Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications
 J. Funct. Anal
, 1994
"... In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = ..."
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Cited by 23 (12 self)
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In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = \Delta H is the GrossLaplacian, (E; H; fl) is an abstract Wiener space and B = \Gammaid E +v where v takes values in the CameronMartin space H . Using Gross' logarithmic Sobolevinequality in an essential way we show that ¯ is always absolutely continuous w.r.t. the Gaussian measure fl and that the square root of the density is in the Malliavin test function space of order 1 in L 2 (fl). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L ¯ = 0. These include results on the "inverse problem", i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove ...
Optimal Hypercontractivity for Fermi Fields and Related NonCommutative Integration Inequalities
, 1992
"... : Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of noncommutativ ..."
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Cited by 15 (1 self)
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: Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of noncommutative integration are established. On leave from School of Math., Georgia Institute of Technology, Atlanta, GA 30332 Work supported by U.S. National Science Foundation grant no. PHY9019433A01. 1 I. INTRODUCTION Observables pertaining to the configuration of a quantum system with n degrees of freedom are operators Q 1 ; Q 2 ; : : : ; Q n which, depending on the system, may or may not commute. Our main concern is with the case in which the configuration variables are amplitudes of certain field modes. For boson fields, these configuration observables do commute, and the state space H can be taken as the space of all complex square integrable functions on their joint spectrum. This is t...
A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices
 Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832
, 2003
"... Summary. We point out a simple argument relying on hypercontractivity to describe tail inequalities on the distribution of the largest eigenvalues of random matrices at the rate given by the Tracy–Widom distribution. The result is illustrated on the known examples of the Gaussian and Laguerre unitar ..."
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Cited by 12 (3 self)
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Summary. We point out a simple argument relying on hypercontractivity to describe tail inequalities on the distribution of the largest eigenvalues of random matrices at the rate given by the Tracy–Widom distribution. The result is illustrated on the known examples of the Gaussian and Laguerre unitary ensembles. The argument may be applied to describe the generic tail behavior of eigenfunction measures of hypercontractive operators.
Strong Haagerup inequalities for free Rdiagonal elements
 J. FUNCT. ANAL
, 2007
"... In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give ..."
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Cited by 11 (6 self)
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In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of Rdiagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for Rdiagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem, generalizing and improving the one in [Bi2].
Hypercontractivity in noncommutative holomorphic spaces
 Commun. Math. Phys
, 2005
"... ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSeg ..."
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Cited by 8 (6 self)
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ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSegalBargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.
Exact Smoothing Properties of Schrödinger Semigroups
, 1997
"... We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generica ..."
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Cited by 7 (0 self)
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We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generically smoother by exactly two derviatives (in given Sobolev spaces) than their potentials. We give applications to the relation between the potential's smoothness and particle kinetic energy in the context of quantum mechanics, and characterize kinetic energies in Coulomb systems. The techniques of proof invove Leibniz and chain rules for fractional derivatives which are of independent interest, as well as a new characterization of the Kato class. 1 Introduction In this paper we attempt a precise study of the action of Schrodinger semigroups in the scale of Sobolev spaces. Work in this area was begun by B. Simon [Si4] in 1985. He proved a number of positive and negative results on such smooth...
Deviation Inequalities on Largest Eigenvalues
 GAFA Seminar Notes
, 2005
"... In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known nonasymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and function ..."
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Cited by 6 (0 self)
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In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known nonasymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and functional analysis to reach some of these quantitative inequalities at the correct small deviation rate of the fluctuation theorems. Results in this direction are rather fragmentary. For simplicity, we mostly restrict ourselves to Gaussian models. 1