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61
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, 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 fo ..."
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Cited by 49 (2 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 34 (16 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
 Comm. Math. Phys
, 1993
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A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices
 S¶eminaire de Probabilit¶es XXXVII. Lecture Notes in Math. 1832
, 2003
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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 15 (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 14 (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.
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 13 (1 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
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 13 (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...