# On Complex Hypercontractivity

@article{Janson1997OnCH, title={On Complex Hypercontractivity}, author={Svante Janson}, journal={Journal of Functional Analysis}, year={1997}, volume={151}, pages={270-280} }

Abstract We give a new proof of a hypercontractivity theorem for the Mehler transform with a complex parameter, earlier proved by Weissler (1979, J. Funct. Anal. 32 , 102–121) and Epperson (1989, J. Funct. Anal. 87 , 1–30). The proof uses stochastic integrals and Ito calculus. The method also yields new proofs of some related results.

## 29 Citations

### Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups

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We show that for a hypoelliptic Dirichlet form operator A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly…

### Strong Hypercontractivity and Relative Subharmonicity

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Abstract A Hermitian metric, g , on a complex manifold, M , together with a smooth probability measure, μ , on M determine minimal and maximal Dirichlet forms, Q D and Q max , given by Q ( f )=∫ M g…

### Optimal holomorphic hypercontractivity for CAR algebras

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For t < t(r, p), e−tAγ is not bounded from Lr to Lp. Gross used this inequality to show that the boson energy operator in a model of 2-dimensional Euclidean quantum field theory has a unique ground…

### Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups

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### Reverse inequalities in μ-deformed Segal-Bargmann analysis

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We prove reverse hypercontractivity inequalities as well as reverse log-Sobolev inequalities in the context of a space of holomorphic functions, which is called the μ-deformed Segal-Bargmann space…

### Reverse hypercontractivity over manifolds

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Suppose thatX is a vector field on a manifoldM whose flow, exptX, exists for all time. If μ is a measure onM for which the induced measuresμt≡(exptX)*μ are absolutely continuous with respect to μ, it…

### Optimal ultracontractivity for R-diagonal dilation semigroups

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- 2007

This paper contains sharp estimates for the small-time behaviour of a natural class of one-parameter semigroups in free probability theory. We prove that the free Ornstein-Uhlenbeck semigroup Ut,…

### Comparison of moments of Rademacher chaoses

- MathematicsArkiv för Matematik
- 2019

We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete…

### Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

- MathematicsCanadian Journal of Mathematics
- 2015

Abstract We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic ( $\text{LSH}$ ) functions. We…

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