# On Complex Hypercontractivity

@article{Janson1997OnCH,
title={On Complex Hypercontractivity},
author={Svante Janson},
journal={Journal of Functional Analysis},
year={1997},
volume={151},
pages={270-280}
}
• S. Janson
• Published 1 December 1997
• Mathematics
• Journal of Functional Analysis
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.
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## References

SHOWING 1-10 OF 18 REFERENCES

LP(#) into Lq(/t), where/z is the Gauss measure, for co real, co2~ p 1 1 < p < q < oo. q l ' This has been extended to imaginary 0) by Beckner [3] (this enabled him to give a sharp version of the
I Preliminaries.- II Semimartingales and Stochastic Integrals.- III Semimartingales and Decomposable Processes.- IV General Stochastic Integration and Local Times.- V Stochastic Differential
1. Gaussian Hilbert spaces 2. Wiener chaos 3. Wick products 4. Tensor products and Fock spaces 5. Hypercontractivity 6. Distributions of variables with finite chaos expansions 7. Stochastic
A Gaussian integral kernelG(x, y) onRn×Rn is the exponential of a quadratic form inx andy; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound ofG as an