On complex hypercontractivity
type: Item
instance of: scholarly article
title: On complex hypercontractivity
zbMATH Open document ID: 0893.46021
author: Svante Janson
published in: Journal of Functional Analysis
publication date: 1998-08-03T00:00:00Z
review text: Let \(X\) be an \(N(0,1)\) random variable defined on a probability space \((\Omega,{\mathcal F},P)\) and regarded as an element of \(L^2(P)\). The standard Gram-Schmidt orthogonalization process applied to the sequence \(1,X,X^2,X^3,\dots\) in \(L^2(P)\) produces the sequence \(h_0= 1\), \(h_1= X\), \(h_2= X^2-1\), \(h_3= X^3- 3X,\dots\) of Hermite polynomials. For \(p\geq 1\), let \(L^p(X)\) denote the closed linear span over \(\mathbb{C}\) of the \(h_n\) with respect to the \(L^p\)-norm. The \(h_n\) form a basic sequence in this space. Given \(z\in\mathbb{C}\), define the Mehler transform \(M_z: L^p(X)\to L^q(X)\) by \(M_z\left(\sum^\infty_{n=0} a_nh_n\right)= \sum^\infty_{n= 0} a_nz^n h_n\). The main result of this paper is a new proof of a theorem by Weissler and Epperson which runs as follows: Let \(1\leq p\leq q<\infty\). Then we have \(\| M_z\|= 1\) if and only if (i) \(| z|^2\leq p/q\), and (ii) \((q- 1)| z|^4- (p+q-2)(\text{Re }z)^2- (pq- p-q+ 2)(\text{Im }z)^2+ (p-1)\geq 0\). Moreover, if these conditions fail, then \(M_z\) is not even a bounded operator. Using the notion of Wick product, the author then greatly extends this result from a single \(X\) to a (closed) subspace of \(L^2(P)\) consisting of centred Gaussian random variables. Such a space is called a Gaussian-Hilbert space [see his ``Gaussian-Hilbert spaces'' (Cambridge, 1997), Chapter III], and it is natural to consider two such spaces, \(H\) and \(H'\), connected by a basic operator \(A: H\to H'\). (With the single \(X\), we have \(H= H'=\) linear span of \(X\), and \(A= zI\), where \(I\) is the identity operator on \(H\)).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 1106597
zbMATH Keywords: stochastic calculus
zbMATH Keywords: random variable
zbMATH Keywords: Gram-Schmidt orthogonalization process
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Mehler transform
zbMATH Keywords: Wick product
zbMATH Keywords: Gaussian-Hilbert space
MaRDI profile type: Publication
OpenAlex ID: W2011198658
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DOI: 10.1006/JFAN.1997.3144
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description: scientific article; zbMATH DE number 1106597