Operators associated with the Hermite semigroup - a survey
type: Item
instance of: scholarly article
title: Operators associated with the Hermite semigroup - a survey
zbMATH Open document ID: 0916.42014
DOI: 10.1007/BF02656487
author: Peter Sjögren
published in: The Journal of Fourier Analysis and Applications
publication date: 1999-07-14T00:00:00Z
full work available at URL: https://eudml.org/doc/59538
review text: Let \(R^n\) be the Euclidean space equipped with the normalized Gaussian measure \(\pi^{-n/2} e^{-x^2} dx\). The symmetric Laplacian \(L\) on this space can be spectrally decomposed using the Hermite polynomials, and the heat operator \(e^{-tL}\) computed; a detailed derivation appears in this paper. The \(L^p\) and weak-type \((1,1)\) for the maximal operator \[ M^* f(x) = \sup_{t > 0} | e^{-tL} f(x)| \] and Riesz operators \(D^\alpha L^{-b} \Pi_0\) are summarized for both finite and infinite dimensions; here \(D\) is a differentiation operator \(b = 2| \alpha| \), and \(\Pi_0\) is the orthogonal projection to the complement of the \(0\) eigenspace. A new proof of the weak (1,1) type of second-order Riesz operators in finite dimensions is then proved, based on a precise computation of the kernel and its decomposition into local and global portions.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1143768
zbMATH Keywords: Hermite semigroup
zbMATH Keywords: Riesz operators
zbMATH Keywords: maximal operators
zbMATH Keywords: heat operator
zbMATH Keywords: weak-type \((1,1)\)
reviewed by: Terence Tao
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1143768