Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator
type: Item
instance of: scholarly article
title: Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator
zbMATH Open document ID: 1094.47046
published in: Annals of Global Analysis and Geometry
publication date: 2005-12-14T00:00:00Z
review text: Let \(Z\) and \(\overline Z\) be the linear partial differential operators on \(\mathbb R^2\) given by \(\overline Z=(\partial/\partial z)+ (1/2)\overline z\) and \(\overline Z=(\partial/\partial{\overline z})-(1/2)z\), \(z=x+iy\), \((x,y)\in\mathbb R^2\). Let \(L=-(1/2)(Z\overline Z+\overline Z Z)\); this operator is closely related to the sub-Laplacian on the Heisenberg group; moreover, it is elliptic but neither globally elliptic nor Shubin globally hypoelliptic [see \textit{M.~A.\ Shubin}, ``Pseudodifferential operators and spectral theory'' (Nauka, Moscow) (1978; Zbl 0451.47064); English translation: (Springer-Verlag, Berlin--New York) (1987; Zbl 0616.47040)]. The author derives a formula for the heat kernel of \(L\) in terms of Weyl transforms of functions in the Schwartz space \({\mathcal S}(\mathbb R)\) and the Fourier--Wigner transforms of Hermite functions, which leads to a formula for the Green function of~\(L\) (the integral kernel of~\(L^{-1}\)) and helps to show that, if \(u\) is a tempered distribution on \(\mathbb R^2\) such that \(Lu\) is in \({\mathcal S}(\mathbb R^2)\), then \(u\in{\mathcal S}(\mathbb R^2)\); the author refers to this very property as the global hypoellipticity. The author also proves that every element of the strongly continuous one-parameter semigroup generated by \(-L\) defines a bounded operator from \(L^2(\mathbb R^2)\) to \(L^\infty(\mathbb R^2)\) (this property of the semigroup is referred to as ultracontractivity) and a bounded operator from \(L^p(\mathbb R^2)\) to \(L^q(\mathbb R^2)\) for \(1\leq p\leq2\) and \(2\leq q\leq\infty\) (which implies the hypercontractivity of the semigroup). The study of the operator \(L\) seems to be of interest from the point of view of representation theory of the Heisenberg group.
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 2238702
zbMATH Keywords: Weyl transforms
zbMATH Keywords: heat kernel
zbMATH Keywords: Green function
zbMATH Keywords: global hypoellipticity
zbMATH Keywords: ultracontractivity
zbMATH Keywords: hypercontractivity
author: M. W. Wong
reviewed by: A. I. Shtern
MaRDI profile type: Publication
full work available at URL: s10455-005-1148-x
OpenAlex ID: W2066537673
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DOI: 10.1007/S10455-005-1148-X
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description: scientific article