Weyl symbolic calculus on any Lie group
type: Item
instance of: scholarly article
title: Weyl symbolic calculus on any Lie group
zbMATH Open document ID: 0779.22005
DOI: 10.1007/BF00992755
author: Dominique Manchon
published in: Acta Applicandae Mathematicae
publication date: 1993-06-29T00:00:00Z
review text: Motivated by the links between the Weyl calculus of pseudodifferential operators on \(\mathbb{R}^ n\) and unitary representations of the \(2n+1\)- dimensional Heisenberg group, the author develops a similar calculus related to an arbitrary Lie group \(G\). In forthcoming work, she will apply it to obtain a Weyl spectral asymptotic formula for elliptic operators. Let \(\mathfrak g\), resp. \({\mathfrak g}^*\), denote the Lie algebra of \(G\), resp. its dual space. Let \(F\) be the classical Fourier transform between functions on \(\mathfrak g\) and \({\mathfrak g}^*\), and let \(\pi\) be a unitary representation of \(G\) on a Hilbert space \(H\). To a (suitable) function \(p(\xi)\) of \(\xi\in {\mathfrak g}^*\), one can associate the operator \(p^ \pi\) on \(H\): \[ p^ \pi =\int_{\mathfrak g}(F^{-1}p)(x)\pi(\exp x)dx, \] called the operator with Weyl symbol \(\pi\). The composed operator \(p^ \pi\circ q^ \pi\) can easily be written as \((p\#q)^ \pi\), where \[ (p\#q)(\xi) =\int_{{\mathfrak g}\times{\mathfrak g}}(F^{-1}p)(x)(F^{- 1}q)(y)e^{-i\langle\xi,x.y\rangle}dxdy; \] here \(x.y =\log(\exp x\exp y)\), given by the Campbell-Hausdorff formula. The main result of the paper states that the product \(\#\) gives a continuous bilinear map between symbol classes of the type \(AS_ \rho^{m,K}\), consisting of \(C^ \infty\) functions \(p\) such that \(\text{supp }F^{-1}p\subset K\) (a compact neighborhood of 0 in \(\mathfrak g\)) and any \(\alpha\)-th derivative of \(p(\xi)\) has polynomial growth of order \(m-\rho|\alpha|\). Asymptotic expansions of symbols can be used in those classes. The proof uses a one parameter family of diffeomorphisms \(S_ t\) of \({\mathfrak g}\times{\mathfrak g}\) onto itself (near the origin), such that \(S_ 0\) is the identity and, setting \(S_ t(x,y) = (x(t),y(t))\) for \(0\leq t\leq 1\): \(x(t)-y(t) = x-y\), \(x(1)+y(1) = x.y\). When \(\mathfrak g\) is solvable, \(S_ t\) is explicitly built by means of an imbedding of \(\mathfrak g\) into the …
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 219669
zbMATH Keywords: Weyl calculus
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: unitary representations
zbMATH Keywords: Weyl spectral asymptotic formula
zbMATH Keywords: elliptic operators
zbMATH Keywords: Lie algebra
zbMATH Keywords: Campbell-Hausdorff formula
zbMATH Keywords: diffeomorphisms
reviewed by: François Rouvière
MaRDI profile type: Publication
Wikidata QID: Q115394641
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description: scientific article; zbMATH DE number 219669