Weyl symbolic calculus on any Lie group
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 Lie algebra of upper triangular matrices; \((S_ t)\) is then a one parameter group of diffeomorphisms. For arbitrary \(\mathfrak g\), a natural construction of \(S_ t\) comes out by contraction of \(\mathfrak g\) onto an abelian subalgebra, strengthening the above conditions to \(x(t)- y(t) = x-y\), \(x(t)+y(t) = x._ ty\), where the last symbol means the Campbell-Hausdorff function for the contracted bracket \(t[x,y]\).
scientific article; zbMATH DE number 219669
scientific article