Analytic pseudodifferential operators for the Heisenberg group and local solvability. (English) Zbl 0695.47051

Geller, Daryl

Mathematical Notes, 37. Princeton, NJ: Princeton University Press. 495 p. $ 29.50 (1990).

The author develops a calculus of pseudodifferential operators for the Heisenberg group \({\mathbb{H}}^ n\), in the (real) analytic setting, and applies this calculus to study of certain operators arising in several complex variables. The main new application is the preservation of analyticity by the Szegö projection. Other new results are a very precise form of an analytic parametrix for the \(\square_ b\) on any nondegenerate analytic CR manifold, an analytic calculus on \({\mathbb{H}}^ n\) natural for dealing with \(\square_ b\) and operators like it, a generalization of the theory of operators like the Folland-Stein operators \(L_{\alpha}\) (\(\alpha\in {\mathbb{C}})\) beyond the study of differential operators, a characterization of the Fourier transform of the space \(\{\) \(K\in {\mathcal S}({\mathbb{R}}^ n):\) K is homogeneous (with respect to a given dilation structure) and analytic away from \(0\}\), a generalization to the study of operators like \(L_{\alpha}\) but for \(\alpha \in \pm \{n,n+2,...\}\). The author places absolute emphasis on the core \(K_ u(w)=(2\pi)^ n\int e^{-iw\cdot \xi}a(u,\xi)d\xi,\) as opposed to the kernel or the symbol, of pseudodifferential operators. The proofs are very detailed and the prerequisites are few.
The contents is as follows: Introduction, Homogeneous distributions, The space \(Z^ q_{q,j}\), Homogeneous partial differential equations, Homogeneous partial differential operators on \({\mathbb{H}}^ n\), Homogeneous singular integral operators on \({\mathbb{H}}^ n\), An analytic Weyl calculus, Analytic pseudodifferential operators on \({\mathbb{H}}^ n\), Analytic parametrices, Applying the calculus, Analytic pseudolocality of the Szegö projection and local solvability, and References.

Reviewer: R.Vaillancourt

MSC:

47Gxx	Integral, integro-differential, and pseudodifferential operators
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

calculus of pseudodifferential operators; Heisenberg group; preservation of analyticity by the Szegö projection; Folland-Stein operators; Fourier transform; symbol; Homogeneous distributions; Homogeneous partial differential equations; Homogeneous singular integral operators; analytic Weyl calculus; Analytic parametrices; Analytic pseudolocality of the Szegö projection; local solvability