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Calculus on Heisenberg manifolds. (English) Zbl 0654.58033
Annals of Mathematics Studies, 119. Princeton, NJ: Princeton University Press. 200 p. (1988).
The most basic elliptic operator on a Riemannian manifold is the Laplacian \(\Delta\). To find a partial inverse one usually considers first, working locally, the operator \(\Delta^ x\) which is \(\Delta\) frozen at x. The intimate relation of \(\Delta^ x\) with the group structure of the coordinate space \({\mathbb{R}}^ n\) shows up in its translational invariance and allows a convolution kernel as a first approximation to the kernel of the parametrix for \(\Delta\). On some manifolds there are other basic operators. The motivating example for the operator calculus which is developed in this book is a CR manifold with the operator \(\square_ b\)- not elliptic but hypoelliptic with loss of one derivation. The authors consider the more general situation where the coordinate space \({\mathbb{R}}^{n+1}\) is isomorphic as a group to \(H_ m\times {\mathbb{R}}^{n-2m}\) with \(H_ m\) a \((2m+1)\)-dimensional Heisenberg group, and this splitting is associated to an operator P whose best approximation \(P^ x\) at x is left invariant and plays a role analogous to \(\Delta^ x.\)
The first chapter of the book introduces this general setting and states conditions under which P is hypoelliptic with loss of one derivative. It turns out that a necessary and sufficient condition is invertibility of the model operator \(P^ x\) at each point x. In chapter 2 the inverse \(Q^ x\) of \(P^ x\) is calculated with expressions for its kernel and its symbol. Partial inverses are also found in special cases even if \(P^ x\) is not invertible.
Chapter 3 starts with a survey of pseudodifferential operator calculus. Then a class of pseudodifferential operators is introduced which contains the parametrices of operators like P from chapter 1. It is a subclass of type (1,1)-operators associated to a codimension one subbundle \(\nu\) of the tangent bundle. As for classical \(\psi\) DO’s there is a complete asymptotic calculus allowing compositions and adjoints to be defined. Parametrices can also be constructed, and lead to a Fredholm theory for such \(\nu\)-operators on a compact manifold.
In the last chapter this theory is applied to CR manifolds. Recalling that a smooth manifold M of dimension \(2n+1\) is a Cauchy-Riemann manifold if TM\(\otimes {\mathbb{C}}\) contains an integrable complex rank n sub-bundle \(T_{1,o}\) with \(T_{1,o}\oplus \bar T_{1,o}=\{0\}\), then a \(\nu\)- structure is naturally defined by taking \(\nu =T_{1,o}\oplus \bar T_{1,o}\). Denoting by N the complementary line bundle one defines \(\Lambda^{o,1}=(T_{1,o}\oplus N)^{\perp}\subset T^*M\otimes {\mathbb{C}}\) and obtains the Cauchy-Riemann operator \({\bar \partial}_ b: C^{\infty}(\Lambda^{o,q})\to C^{\infty}(\Lambda^{o,q+1})\) with formal adjoint \(\vartheta_ b\) and associated Laplacian, the Kohn operator \(\square_{b,q}=\vartheta_ n{\bar \partial}_ b+{\bar \partial}_ b\vartheta_ b\). \(\square_{b,q}\) is proved to be hypoelliptic and a parametrix is constructed using the theory of \(\nu\)- operators if \(\square^ x_{b,q}\) is invertible.
In a final section there is a different approach to the construction of a partial inverse to \(\square_{b,q}\) which can be used to show that the Cauchy-Szegö projection on the boundary of a strictly pseudo-convex domain in \({\mathbb{C}}^ 2\) is a \(\nu\)-operator.
This is an accurate book with all of the technical machinery laid open in detail. A good bibliography refers to previous work on the subject, a great part being contributed by the authors. Needless to say that this theory will have (and to some extent already had) a strong impact on the further development of analysis on CR manifolds, cf. e.g. Getzler’s proof of the Boutet de Monvel index theorem for Toeplitz operators on the boundary of a strictly pseudo-convex domain (to appear).
Reviewer: H.Schröder

58J40 Pseudodifferential and Fourier integral operators on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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