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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

##### MSC:
 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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