A relative index on the space of embedabble CR-structures. II. (English) Zbl 0942.32026

This paper contains the delicate analysis of the Szegő projector and the partial inverse for the \(\square_b\)-operator defined by a strictly pseudoconvex CR-structure on a compact manifold used in Part I [the author, ibid. 147, No. 1, 1–59 (1998; Zbl 0942.32025].
The main tool is the so-called Heisenberg calculus of pseudodifferential operators on the Heisenberg group. Following a review of the basic definitions, the author studies model problems of the form \((\text{ Id}+\beta A)\), where \(A\) is a right-invariant convolution operator of order zero of the class \(\Phi_0^0\) of M. E. Taylor [‘Noncommutative microlocal analysis’, Mem. Am. Math. Soc. 313, 182 p. (1984; Zbl 0554.35025)]. If it is assumed that these operators are elliptic and invertible as maps from \(L^2\) to \(L^2\) for \(|\beta|<r\), then the inverses are also model operators of this class. Then the family of inverses depends smoothly on the parameter \(\beta\) by the results of Rothschild and Stein. Then the method of R. Beals and D. Greiner [‘Calculus on Heisenberg manifolds’, Annals of Mathematics Studies, 119. Princeton, NJ: Princeton University Press (1988; Zbl 0654.58033)] is used to construct both an approximate partial inverse for \(\square_b\) and a Szegő projector for a CR-structure on a compact 3-manifold. The dependence of these operators on the CR-structure is analyzed with care.
Finally the author obtains parametrices for some of the operators introduced in Part I (in the perturbation analysis).
Reviewer: J.S.Joel (Kelly)


32V30 Embeddings of CR manifolds
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32V05 CR structures, CR operators, and generalizations
58J20 Index theory and related fixed-point theorems on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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