The Szego Projection: Sobolev Estimates in Regular Domains

The Szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of Diederich, Fornaess, and Catlin. It preserves local smoothness near boundary points of finite type. The Bergman projection, being linked to the 3 problem, plays an important role in several complex variables, as demonstrated by recent applications to the boundary behavior of holomorphic maps [7,16]. The Szego projection is interesting both by analogy and by its connection with the dh problem and hence with unsolvable equations of Lewy type. While in strictly pseudoconvex domains the Bergman and Szego projections are both well understood, in general much more is known about the Bergman projection. This is no surprise, for the theory of the 3 problem is further advanced than the theory of the 3^ problem. Nonetheless I believe that the Szego projection ought not be any more mysterious than the Bergman projection. This paper is intended to support this contention. The most important domains for which the Bergman projection is known to be regular are domains with a good 3-Neumann operator. Here it is shown that in such domains the Szego projection also is regular. More precisely, in smooth bounded pseudoconvex domains satisfying Catlin's property (P) (the most general condition presently known that guarantees global regularity of the Neumann operator) the Szego projection is shown to be globally regular (Theorem 4.1). Moreover at boundary points of finite type the Szego projection is locally regular (Theorem 5.1). In both cases the Szego projection exactly preserves the degree of differentiability, measured in Sobolev norms. The method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case, for there are serious obstacles to such a generalization. For instance the approach of analysis on the Heisenberg group, used by Phong and Stein [29] to derive estimates for both the Bergman and Szego projections, cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages [28,15]. Similarly one does not know how to carry over the Fourier integral operator approach of Boutet de Monvel and Sjostrand [10] to the weakly pseudoconvex case. Explicit construction of Cauchy-Fantappie forms, Received by the editors March 11, 1986. Paper presented April 11, 1986 to the 826th meeting of the American Mathematical Society, Indianapolis, special session on Several Complex Variables. 1980 Mathematics Subject Classification (1985 Revision). Primary 32A25, 32H10. Partially supported by National Science Foundation grants MCS-8201063 and DMS-8501758. ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page 109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use