Harmonic Integrals on Strongly Pseudo-Convex Manifolds: II

In this paper we prove the basic existence and regularity theorems for the a-Neumann problem (see Theorems 6.6 and 6.14). The results presented here were outlined by the author in 8. In Part I of this work (see 7) we established some of the fundamental properties of the aNeumann problem and, on the assumption of existence and regularity, we obtained several applications. A variant of the 8-Neumann problem was first formulated in 3. D. C. Spencer and the author studied the problem by means of singular integral equations in 5. The starting point for the author's work (see 6 and 7) is the estimate (1.6), a special case of this estimate was first established by C. B. Morrey in 9. In his thesis, (see 1) M. E. Ash has derived estimates relative to moving frames. This method has enabled him to generalize our work (see also 2). The introduction of moving frames is also very useful in the present work. The method of proving regularity by studying the families of norms depending on the parameters a and zwas suggested to the author by L. Hbrmander; essential use is made of some of the results (stated in Ch. 4) which are developed in his book (see 4). The T-norms of Ch. 3 have been also introduced for a different purpose by Andreotti and Vesentini. They have obtained, by the argument of 6 for forms with values in a holomorphic vector bundle over a strongly pseudoconvex domain of Cn, an inequality which contains the inequality of Proposition 3.5 as a particular case. In Ch. 7 we show how the solution of the 8-Neumann problem implies the solution of several boundary value problems which were posed in 5. The methods developed here and in 7 can be used to prove existence and regularity theorems for very general elliptic over determined systems. We shall return to this question in a future publication.