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.