[For the entire collection see Zbl 0342.00011.]
Several recent results involving the \({\bar \partial}\)-Neumann problem and the \({\bar \partial}_ b\) operator are described in this article, which was originally given as a series of five lectures at Williamstown. Care is taken to make the main ideas of the subject accessible to a large audience, and, accordingly, some older material is included, while many of the more complicated proofs are sketched or only cited.
One of the results described here is as follows. Let \(G\) be a domain in \({\mathbb{C}}^ n \)with smooth boundary \(\partial G\), let \(A=\sum a_ j\partial /\partial \bar z_ j\) be an anti-holomorphic vector field which is tangent to \(\partial G\), and let \(f\in L_ 2(\partial G)\). Let \(H_ b\) denote the orthogonal projection of \(L_ 2(\partial G)\) onto the space of boundary values of holomorphic functions, and let \(\tilde H_ bf\) denote the holomorphic function on G with boundary values \(H_ bf\). Then under an analyticity assumption on the Cauchy-Szegö kernel of \(\partial G\), P. C. Greiner, the author and E. M. Stein have proved [Proc. nat. Acad. Sci. USA 72, 3287-3289 (1975; Zbl 0308.35017)] that the equation A \(*u=f\) has a local solution near \(p\in \partial G\) only if \(\tilde H_ bf\) has a holomorphic extension to a neighborhood of p in \({\mathbb{C}}^ n.\) There is also a \(C^{\infty}\) version of this theorem, and in the case of the Lewy equation a necessary and sufficient condition for local solvability is obtained.
Other topics discussed include: pseudoconvexity and the Levi-problem, the Newlander-Nirenberg theorem, local and global regularity for the \({\bar \partial}\)-equation [the author, Trans. Am. Math. Soc. 181, 273-292 (1973; Zbl 0276.35071); Proc. Nat. Acad. Sci. USA 71, 2912-2914 (1974; Zbl 0284.35055)], and a theorem of L. Boutet de Monvel on the imbedding of CR manifolds [Sémin. Goulaouic-Lions-Schwartz 1974-1975: Équat. dériv. part. lin. nonlin., Exposé IX, 13 p. (1975; Zbl 0317.58003)].