Let be a -convex intersection, , , in a complex manifold of complex dimension , , and let be a holomorphic vector bundle of rank over . In this paper, -estimates, , for solutions to the -equation with small loss of smoothness are obtained for -valued -forms on when . In addition, we solve the -equation with a support condition in -spaces. More...
Let be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form where is compact. Assume that the characteristic fiber of has the compact approximation property. Let be the complex dimension of and . Then: If is a holomorphic vector bundle (of finite rank) with , then . In particular, if , then .
We introduce the notion of the Shilov boundary for some subfamilies of upper semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subfamilies with simple structure we show the existence and uniqueness of the Shilov boundary. We provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of...
estimates for convex domains of finite type in are known from [7] for and from [2] for . We want to show the same result for concave domains of finite type. As in the case of strictly pseudoconvex domain, we fit the method used in the convex case to the concave one by switching and in the integral kernel of the operator used in the convex case. However the kernel will not have the same behavior on the boundary as in the Diederich-Fischer-Fornæss-Alexandre work. To overcome this problem...