The aim of this project is the analysis of classes of holomorphic functions of several complex variables. The regularity of holomorphic functions defined in a domain becomes less definite as the independent variable approaches the boundary of the domain. This boundary behavior is the main thrust of this activity. Boundary properties of functions defined in a disc in one complex dimension are well understood. A natural analogue in several dimensions is the ball. Even in this case, information is far from complete. In the study of holomorphic Sobolev and Besov spaces, it can be shown that functions have unique limits on approach to the boundary provided certain exceptional sets are avoided. Some progress has been made in characterizing exceptional sets in terms of their capacity. Further effort will be devoted to refining these results. A related line of study will consider the complex tangential derivative of functions in Lipschitz classes. It appeared that when the Lipschitz constant is equal to one-half, the tangential derivatives in the class are p-th power integrable for all values of the parameter. This turned out to be false, leading to a reconsideration of how functions in this class behave near the boundary. Work will be done on the question. A second line of investigation concerns the restriction of holomorphic Sobolev spaces to complex tangential manifolds. There is ample evidence to suggest that the restricted classes are exactly the real Besov spaces. Work will continue on the problem of determining when a function in such a Besov space can be extended to a holomorphic function of the same type.
