Some Applications of Integral Formulas in Several Complex Variables

Abstract: The thesis gives three examples of how integral formulas can be used in complex analysis.

We say that a set S in Cⁿ is quadratically convex if through any point in its complement there is a quadratic hypersurface that does not intersect S. A quadratically convex set is called strongly quadratically convex if a certain generalization of the Fantappie transform (related to the set) is surjective. In the first paper, we prove that certain sets are strongly quadratically convex, and use the Cauchy-Fantappie-Leray integral representation formula for holomorphic functions to obtain an explicit inversion formula for the related transform.

Let D be a domain in Cⁿ. Assume that the functions g₁,...,g_m are holomorphic and bounded in D, and that there is a constant .delta. such that 0<.delta.<=|g₁|+...+|g_m|. The H^p corona problem in D is the following: Given f in H^p(D), find u_i in H^p(D) such that f=g₁ u₁+...+g_m u_m. In the second paper, we use a weighted integral representation formula for holomorphic functions to solve this problem in the case where D is a non-degenerate analytic polyhedron.

In the third paper, we study solution operators for the d-bar operator that are canonical with respect to a certain metric in strictly pseudoconvex domains. Some features of the metric and the canonical operators are investigated. We show that certain known explicit solution operators in a sense approximately are the canonical ones. This enables us to draw conclusions concerning e.g. regularity properties of the canonical operators.