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Some Applications of Integral Formulas in Several Complex Variables

Doktorsavhandling
Författare Jörgen Boo
Förlag Chalmers University of Technology
Förlagsort Göteborg
Publiceringsår 1996
Publicerad vid Institutionen för matematik
Språk en
Ämneskategorier Matematik

Sammanfattning

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,...,gm are holomorphic and bounded in D, and that there is a constant .delta. such that 0<.delta.<=|g1|+...+|gm|. The Hp corona problem in D is the following: Given f in Hp(D), find ui in Hp(D) such that f=g1 u1+...+gm um. 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.

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