Duality and Hardy Spaces on Levi-Flat Domains with Corners

Abstract

In this thesis we develop a Hardy space theory for piece-wise smooth Levi-flat domains. We use the Cauchy-Fantappie pairing to produce projectively invariant reproducing kernels for such domains, along with families of measures to define Hardy spaces on the original domain and its dual. We study the qualitative properties of the pairing, and show that many of the properties from the smooth, strongly C-convex case continue to hold. However, the maps between Hardy spaces are in general not isomorphisms, but are injective with dense image. Lastly, we demonstrate how to construct projectively invariant Hardy spaces in two variables.