Interpolation problems in Dirichlet type spaces

Abstract

This thesis deals with interpolation problems in spaces of analytic functions of Dirichlet type, that is spaces in which the corresponding norm is defined in terms of some integral involving (partial) derivatives of the function. These problems can be categorized roughly into two categories. The first one is problems of interpolation by sequences where one tries to find a function in a given space such that on prescribed points assumes prescribed values. The second is boundary interpolation where one prescribes the values of the function on a set that sits in the boundary of the natural domain of definition. In the first part we explore simply interpolating sequences for the Dirichlet space. We manage to give a partial answer to a problem first raised by Bishop. We prove that for sequences which satisfy the so called Shapiro–Shields condition there exists a potential theoretic characterization of when they are simply interpolating. In the second part we study random interpolating sequences for the standard weighted Dirichlet spaces. We are able to prove 0–1 Kolmogorov type characterizations for universally interpolating, zero and separated sequences for this class of spaces. Our results generalize and improve upon previous results of Rudowicz Bogdan and Cochran. Finally, in the last part we work with Hardy–Sobolev spaces in the unit ball of n complex variables. Our results concern the equivalence of two notions of “negligibility”of sets on the boundary of the ball. One notion is classical and is that of capacity zero sets and the second notion is that of a totally null set. Our main result shows that at least for compact sets these two notions coincide. Our result can be applied to improve a result on boundary interpolation of Cohn & Verbitsky.