
University of Helsinki, Helsinki 2006 Aspects of atomic decompositions and Bergman projectionsTeemu HänninenDoctoral dissertation, August 2006. The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space Linfinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LVinfinity consisting of measurable and HVinfinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LVinfinity onto HVinfinity and, in some sense, the space HVinfinity is the smallest possible substitute to the space Hinfinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised ForelliRudin estimates instead of integral representations. The minimality of the space LVinfinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HVinfinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HVinfinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe coechelon space. The title page of the publication
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