Extension properties of square integrable holomorphic functions.

We study the extension properties of square integrable holomorphic functions in several complex variables. This has two parts. We begin by characterizing the bounded domains where there is no extension, the bounded L2h -domains of holomorphy. We give a complete characterization of such domains in terms of the boundary points. In the case of Riemann domains we use Grauert's notion of boundary points to give a similar characterization if the projection is a bounded domain. The main tools in this study are results from pluripotential theory and the Ohsawa-Takegoshi extension theorem. The last chapter of this dissertation gives partial results in comparing these domains to domains which are convex with respect to the functions. The second part studies the domains where there is extension of all functions. For bounded domains, the above mentioned results give us a fairly complete description of the L2h -envelopes, the largest (Riemann) domains to which all square integrable holomorphic functions extend holomorphically. We provide examples showing that we cannot hope for a much more complete description. In particular we show that the L2h -envelope does not have to be an L2h -domain of holomorphy.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132122/2/3057972.pd