Reproducing kernels and potential theory for the Bergman spaces

Abstract: The role of weighted biharmonic Green functions in weighted Bergman spaces was first studied in the beginning of the 50's by Paul Garabedian. In 1951 he showed that they are closely related to reproducing kernel functions of weighted Bergman spaces. Half a century later, the properties of biharmonic functions turned out to be crucial to the factorization theory of Bergman spaces on the unit disk. This thesis consists of a summary and three chapters, each one a self-contained article, in which we present some results in weighted Bergman spaces based in the properties of a weighted biharmonic Green function. In Chapter 1, we present the article "Mean value surfaces with prescribe curvature form" , J. Math. Pures Appl. 83 (2004), 1075-1107, by H. Hedenmalm and Y. Perdomo. Chapter 2 is the preprint "A Riesz representation formula for weighted super-biharmonic functions", (2005), also by H. Hedenmalm and Y. Perdomo. And Chapter 3 constitutes the preprint "A monotonicity property of a weighted biharmonic Green function", (2005), by Y. Perdomo.