Multipliers and extremal problems in Bergman spaces.

Abstract

In this thesis we consider two topics in the theory of Bergman spaces $A\sp{p}.$. We study the existence and uniqueness of functions for which the norm of a bounded linear functional on $A\sp{p}$ is attained. In the general case, by means of techniques from approximation theory, we prove a theorem which characterizes the extremal function in terms of an integral condition and relates it to a certain Sobolev space. In some particular cases, we are able to determine the extremal function explicitly by making use of this theorem or of involutive subjective isometries of $A\sp{p}.$. We also study the coefficient multipliers from $A\sp{p}$ into $A\sp{q},$ and obtain some new sufficient and/or necessary conditions. In particular, an analogue of a theorem of Duren about multipliers of $H\sp{p}$ spaces is obtained, as well as the analogue of a known theorem of Hardy-Littlewood-Duren-Shields on Hardy spaces, thus completely characterizing the multipliers from $A\sp1$ into $A\sp2.$ Further results are included.