Toeplitz and related operators in large Bergman spaces

We consider operator theory in Bergman spaces consisting of analytic functions on complex domains.

The aim is to extend known, central results of standard Bergman spaces to the case of large spaces, which are naturally defined by using rapidly decaying, non-doubling weights. The need of weighted estimates is as apparent as anywhere in harmonic analysis and applications.

In the context of Bergman spaces, the case of non-doubling weights is still partially open due to the fact that such weights are not so naturally related with the hyperbolic metric of the underlying domain.

In this context we plan to consider questions of boundedness of the Bergman projection in weighted L^p-norms in relation to the boundedness of Toeplitz and also little Hankel operators.In the case of standard weighted Bergman spaces there is a well-known connection of the theory to the deformation quantization.Another topic of recent interest is formed by the so called localized operator classes.

We aim to extend these studies to the case oflarge Bergman spaces.The methodology comes from the earlier joint works of the researcher and a number of well known experts in the area, on the topicof pointwise estimates of the Bergman kernel among others, and from the techniques of the supervisor and W.Lusky, as well as Fock-spacemethods, which are naturally related to nondoubling measures.