On Bergman type projections in bounded strongly pseudoconvex domains

Abstract

In our note we prove the boundedness of Bergman type projections in two different spaces of analytic functions with mixed norm in general bounded strongly pseudoconvex domains with smooth boundary.The first class of analytic functions was studied previously by many authors,the second function space hovewer is completely new. Our proofs are based on standard known estimates of function space theory in bounded strongly pseudoconvex domains with smooth boundary and on some known estimates of Bergman kernel in such type domains. These estimates are also well- known in the unit disk.This allows us to provide proofs first in one dimensional case which are simpler and then repeating same arguments to show same type results also in more general situation. Note that many results on boundedness of Bergmam type projections are well known and they have also various nice applications in complex function theory in one or several complex variable. Our results may also have various applications in complex function theory in bounded strongly pseudoconvex domains with smooth boundary.