On Bergman projections and sharp decomposition theorems in tubular and related domains in $C^n$

The theory of analytic function spaces in very general tubular domains over symmetric cones is a relatively new interesting research area. Tube domains are very general and very complicated domains. Recently several new results in this research area were provided in papers of B. Sehba and his coauthors concerning Bergman type operators in such type complicated unbounded domains. In this note we expand their results to certain spaces of analytic functions in products of tube domains. We define new integral operators of Bergman type and new analytic mixed norm spaces in such type domains and products of tube domains and provide new results on boundedness of certain Bergman type operators. Our results may have various nice applications in this research area. Our results with very similar proofs may be valid in Siegel domains of second type, in bounded symmetric domains and bounded strongly pseudoconvex domains with the smooth boundary, various matrix domains. We will add at the end of this note a new sharp decomposition theorem for Bergman space. Previously such type sharp decomposition theorems in analytic function spaces were provided by author in other domains. Our sharp result in Bergman type function spaces enlarge that list of previously known such type assertions in analytic function spaces of several variables. We finnaly pose in addition various interesting new problems related to this research area and moreover indicate also some concrete schemes for solutions of these problems. We also provide in the second part of this note many interesting short comments and remarks.