Abstract. In this paper we completely describe traces of holomorphic Bergman classes and Bloch-type classes on polyballs and obtain related estimates generalizing classical Bergman projection theorem.

Abstract. We provide new estimates for distances from fixed analytic functions to their subspaces in the unit disk. We will enlarge the list of previously known assertions of this type obtained recently by R. Zhao and W. Xu.

We prove some sharp extremal distance results for functions in weighted Bergman spaces on the upper halfplane. We also prove new analogous results in the context of bounded strictly pseudoconvex domains with smooth boundary.

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