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In our paper we provide some direct extentions of our recent sharp results on traces in the analytic function spaces, which we proved earlier in case of the unit ball in C, to the case of the bounded strongly pseudoconvex domains with a smooth boundary. To be more precise we consider the analytic Bloch space in the strongly pseudoconvex domains with a smooth boundary, mixed norm spaces and so-called the new Herz type spaces of analytic functions in the domains of the same type. The Bloch spaces, for various complicated domains, were studied by many authors, but the various Herz type spaces are introduced in this paper, as far as we know, for the first time. The role of socalled r-lattices and their new properties are essential for our proofs. These techniques based on the lattices in the strongly pseudoconvex domains were invented and heavily used in the recent papers of Abate and coauthors. The arguments in the proofs in the case of the unit ball and the strongly pseudoconvex domains have some similarity.

ON TRACES OF ANALYTIC HERZ AND BLOCH TYPE SPACES IN BOUNDED STRONGLY PSEUDOCONVEX DOMAINS IN C^n