Mathematics & Physics 2016, 9(4), 510–517 УДК 517.55 On Traces in Hardy Type Analytic Spaces in Bounded Strictly Pseudoconvex Domains and in Tubular Domains over Symmetric Cones Romi F.Shamoyan∗ Sergey M.Kurilenko† Natural Sciences Institute Brjansk State University Bejickaya, 14, Brjansk, 241036 Russia Received 30.01.2016, received in revised form 03.06.2016, accepted 15.08.2016 We provide some new estimates on Traces in new mixed norm Hardy type spaces and related new results on Bergman type intergal operators in Hardy type spaces in tubular domains over symmetric cones and bounded striclty pseudoconvex domains with smooth boundary. <...> We generalize a well-known one dimensional result concerning Traces of Hardy spaces obtained previously in the unit disk by various authors. <...> Keywords: bounded strongly pseudoconvex domain, tubular domains over symmetric cones, Hardy-type spaces. <...> In this note we give new estimates for traces of new mixed norm Hardy type spaces in tubular domains over symmetric cones and in bounded strictly pseudoconvex domains. <...> We generalize a well-known one dimensional result concerning traces of Hardy spaces obtained previously in the unit disk by various authors. <...> This trace problem in Hardy spaces(diagonal map problem) in the unit disk and in the unit polydisk was considered by many authors during last several decades. <...> In polydisk some estimates related with this problem can be seen, for example, in [1, 2]. <...> In polydisk, but in particular values of parameters in [3] and in some papers from list of references of [1, 2]. <...> In this paper we extend a known crucial estimate related with this problem to more general and complicated cases of tubular domains over symmetric cones and pseudoconvex domains with smooth boundary putting a natural condition on Bergman kernel of these domains. <...> This paper can be considered also as continuation of a long series of papers of first author on traces of function spaces (see, for example, [5–8] and various references there). <...> In recent decades many papers appeared where various Hardy and other analytic spaces were studied from various points of views in higher dimension in various domains in Cn. We refer for example <...>