We study the boundedness properties of Rudin-Forelli-type operators associated to tubular domains over symmetric cones. As an application, we give a characterization of the topological dual space of the weighted Bergman space A p,q ν .

We give various equivalent formulations to the (partially) open problem about L pboundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, A p ′ = (A p ) * , and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For p ≥ 2 we identify as a Besov space the range of the Bergman projection acting on L p , and also the dual of A p ′ . For the Bloch space B ∞ we give in addition new necessary conditions on the number of derivatives required in its definition.

We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Cesàro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Cesàro-type operators on weighted Hilbert-Bergman spaces.

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We present here some criteria for Schatten-Von Neumann class membership for the small Hankel operator on Bergman space A 2 (TΩ), when TΩ is the tube over the symmetric cone Ω. Mathematics Subject Classification (2000). Primary 47B35; Secondary 32A37, 46E22, 47B10.