Analytic Besov spaces and Hardy-type inequalities in tube domains over symmetric cones

Abstract

We give various equivalent formulations to the (partially) open problem about Lp-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, Ap? = (Ap)*, 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 Lp, and also the dual of Ap?. For the Bloch space we give in addition new necessary conditions on the number of derivatives required in its definition.