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 L p boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, A p 0 = (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 0 . For the Bloch space B 1 we give in addition new necessary conditions on the number of derivatives required in its deflnition.