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, $A^{p^{\prime }}=(A^p{)}^{\ast }$, 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 $p\ge 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p^{\prime }}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.