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

TL;DR

This paper explores the boundedness of Bergman projections in tube domains over cones, linking it to duality, Hardy inequalities, and introducing analytic Besov spaces to understand these properties.

Contribution

It establishes equivalences between boundedness, duality, and Hardy inequalities, and introduces analytic Besov spaces in tube domains over cones.

Findings

Boundedness of Bergman projections is equivalent to duality identities and Hardy inequalities.

Identified the range of Bergman projection as a Besov space for p ≥ 2.

Provided new necessary conditions for derivatives in Bloch space.

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, $A^{p'}=(A^p)^*$, 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 $p\geq 2$ 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 $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.