Bergman-Lorentz spaces on tube domains over symmetric cones

TL;DR

This paper investigates Bergman-Lorentz spaces on tube domains over symmetric cones, establishing boundedness of Bergman projectors and exploring interpolation, with implications for the boundedness of projectors on L^p spaces.

Contribution

It introduces and analyzes Bergman-Lorentz spaces on symmetric cones, proving boundedness and surjectivity of Bergman projectors and exploring interpolation properties.

Findings

Boundedness of Bergman projectors from Lorentz to Bergman-Lorentz spaces.

Surjectivity of Bergman projectors in this setting.

Open question on extending boundedness to a wider range of p for higher-rank cones.

Abstract

We study Bergman-Lorentz spaces on tube domains over symmetric cones, i.e. spaces of holomorphic functions which belong to Lorentz spaces $L(p, q).$ We establish boundedness and surjectivity of Bergman projectors from Lorentz spaces to the corresponding Bergman-Lorentz spaces and real interpolation between Bergman-Lorentz spaces. Finally we ask a question whose positive answer would enlarge the interval of parameters $p\in (1, \infty)$ such that the relevant Bergman projector is bounded on $L^p$ for cones of rank $r\geq 3.$