Complex interpolation between two weighted Bergman spaces on tubes over symmetric cones

Abstract

We prove that the complex interpolation space [ A ν p 0 , A ν p 1 ] θ , 0< θ<1, between two weighted Bergman spaces A ν p 0 and A ν p 1 on the tube in C n , n⩾3, over an irreducible symmetric cone of R n is the weighted Bergman space A ν p with 1/ p=(1− θ)/ p 0+ θ/ p 1. Here, ν> n/ r−1 and 1⩽ p 0< p 1<2+ ν/( n/ r−1) where r denotes the rank of the cone. We then construct an analytic family of operators and an atomic decomposition of functions, which are related to this interpolation result. To cite this article: D. Békollé et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).