Avatars of Stein's theorem in the complex setting

. In this paper, we establish some variants of Stein’s theorem, which states that a non-negative function belongs to the Hardy space H 1 ( T ) if and only if it belongs to L log L ( T ). We consider Bergman spaces of holomorphic functions in the upper half plane and develop avatars of Stein’s theorem and relative results in this context. We are led to consider weighted Bergman spaces and Bergman spaces of Musielak–Orlicz type. Eventually, we characterize bounded Hankel operators on A 1 ( C + ).