Atomic decomposition of Bergman-Orlicz space on the upper complex half-plane

TL;DR

The paper addresses the problem of representing Bergman-Orlicz spaces on the upper half-plane via atomic decompositions and uses this framework to study Carleson embeddings with loss and composition operators. It develops a discrete sampling approach on a $\delta$-lattice to realize an atomic representation $F(z)=c_{\alpha}\sum \mu_{l,j}K_{\alpha}(z,z_{l,j})2^{j\gamma(\alpha+2)}$ with $\mu\in\ell_{\alpha}^{\Phi}$, and proves a corresponding norm equivalence. These tools yield necessary and sufficient conditions for embedding Bergman-Orlicz spaces into Orlicz spaces with weighted measures and for the boundedness of composition operators via Berezin transforms. The results extend Bergman-Orlicz theory to the upper half-plane, providing a robust analytic and operator-theoretic framework in non-standard growth regimes, and offering explicit duality and sampling mechanisms for these spaces.

Abstract

In this work, we propose an atomic decomposition of the Bergman-Orlicz spaces on the complex upper half-plane. Using this result, we characterize Carleson embeddings with loss between Bergman-Orlicz spaces and certain Orlicz spaces. We also leverage this last result to control the composition operator between two Bergman-Orlicz spaces.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Proposition 2.4
• Proposition 2.5