Area operators on large Bergman spaces

Hicham Arroussi¹ ; Jari Taskinen² ; Cezhong Tong³ ; Zixing Yuan⁴

¹ University of Helsinki, Department of Mathematics and Statistics, and University of Reading, Department of Mathematics and Statistics
² University of Helsinki, Department of Mathematics and Statistics
³ Hebei University of Technology, Institute of Mathematics
⁴ Wuhan University, School of Mathematics and Statistics

Annales Fennici Mathematici, Tome 49 (2024) no. 2, p. 731–749.

Résumé

We completely characterize those positive Borel measures $\mu$ on the open unit disk $\mathbb{D}$ for which the area operator $A_{\mu}\colon A^p_\varphi\rightarrow L^q(\mathbb{T})$ is bounded. Here, the indices $0<p,q<\infty$ are arbitrary and $\varphi$ belongs to a certain class $\mathcal{W}_{0}$ of exponentially decreasing weights. Accordingly, the proofs require techniques adapted to such weights, like tent spaces, Carleson measures for $A^p_\varphi$-spaces, Kahane–Khintchine inequalities, and decompositions of the unit disc by $(\rho,r)$-lattices, which differ from the conventional decompositions into subsets with essentially constant hyperbolic radii.

Détail

DOI : 10.54330/afm.153073

Keywords: Bergman space, tent space, area operator