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Journal of Operator Theory

Volume 91, Issue 1, Winter 2024  pp. 295-318.

Embedding from Bergman spaces into tent spaces

Authors:  Xiaofen Lv (1), Jordi Pau (2)
Author institution: (1) Department of Mathematics, Huzhou University, Huzhou 313000, China
(2) Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

Summary:  Let $A^p_\omega$ denote the Bergman space in the unit disc induced by a radial weight $\omega$ with the doubling property $\int\limits_{r}^1 \omega (s)\mathrm ds\leqslant C\int\limits_{{(1+r)}/{2}}^1 \omega (s)\mathrm ds$. The tent space $T^q_s (\nu, \omega)$ consists of functions such that $ \int\limits_{\mathbb D}\Big(\int\limits_{\Gamma(\zeta)}|f(z)|^s \mathrm d\nu(z)\Big)^{q/s}\cdot$ $\omega(\zeta)\mathrm dA(\zeta)<\infty, $ where $\Gamma(\zeta)$ is a non-tangential approach region with vertex $\zeta$ in the punctured unit disc ${\mathbb D}\setminus\{0\}$. We characterize the positive Borel measures $\nu$ such that $A^p_\omega$ is embedded into the tent space $T^q_s (\nu, \omega)$ for positive and finite $p$ less than $q$, by considering a generalized area operator.

Keywords:  Bergman space, tent space, Carleson measure, area operator

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