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.
DOI: http://dx.doi.org/10.7900/jot.2022apr22.2371
Keywords: Bergman space, tent space, Carleson measure, area operator
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