Embedding Bergman spaces into tent spaces

Abstract

Let $A^p_\omega$ denote the Bergman space in the unit disc $\mathbb {D}$ of the complex plane induced by a radial weight $\omega$ with the doubling property $\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds$. The tent space $T^q_s(\nu ,\omega )$ consists of functions such that

$$\begin{aligned} \begin{aligned} \Vert f\Vert _{T^q_s(\nu ,\omega )}^q =\int _\mathbb {D}\left( \int _{\varGamma (\zeta )}|f(z)|^s\,d\nu (z)\right) ^\frac{q}{s}\omega (\zeta )\,dA(\zeta ) <\infty ,\quad 0<q, \; s<\infty . \end{aligned} \end{aligned}$$

Here $\varGamma (\zeta )$ 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 $T^q_s(\nu ,\omega )$, where $1+\frac{s}{p}-\frac{s}{q}>0$, by considering a generalized area operator. The results are provided in terms of Carleson measures for $A^p_\omega$.