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{equation*} \begin{split} \|f\|_{T^q_s(\nu,\omega)}^q =\int_{\mathbb{D}}\left(\int_{\Gamma(\zeta)}|f(z)|^s\,d\nu(z)\right)^\frac{q}s\omega(\zeta)\,dA(\zeta) <\infty,\quad 0<q,s<\infty. \end{split} \end{equation*} Here $\Gamma(\zeta)$ 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 $T^q_s(\nu,\omega)$, 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 $A^p_\omega$.