Embedding Bergman spaces into tent spaces

Let $A_{\omega }^p$ 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 $T_s^q(\nu ,\omega )$ 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 $\mathrm{\Gamma }(\zeta )$ 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 $A_{\omega }^p$ is embedded into the tent space $T_s^q(\nu ,\omega )$, 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 $A_{\omega }^p$.