Embedding Bergman spaces into tent spaces

Let $A^p_ω$ denote the Bergman space in the unit disc $\mathbb{D}$ of the complex plane induced by a radial weight $ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. The tent space $T^q_s(ν,ω)$ consists of functions such that \begin{equation*} \begin{split} \|f\|_{T^q_s(ν,ω)}^q =\int_{\mathbb{D}}\left(\int_{Γ(ζ)}|f(z)|^s\,dν(z)\right)^\frac{q}sω(ζ)\,dA(ζ) <\infty,\quad 0<q,s<\infty. \end{split} \end{equation*} Here $Γ(ζ)$ is a non-tangential approach region with vertex $ζ$ in the punctured unit disc $\mathbb{D}\setminus\{0\}$. We characterize the positive Borel measures $ν$ such that $A^p_ω$ is embedded into the tent space $T^q_s(ν,ω)$, 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 $A^p_ω$.

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