Embedding theorems for Bergman spaces via harmonic analysis

Let $A_{\omega }^p$ denote the Bergman space in the unit disc 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 positive Borel measures such that the differentiation operator of order $n\in \mathbb{N}\cup \{0\}$ is bounded from $A_{\omega }^p$ into $L^q(\mu )$ are characterized in terms of geometric conditions when $0<p,q<\mathrm{\infty }$. En route to the proof a theory of tent spaces for weighted Bergman spaces is built.