Carleson measures for weighted Bergman--Zygmund spaces

For $0<p<\mathrm{\infty }$, $\mathrm{\Psi }:[0,\mathrm{\infty })\to (0,\mathrm{\infty })$ and a finite positive Borel measure $\mu$ on the unit disc $\mathbb{D}$, the Lebesgue–Zygmund space $L_{\mu ,\mathrm{\Psi }}^p$ consists of all measurable functions $f$ such that $\parallel f{\parallel }_{L_{\mu ,\mathrm{\Psi }}^p}^p={\int }_{\mathbb{D}}\mathrm{\mid }f{\mathrm{\mid }}^p\mathrm{\Psi }(\mathrm{\mid }f\mathrm{\mid })d\mu <\mathrm{\infty }$. For an integrable radial function $\omega$ on $\mathbb{D}$, the corresponding weighted Bergman-Zygmund space $A_{\omega ,\mathrm{\Psi }}^p$ is the set of all analytic functions in $L_{\mu ,\mathrm{\Psi }}^p$ with $d\mu =\omega dA$. The purpose of the paper is to characterize bounded (and compact) embeddings $A_{\omega ,\mathrm{\Psi }}^p\subset L_{\mu ,\mathrm{\Phi }}^q$, when $0<p\le q<\mathrm{\infty }$, the functions $\mathrm{\Psi }$ and $\mathrm{\Phi }$ are essential monotonic, and $\mathrm{\Psi },\mathrm{\Phi },\omega$ satisfy certain doubling properties. The tools developed on the way to the main results are applied to characterize bounded and compact integral operators acting from $A_{\omega ,\mathrm{\Psi }}^p$ to $A_{\nu ,\mathrm{\Phi }}^q$, provided $\nu$ admits the same doubling property as $\omega$.