Carleson measures for weighted Bergman--Zygmund spaces

Abstract: For $0<p<\infty$, $\Psi:[0,\infty)\to(0,\infty)$ and a finite positive Borel measure $\mu$ on the unit disc $\mathbb{D}$, the Lebesgue--Zygmund space $L^p_{\mu,\Psi}$ consists of all measurable functions $f$ such that $\lVert f \rVert_{L_{\mu, \Psi}^{p}}^p =\int_{\mathbb{D}}|f|^p\Psi(|f|)\,d\mu< \infty$. For an integrable radial function $\omega$ on $\mathbb{D}$, the corresponding weighted Bergman-Zygmund space $A_{\omega, \Psi}^{p}$ is the set of all analytic functions in $L_{\mu, \Psi}^{p}$ with $d\mu=\omega\,dA$. The purpose of the paper is to characterize bounded (and compact) embeddings $A_{\omega,\Psi}^{p}\subset L_{\mu, \Phi}^{q}$, when $0<p\le q<\infty$, the functions $\Psi$ and $\Phi$ are essential monotonic, and $\Psi,\Phi,\omega$ 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 $A^p_{\omega,\Psi}$ to $A^q_{\nu,\Phi}$, provided $\nu$ admits the same doubling property as $\omega$.