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