Carleson measures for weighted Bergman--Zygmund spaces

For 0<p<∞, Ψ:[0,∞)→(0,∞) and a finite positive Borel measure μ on the unit disc D, the Lebesgue--Zygmund space Lᵖμ,Ψ consists of all measurable functions f such that f _Lμ, Ψᵖᵖ =∫D|f|ᵖΨ(|f|)\,dμ< ∞. For an integrable radial function ω on D, the corresponding weighted Bergman-Zygmund space Aω, Ψᵖ is the set of all analytic functions in Lμ, Ψᵖ with dμ=ω\,dA. The purpose of the paper is to characterize bounded (and compact) embeddings Aω,Ψᵖ⊂ Lμ, Φq, when 0<p≤ q<∞, 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 Aᵖω,Ψ to Aqν,Φ, provided ν admits the same doubling property as ω.