Carleson measures for weighted Bergman--Zygmund spaces

For 0<p<,: [0, ) (0, ) and a finite positive Borel measure on the unit disc D, the Lebesgue--Zygmund space Lᵖ, consists of all measurable functions f such that f ₋_, ^{p}ᵖ =₃|f|ᵖ (|f|) \, d<. For an integrable radial function on D, the corresponding weighted Bergman-Zygmund space A, ^p is the set of all analytic functions in L, ^p with d=\, dA. The purpose of the paper is to characterize bounded (and compact) embeddings A, ^p L, ^q, when 0<p q<, the functions and are essential monotonic, and, , 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 Aᵖ, to Aq, , provided admits the same doubling property as.