Carleson measures for weighted Bergman–Zygmund spaces

Abstract

Abstract For $$0 0 < p < ∞ , $$\Psi :[0,\infty )\rightarrow (0,\infty )$$ Ψ : [ 0 , ∞ ) → ( 0 , ∞ ) and a finite positive Borel measure $$\mu $$ μ on the unit disc $$\mathbb {D}$$ D , the Lebesgue–Zygmund space $$L^p_{\mu ,\Psi }$$ L μ , Ψ p consists of all measurable functions f such that $$\Vert f \Vert _{L_{\mu , \Psi }^{p}}^p =\int _{\mathbb {D}}|f|^p\Psi (|f|)\,d\mu < \infty $$ ‖ f ‖ L μ , Ψ p p = ∫ D | f | p Ψ ( | f | ) d μ < ∞ . For an integrable radial function $$\omega $$ ω on $$\mathbb {D}$$ D , the corresponding weighted Bergman-Zygmund space $$A_{\omega , \Psi }^{p}$$ A ω , Ψ p is the set of all analytic functions in $$L_{\mu , \Psi }^{p}$$ L μ , Ψ p with $$d\mu =\omega \,dA$$ d μ = ω d A . The purpose of the paper is to characterize bounded (and compact) embeddings $$A_{\omega ,\Psi }^{p}\subset L_{\mu , \Phi }^{q}$$