| Abstract: | Let D be the open unit disk in the complex plane C. Forα>?1, let dAα(z)=(1+α)(1?|z|2)αdA(z) be the weighted Lebesgue measure on D. For a positive func-tion ω∈L1(D,dAα), the generalized weighted Bergman-Orlicz space Aψω(D,dAα) is the space of all analytic functions such that |f|ψω=∫Dψ(|f (z)|)ω(z)dAα(z)<∞, whereψis a strictly convex Orlicz function that satisfies other technical hypotheses. Let G be a measurable subset of D, we say G satisfies the reverse Carleson condition for Aψω(D,dAα) if there exists a positive constant C such that ∫Gψ(|f(z)|)ω(z)dAα(z)≥C∫Dψ(|f (z)|)ω(z)dAα(z), for all f∈Aψω(D,dAα). Let μ be a positive Borel measure, we say μ satisfies the direct Carleson condition if there exists a positive constant M such that for all f∈Aψω(D,dAα), ∫Dψ(|f(z)|)dμ(z)≤M∫Dψ(|f(z)|)ω(z)dAα(z). In this paper, we study the direct and reverse Carleson condition on the generalized weighted Bergman-Orlicz space Aψω(D,dAα). We present conditions on the set G such that the reverse Carleson condition holds. Moreover, we give a sufficient condition for the finite positive Borel measure μ to satisfy the direct carleson condition on the generalized weighted Bergman-Orlicz spaces. |
