From Zygmund space to Bergman–Zygmund space

Let 0<p<∞,α>-1, and β,γ∈R. Let μ be a finite positive Borel measure on the unit disk D. The Zygmund space L^p,β(dμ) consists of all measurable functions f on D such that |f|^plog^β(e+|f|)∈L¹(dμ) and the Bergman–Zygmund space A_α^p,β is the set of all analytic functions in L^p,β(dA_α), where dA_α=c_α(1-|z|²)^αdA. We prove an interpolation theorem for the Zygmund space assuming the weak type estimates on the Zygmund spaces themselves at the end points rather than the weak L^p-L^q type estimates at the end points. We show that the Bergman–Zygmund space is equal to the log^β(e/(1-|z|))dA_α(z) weighted Bergman space as a set and characterize the bounded and compact Carleson measure μ from A_α^p,β into A^p,γ(dμ), respectively. The Carleson measure characterizations are of the same type for any pairs of (β,γ) whether β<γ or γ≤β.