From Zygmund space to Bergman–Zygmund space

Let \(0<p<\infty , \alpha >-1,\) and \(\beta ,\gamma \in {\mathbb {R}}.\) Let \(\mu \) be a finite positive Borel measure on the unit disk \({\mathbb {D}}.\) The Zygmund space \(L^{p,\beta }(d\mu )\) consists of all measurable functions f on \({\mathbb {D}}\) such that \(|f|^p\log ^\beta (e+|f|)\in L^1(d\mu )\) and the Bergman–Zygmund space \(A^{p,\beta }_{\alpha }\) is the set of all analytic functions in \(L^{p,\beta }(dA_\alpha ),\) where \(dA_\alpha =c_\alpha (1-|z|^2)^\alpha dA.\) 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 \(\log ^\beta (e/(1-|z|)) dA_\alpha (z)\) weighted Bergman space as a set and characterize the bounded and compact Carleson measure \(\mu \) from \(A^{p,\beta }_{\alpha }\) into \(A^{p,\gamma }(d\mu ),\) respectively. The Carleson measure characterizations are of the same type for any pairs of \((\beta , \gamma )\) whether \(\beta <\gamma \) or \(\gamma \le \beta .\)