From Zygmund space to Bergman–Zygmund space

Abstract

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 $\gamma \le \beta .$