Carleson measures for weighted Bergman--Zygmund spaces

Abstract

For $0<p<\infty$, $\Psi :[0,\infty )\to (0,\infty )$ and a finite positive Borel measure $\mu$ on the unit disc $\mathbb{D}$, the Lebesgue--Zygmund space $L_{\mu ,\Psi }^p$ consists of all measurable functions $f$ such that $\Vert f{\Vert }_{L_{\mu ,\Psi }^p}^p={\int }_{\mathbb{D}}|f{|}^p\Psi (|f|)d\mu <\infty$. For an integrable radial function $\omega$ on $\mathbb{D}$, the corresponding weighted Bergman-Zygmund space $A_{\omega ,\Psi }^p$ is the set of all analytic functions in $L_{\mu ,\Psi }^p$ with $d\mu =\omega dA$. The purpose of the paper is to characterize bounded (and compact) embeddings $A_{\omega ,\Psi }^p\subset L_{\mu ,\Phi }^q$, when $0<p\le q<\infty$, the functions $\Psi$ and $\Phi$ are essential monotonic, and $\Psi ,\Phi ,\omega$ satisfy certain doubling properties. The tools developed on the way to the main results are applied to characterize bounded and compact integral operators acting from $A_{\omega ,\Psi }^p$ to $A_{\nu ,\Phi }^q$, provided $\nu$ admits the same doubling property as $\omega$.