Embedding theorems for Bergman-Zygmund spaces induced by doubling weights

Abstract

Let $0<p<\infty$ and $\Psi: [0,1) \to (0,\infty)$, and let $\mu$ be a finite positive Borel measure on the unit disc $\mathbb{D}$ of the complex plane. We define the Lebesgue-Zygmund space $L^p_{\mu,\Psi}$ as the space of all measurable functions $f$ on $\mathbb{D}$ such that $\int_{\mathbb{D}}|f(z)|^p\Psi(|f(z)|)\,d\mu(z)<\infty$. The weighted Bergman-Zygmund space $A^p_{\omega,\Psi}$ induced by a weight function $\omega$ consists of analytic functions in $L^p_{\mu,\Psi}$ with $d\mu=\omega\,dA$. Let $0<q<p<\infty$ and let $\omega$ be radial weight on $\mathbb{D}$ which has certain two-sided doubling properties. In this study, we will characterize the measures $\mu$ such that the identity mapping $I: A^p_{\omega,\Psi} \to L^q_{\mu,\Phi}$ is bounded and compact, when we assume $\Psi,\Phi$ to be almost monotonic and to satisfy certain doubling properties. In addition, we apply our result to characterize the measures for which the differentiation operator $D^{(n)}: A^p_{\omega,\Psi} \to L^q_{\mu,\Phi}$ is bounded and compact.