Embedding theorems for Bergman-Zygmund spaces induced by doubling weights
Atte Pennanen
TL;DR
This work studies embedding and differentiation-operator problems for Bergman-Zygmund spaces $A^p_{\omega,\Psi}$ induced by radial doubling weights, with $0<q<p<\infty$ and inducing functions $\Psi,\Phi\in\mathcal{L}$. It introduces a Carleson-measure framework via the test function $\Upsilon^{\mu,\omega}_{\Phi,\Psi}$ and proves a sharp criterion: $I: A^p_{\omega,\Psi} \to L^q_{\mu,\Phi}$ is bounded/compact if and only if $\mu$ is a $(q,\Phi)$-Carleson measure for $A^p_{\omega,\Psi}$, which is equivalent to $\Upsilon^{\mu,\omega}_{\Phi,\Psi} \in L^{\frac{p}{p-q}}_{\widetilde{\omega}}$. This criterion is extended to the $n$-th differentiation operator, with an additional factor $(1-|z|)^{nq}$ in the test function. The results generalize prior Carleson-measure characterizations to Bergman-Zygmund spaces with general doubling radial weights and $(\Psi,\Phi)$ in $\mathcal{L}$, leveraging Littlewood-Paley estimates and weight-atomic decompositions. Together, they provide a precise embedding theory and operator-compactness criteria in this setting.
Abstract
Let $0<p<\infty$ and $Ψ: [0,1) \to (0,\infty)$, and let $μ$ be a finite positive Borel measure on the unit disc $\mathbb{D}$ of the complex plane. We define the Lebesgue-Zygmund space $L^p_{μ,Ψ}$ as the space of all measurable functions on $\mathbb{D}$ such that $\int_{\mathbb{D}}|f(z)|^pΨ(|f(z)|)\,dμ(z)<\infty$. The weighted Bergman-Zygmund space $A^p_{ω,Ψ}$ induced by a weight function $ω$ consists of analytic functions in $L^p_{μ,Ψ}$ with $dμ=ω\,dA$. Let $0<q<p<\infty$ and let $ω$ be radial weight on $\mathbb{D}$ which has certain two-sided doubling properties. In this study, we will characterize the measures $μ$ such that the identity mapping $I: A^p_{ω,Ψ} \to L^q_{μ,Φ}$ is bounded and compact, when we assume $Ψ,Φ$ to be essentially monotonic and to satisfy certain doubling properties. In addition, we apply our result to characterize the measures for which the differentiation operator $D^{(n)}: A^p_{ω,Ψ} \to L^q_{μ,Φ}$ is bounded and compact.