Embedding theorems for Bergman-Zygmund spaces induced by doubling weights

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 $f$ 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 almost monotonic and to satisfy certain doubling properties. In addition, we apply our result to characterize the measures for which the differentiation operator $D^{(n)}: A^p_{ω,Ψ} \to L^q_{μ,Φ}$ is bounded and compact.

💡 Research Summary

The paper investigates embedding properties of weighted Bergman–Zygmund spaces on the unit disc 𝔻, focusing on the case where the exponents satisfy 0 < q ≤ p < ∞. The authors introduce the Lebesgue–Zygmund space L⁽ᵖ⁾_{μ,Ψ} consisting of measurable functions f for which
∫_𝔻 |f(z)|⁽ᵖ⁾ Ψ(|f(z)|) dμ(z) < ∞,
where Ψ: