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 $L_{\mu ,\Psi }^p$ 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 $A_{\omega ,\Psi }^p$ induced by a weight function $\omega$ consists of analytic functions in $L_{\mu ,\Psi }^p$ 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 $I:A_{\omega ,\Psi }^p\to L_{\mu ,\Phi }^q$ 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 $D^{(n)}:A_{\omega ,\Psi }^p\to L_{\mu ,\Phi }^q$ is bounded and compact.

12 pages

References in corpus (2)

• Moduli of holomorphic functions and logarithmically convex radial weights
• Littlewood-Paley inequalities for fractional derivative on Bergman spaces