Let \(0<p<\infty \) and \(\Psi : [0,1) \rightarrow (0,\infty )\) , and let \(\mu \) be a finite positive Borel measure on the unit disk \({\mathbb {D}}\) 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 \(I: A^p_{\omega ,\Psi } \rightarrow L^q_{\mu ,\Phi }\) 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 \(D^{(n)}: A^p_{\omega ,\Psi } \rightarrow L^q_{\mu ,\Phi }\) is bounded and compact.
