On the Li-Stević integral type operators from weighted Bergman spaces into \(\beta\)-Zygmund spaces
Let \(H(D)\) be the space of all holomorphic functions on the open unit disk \(D\) in the complex plane. For \(\alpha>-1\) and \(0<p<\infty\), let \(L^p_a(dA_\alpha)=L^p(dA_\alpha)\cap H(D)\) be the standard weighted Bergman space, where \(dA_\alpha(z)= (\alpha+1)(1-|z|^2)^\alpha dA(z)\), with \(dA\) denoting the normalized area measure on \(D\). For \(\beta>0\), the author means by the \(\beta\)-Zygmund space \(\mathcal Z_\beta\) in the title the space of all \(f\in H(D)\) such that the norm \[ \|f\|_{\mathcal Z_\beta}:=|f(0)|+|f'(0)|+\sup_{z\in D}(1-|z|^2)^\beta |f''(z)| \] is finite. For an holomorphic self-map \(\varphi\) of \(D\) and \(g\in H(D)\), the author considers two types of integral type operators, namely, \[ T_\varphi^g f(z) := \int_0^z f(\varphi(\zeta)) g(\zeta)\, d\zeta\,\, \text{and} \,\, C_\varphi^g f(z) := \int_0^z f'(\varphi(\zeta)) g(\zeta)\, d\zeta \] for \(f\in H(D)\). The author first characterizes the boundedness and compactness of \(T_\varphi^g:L^p_a(dA_\alpha)\to \mathcal Z_\beta\) when \(1<p<\infty\, \) and \( \|\varphi\|_\infty=1\). His characterizations are a bit too technical to describe here. Analogous results are obtained when the target space is replaced by the little space \({\mathcal Z}_{\beta,0}\) consisting of all \(f\in {\mathcal Z}_\beta\) such that \((1-|z|^2)^\beta |f''(z)|\to 0\) as \(|z|\to 1\). Also, when \(\|\varphi\|_\infty<1\), a simpler characterization is observed. Similar results for the operator \(C_\varphi^g\) are also obtained. Finally, several applications to Volterra type integral operators and composition operators are given.
scientific article; zbMATH DE number 6149038
scientific article