Volterra type operators on Bergman spaces with exponential weights
integration operators on Bergman spaces with exponential weights
For a holomorphic function \(g\) on the disc \(\mathbb{D}\) the authors consider the Volterra-type operator acting on holomorphic functions: NEWLINE\[NEWLINE J_g(f)(z)=\int_{0}^z f(\xi) g'(\xi) d\xi. NEWLINE\]NEWLINE A weight function \(w\) is a positive function on \(0\leq r<1\) that is integrable on \((0,1)\). For \(z\in\mathbb{D}\) set \(w(z)=w(|z|)\). For \(0<p<\infty\), the weighted Bergman space \(A^p(w)\) is the space of holomorphic functions such that NEWLINE\[NEWLINE \int_{\mathbb{D}} |f(z)|^p w(z) dA(z)<\infty. NEWLINE\]NEWLINE For the weights NEWLINE\[NEWLINE w_{\gamma,\alpha}(r)=(1-r)^{\gamma}\exp\left(-\frac{c}{(1-r)^{\alpha}}\right) NEWLINE\]NEWLINE with \(\gamma\geq 0\), \(\alpha>0\) and \(c>0\), the authors consider necessary and sufficient conditions on the symbol \(g\) so that \(J_g\) is bounded and compact. In particular, they show that \(J_g:A^{p}(w_{\gamma,\alpha})\to A^{p}(w_{\gamma,\alpha})\) is bounded if and only if NEWLINE\[NEWLINE \sup_{z\in\mathbb{D}}(1-|z|)^{1+\alpha}|g'(z)|<\infty. NEWLINE\]NEWLINE Similar results for compactness and Schatten ideal membership are also provided.NEWLINENEWLINEFor the entire collection see [Zbl 1232.30005].
scientific article; zbMATH DE number 6077516