An application of atomic decomposition in Bergman spaces to the study of differences of composition operators (Q418706)

Let \(\varphi\) and \(\psi\) be analytical self maps of the unit disc \(\mathbb D\) and \(\sigma (z)=\frac{\varphi(z) -\psi(z)}{1-\overline{ \varphi(z)}\psi(z)}\). Let \(A^p_{\alpha}\) be the standard weighted Bergman space, \(\alpha >-1\) and \(\mu\) be a positive measure on \(\mathbb D\). Assume that \(0<q<p\) and set \(s=\frac{p}{p-q}\). The main result in the paper is the equivalence ofNEWLINENEWLINE(1) the operator \(C_{\varphi}-C_{\psi}\) maps \(A^p_{\alpha}\) into \(L^q(\mu);\)NEWLINENEWLINE(2) the operators \(\sigma C_{\varphi}\) and \(\sigma C_{\psi}\) map \(A^p_{\alpha}\) into \(L^q(\mu);\)NEWLINENEWLINE(3) the function \(K_{\varphi,\psi}(z)=\int_{\mathbb D}\left|\left(\frac{1-|z|^2}{(1-\overline z\varphi(w))^2} \right)^{\frac{\alpha +2}{q}}- \left(\frac{1-|z|^2}{(1-\overline z\psi(w))^2}\right)^{\frac{\alpha +2}{q}}\right |^q d\mu(w)\) belongs to \(L^s(A^p_{\alpha})\).NEWLINENEWLINEThis result allows the author to complete a characterization of bounded and compact differences of weighted composition operators between standard weighted Bergman spaces, which he initiated in [J. Math. Anal. Appl. 381, No. 2, 789--798 (2011; Zbl 1244.47024)].