Diagonal mappings in bounded symmetric domains (Q2725212)

Let \(\Omega\subset {\mathbb C}^m\) be an irreducible bounded symmetric space and let \(\Omega^n\) be the \(n\)-fold Cartesian product of \(\Omega\). Let \({\mathcal D}\) be a diagonal mapping defined on \(\Omega^n\) as follows: for \(F\in {\mathcal H}(\Omega^n)\), NEWLINE\[NEWLINE {\mathcal D}F(z)=F(z,\dots, z)\qquad z\in \Omega. NEWLINE\]NEWLINE The authors prove that for \(0<p\leq 1\) and \(\alpha>-1\), \(F\in L^p_a(\Omega^n,d\mu_\alpha)\) if and only if \({\mathcal D}(F)\in L^p_a(\Omega^n,d\lambda_{|\alpha|+(n-1)N})\). Here \(L^p_a(\Omega^n,d\mu_\alpha)\) is the weighted Bergman space.