Weighted composition operators between different Bergman spaces of bounded symmetric domains
Let \(\Omega\) be a bounded symmetric domain in \(\mathbb C^n\) with Bergman kernel \(K(z,w)\), and consider the measures \(dv_\alpha:=K(z,z)^{1-\alpha} \,dv(z)\), where \(dv\) stands for the Lebesgue volume measure and \(\alpha> \frac{N-1}N\), \(N\)~being the genus of \(\Omega\). The weighted Bergman space \(A^p_\alpha\), \(0<p<\infty\), consists, by definition, of all holomorphic functions in \(L^p(\Omega,dv_\alpha)\). For \(\delta>0\), a measure \(\mu\) on \(\Omega\) is called \(\delta\)-Carleson if the expression \(\mu(E(z,r))/ v(E(z,r))^\delta\) is bounded as \(z\) ranges over~\(\Omega\); if~this expression even tends to zero as \(z\to\partial\Omega\), then one calls \(\mu\) a vanishing \(\delta\)-Carleson measure. Here \(E(z,r)\) is the ball with center \(z\) and radius \(r\) with respect to the Bergman metric. In the paper under review, the authors study the weighted composition operators \[ W_{\phi,\psi} f(z) := \psi(z) \, f(\phi(z)) \] on \(\Omega\), where \(\phi:\Omega\to\Omega\) and \(\psi:\Omega\to\mathbb C\) are holomorphic. The main results are that for \(0<p\leq q<\infty\), \(W_{\phi,\psi}:A^p_\alpha\to A^q_\beta\) is bounded if and only~if \[ \mu_{\phi,\psi,q,\beta}(E):=\int_{\phi^{-1}(E)} |\psi|^q \,dv_\beta \] is an \(\frac{\alpha q}p\)-Carleson measure (and similarly for ``compact'' and ``vanishing Carleson''); while for \(0<q<p<\infty\), \(W_{\phi,\psi}: A^p_\alpha\to A^q_\beta\) is bounded if and only if it is compact if and only if \(\mu_{\phi,\psi,q,\beta}(E(\cdot,r))/v(E(\cdot,r))^\alpha\) belongs to \(L^s(\Omega,dv_\alpha)\), where \(s=\frac p{p-q}\). Applications to multiplication and composition operators are given.
scientific article; zbMATH DE number 5644692