Composition operators from the Bloch space into the spaces \(Q_T\) (Q1811959)

The paper studies composition operators from the Bloch space of the unit disk \(D\), consisting of analytic functions \(f\) such that \[ \sup_{z\in D} \bigl(1-| z|^2\bigr)| f'(z)|<\infty, \] into a class of \(Q_T\) spaces, consisting of analytic functions \(f\) such that \[ \sup_{a\in D} \int_D\bigl| f'(z)\bigr|^2T \bigl(g(z,a)\bigr) dA(z)<\infty, \] where \(T\) is a nonnegative, nonincreasing function \([0,\infty)\), \(dA\) is an area measure on \(D\), and \[ g(z,a) =\log \frac{1-\overline az}{a-z} \] is the Green function at \(a\in D\). Characterizations are obtained for the boundedness and compactness of such composition operators.