Norm-attaining integral operators on analytic function spaces

Abstract

If f and g are analytic functions in the unit disc D we define Sg(f)(z)=∫0zf′(w)g(w)dw,(z∈D). If g is bounded then the integral operator Sg is bounded on the Bloch space, on the Dirichlet space, and on BMOA. We show that Sg is norm-attaining on the Bloch space and on BMOA for any bounded analytic function g, but does not attain its norm on the Dirichlet space for non-constant g. Some results are also obtained for Sg on the little Bloch space, and for another integral operator Tg from the Dirichlet space to the Bergman space.