A Nonlinear Functional on the Dirichlet Space

Abstract

The nonlinear functional Λ1(f) = (1/2π) ∫2π0e|f(eiθ|2dθ was shown by Chang and Marshall to be bounded on the unit ball B of the space D of analytic functions in the unit disk with finite Dirichlet integral. We show that Λ1 is weakly continuous on B except at zero and that Λ1 attains its maximum over a subset of B determined by kernel functions.