Volterra-type operators mapping weighted Dirichlet space into H^infty

Abstract

The problem of describing the analytic functions g on the unit disc such that the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H^infty is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of T_g acting from a weighted Dirichlet space D^p_omega , induced by an upper doubling weight omega , to H^infty . We also characterize, in terms of neat conditions on omega , the upper doubling weights for which T_g: D^p_omega rightarrow H^infty is bounded (or compact) only if g is constant.