Volterra-type operators mapping weighted Dirichlet space into H8

The problem of describing the analytic functions g on the unit disc such that the integral operator T-g(f)(z) = ?(z)(0) f (?)g'(?)d? is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H-8 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), induced by an upper doubling weight ?, to H-8. We also characterize, in terms of neat conditions on ?, the upper doubling weights for which T-g : D-?(p) ? H-8 is bounded (or compact) only if g is constant.