Bergman projections and operators on hardy spaces

We show that ''Toeplitz like'' operators of the form T-u(s) f = P-s(uf), where P-s is a weighted Bergman projection, are bounded on the Hardy spaces H-p, for 1 less than or equal to p < infinity for certain ''symbols'' u defined on the unit disk. In particular, T-u(s) is bounded if u is of the form u = h + G mu where h is a bounded harmonic function and G mu is the Green potential of a Borel measure mu satisfying the condition that (1 - \z\) d mu(z) is a Carleson measure. If u is of the form u = h + G mu where mu is a positive measure then these two conditions are also necessary that T-u(s), be bounded on H-p. We also consider the cases 0 < p < 1. (C) 1997 Academic Press.