Bergman Projections and Operators on Hardy Spaces

We show that “Toeplitz like” operators of the formTsuf=Ps(uf), wherePsis a weighted Bergman projection, are bounded on the Hardy spacesHp, for 1⩽p<∞ for certain “symbols”udefined on the unit disk. In particular,Tsuis bounded ifuis of the formu=h+Gμwherehis a bounded harmonic function andGμis the Green potential of a Borel measureμsatisfying the condition that (1−|z|) dμ(z) is a Carleson measure. Ifuis of the formu=h+Gμwhereμis a positive measure then these two conditions are also necessary thatTsube bounded onHp. We also consider the cases 0<p<1.