Bergman projections and operators on Hardy spaces (Q674724)

Let \(D\) denote the unit disk in the complex plane. For \(s>0\) let \(z= re^{i\theta}\) be a point in \(D\) and let \(dm_s\) be the measure \(dm_s(z)=(s/\pi)(1- r^2)^{s-1}dm(z)\), where \(dm\) is a two-dimensional Lebesgue measure. \(P_s\) denotes the orthogonal projection of \(L^2(dm_s)\) to \(L^2_a(dm_s)\), then \(\lim_{s\to 0} P_sF(z)= P_+F(z)\), where \(F\) is continuous on \(\partial D\) and \(P_+\) is the Szegö projection. The author shows that ``Toeplitz like'' operators of the form \(T^s_uf= P_s(uf)\) are bounded on the Hardy spaces \(H^p\), for \(1\leq p\leq\infty\) 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. He also shows the converse is true and considers the cases \(0<p<1\).