Generalized area operators on Hardy spaces

We show that if 0 < p < infinity then the operator Gf(zeta) = integral(Gamma(zeta))\f(z)\d mu/(1 - \z\) maps the Hardy space H-p to L-p(\d zeta\) if and only if mu is a Carleson measure. Here Gamma(zeta) is the usual nontangential approach region with vertex zeta on the unit circle Gamma(zeta) = {z <epsi is an element of D:\1 - z\ less than or equal to 1 - \z\(2)}, and \d zeta\ is arclength measure on the circle. We also show that if 0 < p less than or equal to 1, beta > 0, and 1 - beta p > 0 then the operator Gf maps the Hardy-Sobolev space H-beta(p) into L-p(\d zeta\) if and only if the function G(mu)(zeta) = integral(Gamma(zeta))d mu/(1 - \z\) belongs to the Morrey space L-p,L-1-beta p. In case p = 1, this condition is equivalent to the condition that mu(T(I)) less than or equal to C\I\(1-beta) for all arcs I contained in the circle, where T(I) is the tent over I contained in the unit disk. (C) 1997 Academic Press.