Generalized area operators on Hardy spaces

[摘要] 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 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.

[发布日期] 1997-12-01