Area operators on Hardy spaces.

摘要

We define the operator Gm by Gmf t=Ga t fzdm , where m is a non-negative measure on the upper half-plane and Gat is a cone in the upper-half plane, with the vertex t on R , defined by a curve y = ax . We show that if 0 p ≤ 1 ≤ q infinity then a necessary and sufficient condition that Gm is bounded from Hp to L q is that SI G*g zdm≤C' I1p for all g ∈ Lq' and g ≥ 0; where gq '≤1;1q +1q'=1 ; and G*g z=x-a-1 yx+a- 1yg tdt.; Moreover, Gm is bounded from H1 to L 1 if and only if a-1ydm is a Carleson measure. If a-1ydm is a Carleson measure, and Gat is a nontangential cone, then Gm is bounded from Hp to L p, 0 p infinity. For a function f holomorphic on the unit disk, we define the operator Gm by Gmf&parl0; eit&parr0;=G&parl0;e it&parr0;fz dm , where G&parl0;eit&parr0; is the usual nontangential cone inside D with the vertex eit on T . We show that a necessary and sufficient condition that Gm is a compact operator is that m is a compact Carleson measure. We show that if 1 p infinity, then Gm is a bounded operator in generalized Nevanlinna class if and only if Tlog 1+mGe it pdtinfinity . We show that Gm is bounded from Hinfinity to Lp, for 0 p ≤ infinity, if and only if m&parl0;G&parl0;eit&parr0;&parr0; ∈ Lp. The operator Gm is also bounded from Hinfinity to BMO if m&parl0;G&parl0;eit&parr0;&parr0; ∈ Linfinity.