Generalized Area Operators on Hardy Spaces

AbstractWe show that if 0<p<∞ then the operatorGf(ζ)=∫Γ(ζ)|f(z)|dμ/(1−|z|) maps the Hardy spaceHptoLp(|dζ|) if and only if μ is a Carleson measure. Here Γ(ζ) is the usual nontangential approach region with vertex ζ on the unit circle Γ(ζ)={z∈D:|1−z|≤1−|z|2}, and |dζ| is arclength measure on the circle. We also show that if 0<p≤1, β>0, and 1−βp>0 then the operatorGfmaps the Hardy–Sobolev spaceHpβintoLp(|dζ|) if and only if the functionGμ(ζ)=∫Γ(ζ)dμ/(1−|z|) belongs to the Morrey spaceLp,1−βp. In casep=1, this condition is equivalent to the condition that μ(T(I))≤C|I|1−βfor all arcsIcontained in the circle, whereT(I) is the tent overIcontained in the unit disk