Compostion of Analytic Paraproducts

Abstract

For a fixed analytic function $g$ on the unit disc $\D$, we consider the analytic paraproducts induced by $g$, which are defined by $T_gf(z)= \int_0^z f(\z)g'(\z)\,d\z$, $S_gf(z)= \int_0^z f'(\z)g(\z)\,d\z$, and $M_gf(z)= f(z)g(z)$. The boundedness of these operators on various spaces of analytic functions on $\D$ is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example $T_g^2, \,T_gS_g,\, M_gT_g$, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol $g$. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol $g$ than the case of a single paraproduct.