Composition of analytic paraproducts

Abstract

For a fixed analytic function $g$ on the unit $\operatorname{disc} \mathbb{D}$, we consider the analytic paraproducts induced by $g$, which are defined by $T_g f(z)=\int_0^z f(\zeta) g^{\prime}(\zeta) d \zeta, S_g f(z)=\int_0^z f^{\prime}(\zeta) g(\zeta) d \zeta$, and $M_g f(z)=$ $f(z) g(z)$. The boundedness of these operators on various spaces of analytic functions on $\mathbb{D}$ is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example $T_g^2, T_g S_g, M_g T_g$, 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.