Words of analytic paraproducts on Hardy and weighted Bergman spaces

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by $T_{g}f\left(z\right)={\int }_0^{z}f\left(\zeta \right)g^{\prime }\left(\zeta \right)d\zeta$, $S_{g}f\left(z\right)={\int }_0^z{f}^{\prime }\left(\zeta \right)g\left(\zeta \right)d\zeta$, and $M_{g}f\left(z\right)=g\left(z\right)f\left(z\right)$. We are concerned with the study of the boundedness of operators in the algebra $A_g$ generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in $A_g$ are finite linear combinations of finite products (words) of $T_g,S_g,M_g$ which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of $A_g$ can be characterized by one of the conditions $g\in H^{\infty }$, or $g^n$ belongs to $BMOA$ or the Bloch space, for some integer $n>0$. However, it is also proved that there are many operators, even single words in $A_g$ whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in $A_g$ in terms of a “fractional power” of the symbol g, that only depends on the number of appearances of each of the letters $T_g,S_g,M_g$ in the given word.