Words of analytic paraproducts on 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 $T_{g}f(z)={\int }_0^{z}f(\zeta )g^{\mathrm{\prime }}(\zeta )d\zeta$, $S_{g}f(z)={\int }_0^z{f}^{\mathrm{\prime }}(\zeta )g(\zeta )d\zeta$, and $M_{g}f(z)=g(z)f(z)$. An $N$-letter $g$-word is an operator of the form $L=L_1\cdots L_N$, where each $L_j$ is either $M_g$, $S_g$ or $T_g$. It has been recently proved, in a recent paper by A. Aleman and the authors of this paper, that understanding the boundedness of a $g$-word on classical Hardy and Bergman spaces is a challenging problem due to the potential cancellations involved. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary $g$-word on a weighted Bergman space $A_{{\omega }^{p\mathrm{/}2}}^p$, where $\omega =e^{-2\phi }$ is a smooth rapidly decreasing weight. In particular, it states that any $N$-letter $g$-word such that $\mathrm{\#}\{j:L_j=T_g\}=n\ge 1$ is bounded on $A_{{\omega }^{p\mathrm{/}2}}^p$ if and only if $g$ satisfies the “fractional” Bloch-type condition $$\mathrm{\parallel }g{\mathrm{\parallel }}_{{\mathcal{B}}_{\phi }^s}^s=\underset{z\in \mathbb{D}}{\sup }\frac{s\mathrm{\mid }g(z){\mathrm{\mid }}^{s-1}\mathrm{\mid }{g}^{\mathrm{\prime }}(z)\mathrm{\mid }}{1+{\phi }^{\mathrm{\prime }}(\mathrm{\mid }z\mathrm{\mid })}<\mathrm{\infty },$$ where $s=\frac{N}{n}$, and $\mathrm{\parallel }L{\mathrm{\parallel }}_{A_{{\omega }^{p\mathrm{/}2}}^p}\simeq \mathrm{\parallel }g{\mathrm{\parallel }}_{{\mathcal{B}}_{\phi }^s}^N$. The class of smooth rapidly decreasing weights contains the radial weights \begin{equation*} ω_n(z)=e^{-2\exp_{n}(g_{α,c}(|z|))}, \quad\mbox{where}\quad g_{α,c}®=\tfrac{c}{(1-r^2)^α}, \quad\mbox{for $c,\alpha >0$,} \end{equation*} ${\exp }_0(x)=x$ and ${\exp }_n(x)=e^{{\exp }_{n-1}(x)}$, for $n\in \mathbb{N}$. Therefore it contains weights which decrease arbitrarily rapidly to zero as $\mathrm{\mid }z\mathrm{\mid }\to 1^{-}$.