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For a fixed analytic function g on the unit disc, we consider the analyticparaproducts induced by g, which are formally defined by$T_gf(z)=\int_0^zf(\zeta)g'(\zeta)d\zeta$,$S_gf(z)=\int_0^zf'(\zeta)g(\zeta)d\zeta$, and $M_gf(z)=g(z)f(z)$. We areconcerned with the study of the boundedness of operators in the algebra$\mathcal{A}_g$ generated by the above operators acting on Hardy, or standardweighted Bergman spaces on the disc. The general question is certainly verychallenging, since operators in $\mathcal{A}_g$ are finite linear combinationsof finite products (words) of $T_g,S_g,M_g$ which may involve a large amount ofcancellations to be understood. The results in the paper "Composition ofanalytic paraproducts, J. Math. Pures Appl. 158 (2022) 293--319" show thatboundedness of operators in a fairly large subclass of $\mathcal{A}_g$ can becharacterized 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 provedthat there are many operators, even single words in $\mathcal{A}_g$ whoseboundedness cannot be described in terms of these conditions. The present paperprovides a considerable progress in this direction. Our main result provides acomplete quantitative characterization of the boundedness of an arbitrary wordin $\mathcal{A}_g$ in terms of a ``fractional power'' of the symbol $g$, thatonly depends on the number of appearances of each of the letters $T_g,S_g,M_g$in the given word.

Words of analytic paraproducts on Hardy and weighted Bergman spaces