For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by , , and . We are concerned with the study of the boundedness of operators in the algebra 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 are finite linear combinations of finite products (words) of 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 can be characterized by one of the conditions , or belongs to or the Bloch space, for some integer . However, it is also proved that there are many operators, even single words in 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 in terms of a “fractional power” of the symbol g, that only depends on the number of appearances of each of the letters in the given word.
