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 (z) = ∫ 0 z f (ζ) g ′ (ζ) d ζ , S g f (z) = ∫ 0 z f ′ (ζ) g (ζ) d ζ , and M g f (z) = g (z) f (z). 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 ∈ H ∞ , or g n belongs to B M O A 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. [ABSTRACT FROM AUTHOR]