Weighted inequalities for multivariable dyadic paraproducts

Abstract

Using Wilson�s Haar basis in Rn, which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in Rn. We can then extend �trivially� Beznosova�s Bellman function proof of the linear bound in L2(w) with respect to [w]A2 for the 1-dimensional dyadic paraproduct. Here trivial means that each piece of the argument that had a Bellman func- tion proof has an n-dimensional counterpart that holds with the same Bellman function. The lemma that allows for this painless extension we call the good Bellman function Lemma. Further- more the argument allows to obtain dimensionless bounds in the anisotropic case.