Weighted inequalities for multivariable dyadic paraproducs

TL;DR

This paper extends Bellman function techniques to establish weighted inequalities for multivariable dyadic paraproducts using Wilson's Haar basis, achieving dimensionally robust bounds in higher dimensions.

Contribution

It introduces an n-dimensional Bellman function approach for dyadic paraproducts based on Wilson's Haar basis, generalizing 1D results to multiple dimensions.

Findings

Established linear bounds in L^2(w) with respect to [w]_{A_2} in higher dimensions.

Extended Bellman function proof technique to multivariable dyadic paraproducts.

Achieved dimensionless bounds in anisotropic settings.

Abstract

Using Wilson's Haar basis in $\R^n$, which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in $\R^n$. We can then extend "trivially" Beznosova's Bellman function proof of the linear bound in $L^2(w)$ with respect to $[w]_{A_2}$ for the 1-dimensional dyadic paraproduct. Here trivial means that each piece of the argument that had a Bellman function 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. Furthermore the argument allows to obtain dimensionless bounds in the anisotropic case.