Weighted estimates for dyadic paraproducts and t-Haar multipiers with complexity (m,n)

TL;DR

This paper extends the analysis of dyadic paraproducts and t-Haar multipliers to operators with complexity (m,n), establishing bounds that depend linearly or polynomially on complexity, and providing new proofs for known cases.

Contribution

It introduces new bounds for dyadic paraproducts and t-Haar multipliers with complexity (m,n), generalizing previous results and offering alternative proofs.

Findings

Weighted L^2(w)-norm of paraproducts depends linearly on A_2-characteristic and BMO-norm, polynomially on complexity.

L^2-norm of t-Haar multipliers depends on square roots of C_{2t} and A_2-characteristics, polynomially on complexity.

New proof of linear bound for dyadic paraproduct with zero complexity.

Abstract

We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m,n), for m and n positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted L^2(w)-norm of a paraproduct with complexity (m,n) associated to a function b\in BMO, depends linearly on the A_2-characteristic of the weight w, linearly on the BMO-norm of b, and polynomially in the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity (0,0)). Also we prove that the L^2-norm of a t-Haar multiplier for any t and weight w depends on the square root of the C_{2t}-characteristic of w times the square root of the A_2-characteristic of w^{2t} and polynomially in the complexity (m,n), recovering a result of Beznosova for the (0,0)-complexity case.