The Linear Bound for the Natural Weighted Resolution of the Haar Shift

TL;DR

This paper establishes a linear bound for the weighted resolution of Haar shifts, extending the Hilbert transform's known bounds to a broader class of operators using harmonic analysis techniques.

Contribution

It extends the linear $A_2$ bound to the natural weighted resolution of Haar shifts, employing advanced harmonic analysis tools and decompositions.

Findings

Proves linear bounds for nine operators in Haar shift resolution.

Extends Hilbert transform bounds to Haar shift operators.

Uses paraproduct composition and Carleson Embedding Theorem.

Abstract

The Hilbert transform has a linear bound in the $A_{2}$ characteristic on weighted $L^{2}$, \begin{equation*} \left\Vert H\right\Vert _{L^{2}(w)\rightarrow L^{2}(w)}\lesssim \left[ w \right] _{A_{2}}, \end{equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{w^{\frac{1}{2}}}\mathcal{S }M_{w^{-\frac{1}{2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \begin{equation*} M_{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \end{equation*} The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function.