Quantitative weighted estimates for the Littlewood-Paley square function and Marcinkiewicz multipliers

TL;DR

This paper derives sharp quantitative weighted bounds for the Littlewood-Paley square function and Marcinkiewicz multipliers, highlighting their dependence on weight characteristics in $L^p(w)$ spaces.

Contribution

It provides the first sharp weighted estimates for these operators, especially detailing the dependence on the $A_p$ characteristic.

Findings

Sharp $A_p$ dependence for $L^p(w)$ operator norm of $S$ for $1<p extless=2

Quantitative bounds for Marcinkiewicz multipliers

Improved understanding of weighted inequalities in harmonic analysis

Abstract

Quantitative weighted estimates are obtained for the Littlewood-Paley square function $S$ associated with a lacunary decomposition of ${\mathbb R}$ and for the Marcinkiewicz multiplier operator. In particular, we find the sharp dependence on $[w]_{A_p}$ for the $L^p(w)$ operator norm of $S$ for $1<p\le 2$.