Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type

TL;DR

This paper establishes endpoint sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions, leading to new weighted norm inequalities and sharp growth estimates in the Walsh setting.

Contribution

It provides the first endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type and derives sharp weighted norm inequalities, extending Lerner's conjectures to the Walsh context.

Findings

Sharp growth rate of $L^p$ weighted operator norm in terms of $A_p$ characteristic.

Endpoint sparse bounds for Walsh-Fourier Marcinkiewicz multipliers.

Novel weighted inequalities for Walsh-Littlewood-Paley square functions.

Abstract

We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{{\frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.