Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type

Amalia Culiuc, Francesco Di Plinio, Michael Lacey, Yumeng Ou

Published 2018-05-15Version 1

We prove sharp 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 a sharp range of weighted norm inequalities for these operators. In particular, we obtain 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 both Marcinkiewicz multipliers and Walsh-Littlewood-Paley square functions. Zygmund's $L{(\log L)^{{\frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof.