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arXiv · 1805.06060

Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type

Wei Chen·Amalia Culiuc·Francesco Di Plinio·Michael Lacey·Yumeng Ou
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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.

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Wei Chen, Amalia Culiuc, Francesco Di Plinio, Michael Lacey, Yumeng Ou. 2019-05-26. Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type. https://arxiv.org/abs/1805.06060

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