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

Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type