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Dimension-Free $L_2$ Maximal Inequality for Spherical Means in the Hypercube
Received: February 21, 2013
Revised: November 12, 2013
Published: May 23, 2014
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Keywords: maximal inequality, Fourier analysis, Boolean hypercube
ACM Classification: G.3
AMS Classification: 42B25

Abstract: [Plain Text Version]

$ \newcommand{\ep}{\varepsilon} $

We establish the maximal inequality claimed in the title. In combinatorial terms this has the implication that for sufficiently small $\ep>0$, for all $n$, any marking of an $\ep$ fraction of the vertices of the $n$-dimensional hypercube necessarily leaves a vertex $x$ such that marked vertices are a minority of every sphere centered at $x$.