Dimension-Free L p-Maximal Inequalities for Spherical Means in ℤm+1N

Abstract

We extend the main result of Krause—the existence of dimension-free $L^p$-bounds, $p > 1$, for the spherical maximal function in the hypercube, $\{0,1\}^N$—to all cyclic groups, $\mathbb{Z}_{m+1}^N, \ m \geq 1$. Our approach follows that of Krause, which grew out of the arguments of Harrow, Kolla, and Schulman, which were in turn motivated by the spectral technique developed by Nevo and Stein, and by Stein, in the context of pointwise ergodic theorems on general groups.