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We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein, and Wainger) corresponding to the Euclidean spheres in $\mathbb {Z}^{d}$ with dyadic radii have $\ell ^{p}(\mathbb {Z}^{d})$ bounds for all $p\in [2, \infty ]$ independent of the dimensions $d\ge 5$. An important part of our argument is the asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term. By considering new approximating multipliers, we will show how to absorb an exponential in dimension (like $C^{d}$ for some $C>1$) growth in norms arising from the sampling principle of Magyar, Stein, and Wainger and ultimately deduce dimension-free estimates for the discrete spherical maximal functions.

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

Greenblatt

Kolla

Krause

2019

International Mathematics Research Notices

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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.

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Cited by 3 publications

References 33 publications

On Discrete Hardy–Littlewood Maximal Functions over the Balls in $${\boldsymbol {\mathbb {Z}^d}}$$ : Dimension-Free Estimates

On Discrete Hardy–Littlewood Maximal Functions over the Balls in $${\boldsymbol {\mathbb {Z}^d}}$$ : Dimension-Free Estimates

Dimension-Free Estimates for the Discrete Spherical Maximal Functions

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

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