A note on discrete spherical averages over sparse sequences

Brian Cook

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arXiv · 1808.03822

A note on discrete spherical averages over sparse sequences

Brian Cook

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Abstract

This note presents an example of an increasing sequence $(\lambda_l)_{l=1}^\infty$ such that the maximal operators associated to normalized discrete spherical convolution averages \[ \sup_{l\geq 1}\frac{1}{r(\lambda_l)}\left|\sum_{|x|^2=\lambda_l}f(y-x)\right|\] for functions $f:\mathbb{Z}^n\to\mathbb{C}^n$ are bounded on $\ell^p$ for all $p>1$ when the ambient dimension $n$ is at least five.

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Brian Cook. 2018-08-11. A note on discrete spherical averages over sparse sequences. https://arxiv.org/abs/1808.03822

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