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p-Improving Inequalities for Discrete Spherical Averages

Abstract

Let λ2 ∈ ℕ, and in dimensions d ≥ 5, let Aλf(x) denote the average of f: ℤd → ℝ over the lattice points on the sphere of radius λ centered at x. We prove ℓp improving properties of Aλ:

$$\begin{array}{*{20}{c}} {{{\left\| {{A_\lambda }} \right\|}_{{\ell ^{p \to }}{\ell ^{p'}}}} \leqslant {C_{d,p,\omega ({\lambda ^2})}}{\lambda ^{(1 - \tfrac{2}{p})}},}&{\frac{{d - 1}}{{d + 1}} < p \leqslant \frac{d}{{d - 2}}} \end{array}.$$

It holds in dimension d = 4 for odd λ2. The dependence is in terms of ω2), the number of distinct prime factors of λ2. These inequalities are discrete versions of a classical inequality of Littman and Strichartz on the Lp improving property of spherical averages on ℝd. In particular they are scale free, in a natural sense. The proof uses the decomposition of the corresponding multiplier whose properties were established by Magyar–Stein–Wainger, and Magyar. We then use a proof strategy of Bourgain, which dominates each part of the decomposition by an endpoint estimate.

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Author information

Correspondence to M. T. Lacey.

Additional information

Research was supported in part by grant from the US National Science Foundation, DMS-1600693 and the Australian Research Council ARC DP160100153.

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Kesler, R., Lacey, M.T. ℓp-Improving Inequalities for Discrete Spherical Averages. Anal Math 46, 85–95 (2020). https://doi.org/10.1007/s10476-020-0019-9

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Key words and phrases

  • discrete spherical average
  • maximal function

Mathematics Subject Classification

  • 42B25