Abstract
We show that the discrete Hardy–Littlewood maximal functions associated with the Euclidean balls in \(\mathbb Z^d\) with dyadic radii have bounds independent of the dimension on \(\ell ^p(\mathbb Z^d)\) for p ∈ [2, ∞].
Jean Bourgain was supported by NSF grant DMS-1800640. Mariusz Mirek was partially supported by the Schmidt Fellowship and the IAS School of Mathematics and by the National Science Center, Poland grant DEC-2015/19/B/ST1/01149. Elias M. Stein was partially supported by NSF grant DMS-1265524. Błażej Wróbel was partially supported by the National Science Centre, Poland grant Opus 2018/31/B/ST1/00204.
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Acknowledgements
The authors are grateful to the referees for careful reading of the manuscript and useful remarks that led to the improvement of the presentation. We also thank for pointing out a simple proof of Lemma 2.5.
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Bourgain, J., Mirek, M., Stein, E.M., Wróbel, B. (2020). On Discrete Hardy–Littlewood Maximal Functions over the Balls in \({\boldsymbol {\mathbb {Z}^d}}\): Dimension-Free Estimates. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2256. Springer, Cham. https://doi.org/10.1007/978-3-030-36020-7_8
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