A note on discrete spherical averages over sparse sequences

Cook, Brian

doi:10.1090/proc/14575

A note on discrete spherical averages over sparse sequences
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Proc. Amer. Math. Soc. 150 (2022), 4303-4314 Request permission

Abstract:

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

References

B. J. Birch, Forms in many variables, Proc. Roy. Soc. London Ser. A 265 (1961/62), 245–263. MR 150129, DOI 10.1098/rspa.1962.0007
Jean Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein. MR 1019960
J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 874045, DOI 10.1007/BF02792533
Calixto P. Calderón, Lacunary spherical means, Illinois J. Math. 23 (1979), no. 3, 476–484. MR 537803
R. R. Coifman and Guido Weiss, Book Review: Littlewood-Paley and multiplier theory, Bull. Amer. Math. Soc. 84 (1978), no. 2, 242–250. MR 1567040, DOI 10.1090/S0002-9904-1978-14464-4
Brian Cook, Maximal function inequalities and a theorem of Birch, Israel J. Math. 231 (2019), no. 1, 211–241. MR 3960006, DOI 10.1007/s11856-019-1853-y
Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, DOI 10.1007/BF01388746
K. Hughes, The discrete spherical averages over a family of sparse sequences, J. Anal. Math. (to appear).
R. Kesler and M. T. Lacey, $\ell ^p$-improving inequalities for discrete spherical averages, Anal. Math. 46 (2020), no. 1, 85–95. MR 4064582, DOI 10.1007/s10476-020-0019-9
Akos Magyar, Diophantine equations and ergodic theorems, Amer. J. Math. 124 (2002), no. 5, 921–953. MR 1925339
A. Magyar, E. M. Stein, and S. Wainger, Discrete analogues in harmonic analysis: spherical averages, Ann. of Math. (2) 155 (2002), no. 1, 189–208. MR 1888798, DOI 10.2307/3062154
Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174