Bounds for Lacunary maximal functions given by Birch–Magyar averages

Cook, Brian; Hughes, Kevin

doi:10.1090/tran/8152

Bounds for Lacunary maximal functions given by Birch–Magyar averages
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by Brian Cook and Kevin Hughes PDF

Trans. Amer. Math. Soc. 374 (2021), 3859-3879

Abstract:

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar.

In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near $\ell ^1$. For our negative results we generalize an argument of Zienkiewicz.

References

Theresa Anderson, Brian Cook, Kevin Hughes, and Angel Kumchev, Improved $\ell ^p$-boundedness for integral $k$-spherical maximal functions, Discrete Anal. , posted on (2018), Paper No. 10, 18. MR 3819049, DOI 10.19086/da.3675
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
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
J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), no. 1, 39–72. MR 937581, DOI 10.1007/BF02776301
J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. I.H.E.S. 69 (1989), 5–45.
Zoltán Buczolich and R. Daniel Mauldin, Divergent square averages, Ann. of Math. (2) 171 (2010), no. 3, 1479–1530. MR 2680392, DOI 10.4007/annals.2010.171.1479
Calixto P. Calderón, Lacunary spherical means, Illinois J. Math. 23 (1979), no. 3, 476–484. MR 537803
M. Christ, Weak type $(1,1)$ bounds for rough operators, Ann. of Math. (2), 128 19–42 (1988), no. 1, ISSN 0003-486X. DOI 10.2307/1971461.
M. Christ, A weak type (1,1) inequality for maximal averages over certain sparse sequences, arXiv e-prints, art. arXiv:1108.5664, Aug 2011.
L. Cladek and B. Krause, Improved endpoint bounds for the lacunary spherical maximal operator, arXiv e-prints, art. arXiv:1703.01508, Mar 2017.
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
B. Cook, A note on discrete spherical averages over sparse sequences, arXiv pre-print: arXiv:1808.03822.
Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
H. Davenport, On a principle of Lipschitz, J. London Math. Soc. 26 (1951), 179–183. MR 43821, DOI 10.1112/jlms/s1-26.3.179
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
Kevin J. Hughes Jr, Arithmetic analogues in harmonic analysis: Results related to Waring’s problem, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Princeton University. MR 3122031
Kevin Hughes, The discrete spherical averages over a family of sparse sequences, J. Anal. Math. 138 (2019), no. 1, 1–21. MR 3996030, DOI 10.1007/s11854-019-0020-z
Kevin Hughes, Restricted weak-type endpoint estimates for k-spherical maximal functions, Math. Z. 286 (2017), no. 3-4, 1303–1321. MR 3671577, DOI 10.1007/s00209-016-1802-y
K. Hughes, $\ell ^p$-improving for discrete spherical averages, Annales Henri Lebesgue, Vol. 3 (2020), 959-980. DOI 10.5802/ahl.50
K. Hughes, J. Wright, and J. Zienkiewicz, An ergodic Hasse principle, pre-print.
Alexandru D. Ionescu, An endpoint estimate for the discrete spherical maximal function, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1411–1417. MR 2053347, DOI 10.1090/S0002-9939-03-07207-1
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
Robert Kesler, Michael T. Lacey, and Darío Mena Arias, Lacunary discrete spherical maximal functions, New York J. Math. 25 (2019), 541–557. MR 3982252
Patrick LaVictoire, Universally $L^1$-bad arithmetic sequences, J. Anal. Math. 113 (2011), 241–263. MR 2788358, DOI 10.1007/s11854-011-0006-y
Akos Magyar, Diophantine equations and ergodic theorems, Amer. J. Math. 124 (2002), no. 5, 921–953. MR 1925339
Akos Magyar, $L^p$-bounds for spherical maximal operators on $\mathbf Z^n$, Rev. Mat. Iberoamericana 13 (1997), no. 2, 307–317. MR 1617657, DOI 10.4171/RMI/222
Akos Magyar, On the distribution of lattice points on spheres and level surfaces of polynomials, J. Number Theory 122 (2007), no. 1, 69–83. MR 2287111, DOI 10.1016/j.jnt.2006.03.006
Mariusz Mirek, Weak type $(1,1)$ inequalities for discrete rough maximal functions, J. Anal. Math. 127 (2015), 247–281. MR 3421994, DOI 10.1007/s11854-015-0030-4
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
Andreas Seeger, Terence Tao, and James Wright, Pointwise convergence of lacunary spherical means, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001) Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 341–351. MR 1979950, DOI 10.1090/conm/320/05617
S. B. Stečkin, An estimate of a complete rational trigonometric sum, Trudy Mat. Inst. Steklov. 143 (1977), 188–207, 211 (Russian). Analytic number theory, mathematical analysis and their applications (dedicated to I. M. Vinogradov on his 85th birthday). MR 0480376
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
Roman Urban and Jacek Zienkiewicz, Weak type $(1,1)$ estimates for a class of discrete rough maximal functions, Math. Res. Lett. 14 (2007), no. 2, 227–237. MR 2318621, DOI 10.4310/MRL.2007.v14.n2.a6
J. Zienkiewicz, Personal communication, 2014.