Pure and Applied Analysis

Sparse bounds for the discrete spherical maximal functions

Abstract

We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy–Littlewood circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.

Article information

Source
Pure Appl. Anal., Volume 2, Number 1 (2020), 75-92.

Dates
Accepted: 5 July 2019
First available in Project Euclid: 13 December 2019

https://projecteuclid.org/euclid.paa/1576206323

Digital Object Identifier
doi:10.2140/paa.2020.2.75

Mathematical Reviews number (MathSciNet)
MR4041278

Zentralblatt MATH identifier
07159297

Citation

Kesler, Robert; Lacey, Michael T.; Mena, Darío. Sparse bounds for the discrete spherical maximal functions. Pure Appl. Anal. 2 (2020), no. 1, 75--92. doi:10.2140/paa.2020.2.75. https://projecteuclid.org/euclid.paa/1576206323

References

• J. Bourgain, “Estimations de certaines fonctions maximales”, C. R. Acad. Sci. Paris Sér. I Math. 301:10 (1985), 499–502.
• J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations”, Geom. Funct. Anal. 3:2 (1993), 107–156.
• B. Cook, “A note on discrete spherical averages over sparse sequences”, 2018. To appear in Proc. Amer. Math. Soc.
• K. Hughes, “$\ell^p$-improving for discrete spherical averages”, preprint, 2018.
• K. Hughes, “The discrete spherical averages over a family of sparse sequences”, J. Anal. Math. (online publication July 2019).
• A. D. Ionescu, “An endpoint estimate for the discrete spherical maximal function”, Proc. Amer. Math. Soc. 132:5 (2004), 1411–1417.
• R. Kesler, “$\ell^p(\mathbb{Z}^d)$-improving properties and sparse bounds for discrete spherical maximal averages”, 2018. To appear in J. Anal. Math.
• R. Kesler, “$\ell^p(\mathbb{Z}^d)$-improving properties and sparse bounds for discrete spherical maximal means, revisited”, preprint, 2018.
• R. Kesler and M. T. Lacey, “$\ell^{p}$-improving inequalities for discrete spherical averages”, preprint, 2018.
• R. Kesler, M. T. Lacey, and D. Mena Arias, “Lacunary discrete spherical maximal functions”, New York J. Math. 25 (2019), 541–557.
• M. T. Lacey, “Sparse bounds for spherical maximal functions”, 2017. To appear in J. Anal. Math.
• S. Lee, “Endpoint estimates for the circular maximal function”, Proc. Amer. Math. Soc. 131:5 (2003), 1433–1442.
• A. Magyar, “$L^p$-bounds for spherical maximal operators on $\mathbb{Z}^n$”, Rev. Mat. Iberoam. 13:2 (1997), 307–317.
• A. Magyar, E. M. Stein, and S. Wainger, “Discrete analogues in harmonic analysis: spherical averages”, Ann. of Math. $(2)$ 155:1 (2002), 189–208.
• W. Schlag, “A generalization of Bourgain's circular maximal theorem”, J. Amer. Math. Soc. 10:1 (1997), 103–122.
• W. Schlag and C. D. Sogge, “Local smoothing estimates related to the circular maximal theorem”, Math. Res. Lett. 4:1 (1997), 1–15.