Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

p-Improving Inequalities for Discrete Spherical Averages


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.

This is a preview of subscription content, log in to check access.


  1. [1]

    T. Anderson, B. Cook, K. Hughes and A. Kumchev, Improved ℓp-boundedness for integral k-spherical maximal functions, Discrete Anal. (2018), Paper No. 10, 18 pp.

  2. [2]

    J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris S´er. I Math., 301 (1985), 499–502.

  3. [3]

    M. Christ, Convolution, curvature, and combinatorics: a case study, Internat. Math. Res. Notices, 19 (1998), 1033–1048.

  4. [4]

    B. Cook, Maximal function inequalities and a theorem of Birch, Israel J. Math., 231 (2019), 211–241.

  5. [5]

    A. Culiuc, R. Kesler and M. T. Lacey, Sparse bounds for the discrete cubic Hilbert transform, Anal. PDE, 12 (2019), 1259–1272.

  6. [6]

    K. Hughes, Restricted weak-type endpoint estimates for k-spherical maximal functions, Math. Z., 286 (2017), 1303–1321.

  7. [7]

    K. Hughes, The discrete spherical averages over a family of sparse sequences, (2016), ArXiv: 1609.04313.

  8. [8]

    K. Hughes, ℓp-improving for discrete spherical averages, (2018), ArXiv: 1804.09260.

  9. [9]

    A. D. Ionescu, An endpoint estimate for the discrete spherical maximal function, Proc. Amer. Math. Soc., 132 (2004), 1411–1417.

  10. [10]

    A. D. Ionescu, An endpoint estimate for the discrete spherical maximal function,Proc. Amer. Math. Soc., 132 (2004), 1411–1417.

  11. [11]

    R. Kesler and D. M. Arias, Uniform sparse bounds for discrete quadratic phase Hilbert transforms, Anal. Math. Phys., 9 (2019), 263–274.

  12. [12]

    R. Kesler, M. T. Lacey and D. M. Arias, Sparse bound for the discrete spherical maximal functions, Pure Appl. Anal. (2018) (to appear).

  13. [13]

    H. D. Kloosterman, On the representation of numbers in the form ax2 + by2 + cz2 + dt2, Acta Math., 49 (1927), 407–464.

  14. [14]

    W. Littman, LpLq -estimates for singular integral operators arising from hyperbolic equations, in: Partial Differential Equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc. (Providence, R.I., 1973), pp. 479–481.

  15. [15]

    Á. Magyar, Lp-bounds for spherical maximal operators on Zn, Rev. Mat. Iberoam., 13 (1997), 307–317.

  16. [16]

    Á. Magyar, Diophantine equations and ergodic theorems, Amer. J. Math.,124 (2002), 921–953.

  17. [17]

    Á. Magyar, On the distribution of lattice points on spheres and level surfaces of polynomials, J. Number Theory, 122 (2007), 69–83.

  18. [18]

    Á. Magyar, E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis: spherical averages, Ann. of Math. (2), 155 (2002), 189–208.

  19. [19]

    Á. Magyar, E. M. Stein and S. Wainger, Maximal operators associated to discrete subgroups of nilpotent Lie groups, J. Anal. Math., 101 (2007), 257–312.

  20. [20]

    D. M. Oberlin, Two discrete fractional integrals, Math. Res. Lett., 8 (2001), 1–6.

  21. [21]

    L. B. Pierce, Discrete fractional Radon transforms and quadratic forms, Duke Math. J., 161 (2012), 69–106.

  22. [22]

    E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis. II. Fractional integration, J. Anal. Math., 80 (2000), 335–355.

  23. [23]

    E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A., 73 (1976), 2174–2175.

  24. [24]

    E. M. Stein and S. Wainger, Two discrete fractional integral operators revisited, J. Anal. Math., 87 (2002), 451–479.

  25. [25]

    R. S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc., 148 (1970), 461–471.

  26. [26]

    T. Tao and J. Wright, Lp improving bounds for averages along curves,J. Amer. Math. Soc., 16 (2003), 605–638.

  27. [27]

    A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34 (1948), 204–207.

Download references

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kesler, R., Lacey, M.T. ℓp-Improving Inequalities for Discrete Spherical Averages. Anal Math 46, 85–95 (2020).

Download citation

Key words and phrases

  • discrete spherical average
  • maximal function

Mathematics Subject Classification

  • 42B25