Volume 3 (2020)
Hughes, Kevin
ℓ p -improving for discrete spherical averages
Annales Henri Lebesgue, Volume 3 (2020), pp. 959-980.

Metadata

DOI10.5802/ahl.50
Keywords $L^p$-improving, discrete averages, discrete maximal functions, circle method, Littlewood–Paley theory

Abstract

We initiate the theory of ℓ p -improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove ℓ p -improving estimates for the discrete spherical averages and some of their generalizations. As an application of our ℓ p -improving inequalities for the dyadic discrete spherical maximal function, we give a new estimate for the full discrete spherical maximal function in four dimensions. Our proofs are analogous to Littman’s result on Euclidean spherical averages. One key aspect of our proof is a Littlewood–Paley decomposition in both the arithmetic and analytic aspects. In the arithmetic aspect this is a major arc-minor arc decomposition of the circle method.


References

[ACHK18] Anderson, Theresa; Cook, Brian; Hughes, Kevin; Kumchev, Angel Improved l p boundedness for integral k-spherical maximal functions, Discrete Anal., Volume 2018 (2018), 10, 18 pages | DOI | MR | Zbl

[BDG16] Bourgain, Jean; Demeter, Ciprian; Guth, Larry Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. Math., Volume 184 (2016) no. 2, pp. 633-682 | DOI | MR | Zbl

[Bir62] Birch, Bryan J. Forms in many variables, Proc. R. Soc. Lond. Ser. A, Volume 265 (1961/62), pp. 245-263

[BNW88] Bruna, Joaquim; Nagel, Alexander; Wainger, Stephen Convex hypersurfaces and Fourier transforms, Ann. Math., Volume 127 (1988) no. 2, pp. 333-365 | DOI | MR | Zbl

[Bou85] Bourgain, Jean Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math., Volume 301 (1985) no. 10, pp. 499-502 | MR

[Bou17] Bourgain, Jean On the Vinogradov mean value, Proc. Steklov Inst. Math., Volume 296 (2017), pp. 30-40 | DOI | Zbl

[BR15] Brüdern, Jörg; Robert, Olivier Rational points on linear slices of diagonal hypersurfaces, Nagoya Math. J., Volume 218 (2015), pp. 51-100 | DOI | MR | Zbl

[Gra08] Grafakos, Loukas Classical Fourier Analysis, Graduate Texts in Mathematics, Volume 249, Springer, 2008 | MR | Zbl

[HL14] Hu, Yi; Li, Xiaochun Discrete Fourier restriction associated with Schrödinger equations, Rev. Mat. Iberoam., Volume 30 (2014) no. 4, pp. 1281-1300 | DOI | Zbl

[Hug12] Hughes, Kevin Problems and results related to waring’s problem: Maximal functions and ergodic averages (2012) (Ph. D. Thesis)

[Hug17] Hughes, Kevin Restricted weak-type endpoint estimates for k-spherical maximal functions, Math. Z., Volume 286 (2017) no. 3-4, pp. 1303-1321 | DOI | MR | Zbl

[Hug19] Hughes, Kevin The discrete spherical averages over a family of sparse sequences, J. Anal. Math., Volume 138 (2019) no. 1, pp. 1-21 | DOI | MR | Zbl

[Ion04] Ionescu, Alexandru D. An endpoint estimate for the discrete spherical maximal function, Proc. Am. Math. Soc., Volume 132 (2004) no. 5, pp. 1411-1417 | DOI | MR | Zbl

[KL18] Kesler, Robert; Lacey, Michael T. ℓ p -improving inequalities for discrete spherical averages (2018) (https://arxiv.org/abs/1804.09845)

[Lee03] Lee, Sanghyuk Endpoint estimates for the circular maximal function, Proc. Am. Math. Soc., Volume 131 (2003) no. 5, pp. 1433-1442 | MR | Zbl

[Lit73] Littman, Walter L p -L q -estimates for singular integral operators arising from hyperbolic equations, Partial differential Equations, Berkeley 1971 (Proceedings of Symposia in Pure Mathematics) Volume 23 (1973), pp. 479-481 | Zbl

[Mag02] Magyar, Ákos Diophantine equations and ergodic theorems, Am. J. Math., Volume 124 (2002) no. 5, pp. 921-953 | DOI | MR | Zbl

[Mag07] Magyar, Ákos On the distribution of lattice points on spheres and level surfaces of polynomials, J. Number Theory, Volume 122 (2007) no. 1, pp. 69-83 | DOI | MR | Zbl

[MSW02] Magyar, Ákos; Stein, Elias M.; Wainger, Stephen Discrete analogues in harmonic analysis: spherical averages, Ann. Math., Volume 155 (2002) no. 1, pp. 189-208 | DOI | MR | Zbl

[RdF86] Rubio de Francia, José L. Maximal functions and Fourier transforms, Duke Math. J., Volume 53 (1986) no. 2, pp. 395-404 | DOI | MR | Zbl

[Woo12] Wooley, Trevor D. The asymptotic formula in Waring’s problem, Int. Math. Res. Not., Volume 2012 (2012) no. 7, pp. 1485-1504 | DOI | MR | Zbl