${\ell }^p$-improving for discrete spherical averages

Hughes, Kevin

Annales Henri Lebesgue, Volume 3 (2020), pp. 959-980.

We initiate the theory of ${\ell }^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove ${\ell }^p$-improving estimates for the discrete spherical averages and some of their generalizations. As an application of our ${\ell }^p$-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.

Nous initions la théorie sur les inégalités de gain d’intégrabilité ${\ell }^p$ pour des moyennes arithmétiques sur des hypersurfaces ainsi que pour leurs fonctions maximales. En particulier, nous prouvons un gain d’intégrabilité pour les moyennes sphériques discrètes et certaines généralisations. Comme application de notre résultat, nous obtenons une nouvelle estimation pour la fonction maximale sphérique discrète en dimension 4. Nos preuves sont analogues à celle de Littman sur les moyennes sphériques Euclidiennes. Un aspect important de la preuve est une décomposition de Littlewood–Paley reposant simultanément sur des aspects arithmétiques et analytiques. L’aspect arithmétique est encodé par une méthode de décomposition du cercle en arcs mineurs et majeurs.

Classification: 37A45, 42B25, 11P05, 11P55, 11L07
Keywords: $L^p$-improving, discrete averages, discrete maximal functions, circle method, Littlewood–Paley theory

[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, 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, Ph. D. Thesis, Princeton University, USA (2012)

[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. ${\ell }^p$-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, American Mathematical Society (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