### Metadata

### Abstract

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.

### References

[ACHK18] Improved ${l}^{p}$ boundedness for integral k-spherical maximal functions, Discrete Anal., Volume 2018 (2018), 10, 18 pages | Article | MR 3819049 | Zbl 1404.42034

[BDG16] 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 | Article | MR 3548534 | Zbl 1408.11083

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

[BNW88] Convex hypersurfaces and Fourier transforms, Ann. Math., Volume 127 (1988) no. 2, pp. 333-365 | Article | MR 932301 | Zbl 0666.42010

[Bou85] Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math., Volume 301 (1985) no. 10, pp. 499-502 | MR 812567

[Bou17] On the Vinogradov mean value, Proc. Steklov Inst. Math., Volume 296 (2017), pp. 30-40 | Article | Zbl 1371.11138

[BR15] Rational points on linear slices of diagonal hypersurfaces, Nagoya Math. J., Volume 218 (2015), pp. 51-100 | Article | MR 3345624 | Zbl 1371.11139

[Gra08] Classical Fourier Analysis, Graduate Texts in Mathematics, Volume 249, Springer, 2008 | MR 2445437 | Zbl 1220.42001

[HL14] Discrete Fourier restriction associated with Schrödinger equations, Rev. Mat. Iberoam., Volume 30 (2014) no. 4, pp. 1281-1300 | Article | Zbl 1314.42010

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

[Hug17] Restricted weak-type endpoint estimates for k-spherical maximal functions, Math. Z., Volume 286 (2017) no. 3-4, pp. 1303-1321 | Article | MR 3671577 | Zbl 1375.42032

[Hug19] The discrete spherical averages over a family of sparse sequences, J. Anal. Math., Volume 138 (2019) no. 1, pp. 1-21 | Article | MR 3996030 | Zbl 1423.42035

[Ion04] An endpoint estimate for the discrete spherical maximal function, Proc. Am. Math. Soc., Volume 132 (2004) no. 5, pp. 1411-1417 | Article | MR 2053347 | Zbl 1076.42014

[KL18] ${\ell}^{p}$-improving inequalities for discrete spherical averages (2018) (https://arxiv.org/abs/1804.09845)

[Lee03] Endpoint estimates for the circular maximal function, Proc. Am. Math. Soc., Volume 131 (2003) no. 5, pp. 1433-1442 | MR 1949873 | Zbl 1042.42007

[Lit73] ${L}^{p}-{L}^{q}$-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 0263.44006

[Mag02] Diophantine equations and ergodic theorems, Am. J. Math., Volume 124 (2002) no. 5, pp. 921-953 | Article | MR 1925339 | Zbl 1042.37004

[Mag07] On the distribution of lattice points on spheres and level surfaces of polynomials, J. Number Theory, Volume 122 (2007) no. 1, pp. 69-83 | Article | MR 2287111 | Zbl 1119.11046

[MSW02] Discrete analogues in harmonic analysis: spherical averages, Ann. Math., Volume 155 (2002) no. 1, pp. 189-208 | Article | MR 1888798 | Zbl 1036.42018

[RdF86] Maximal functions and Fourier transforms, Duke Math. J., Volume 53 (1986) no. 2, pp. 395-404 | Article | MR 850542 | Zbl 0612.42008

[Woo12] The asymptotic formula in Waring’s problem, Int. Math. Res. Not., Volume 2012 (2012) no. 7, pp. 1485-1504 | Article | MR 2913181 | Zbl 1267.11104