ℓ^p-improving for discrete spherical averages

2018/04/24 by Kevin Hughes, Hughes, Kevin · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1804.09260

openalex publication_date 2018/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Supercritical discrete restriction estimates for forms in many variables

Related

• ℓ^p-improving inequalities for Discrete Spherical Averages
• ℓ^p(ℤ^d)-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages
• Quantitative l^p-improving for discrete spherical averages along the primes
• Lacunary Discrete Spherical Maximal Functions
• A framework for discrete bilinear spherical averages and applications to ℓ^p-improving estimates