Discrete analogues in harmonic analysis: Spherical averages

2004/09/20 by A. Magyar, E. M. Stein, S. Wainger · 2 citations
Mathematics · #math.CA

paper · pdf

published as Ann. of Math. (2), Vol. 155, (2002), no. 1, 189--208 · 20 pages, published version

arxiv created 2004/09/20 · arxiv updated 2009/12/01

Abstract

In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the application of certain "sampling" techniques, and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular, the "circle method". The results we obtained are by necessity limited to d ≥ 5, and moreover the range of p for the L^p estimates differs from its analogue in \Bbb R^d.

Cited by

• Pointwise ergodic theorem along primes of the form x² + ny²
• Dimension-free estimate for semi-commutative discrete spherical maximal operator

Related

• ℓ^p-improving for discrete spherical averages
• Quantitative l^p-improving for discrete spherical averages along the primes
• ℓ^p(ℤ^d)-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages
• The discrete spherical averages over a family of sparse sequences
• Restricted weak-type endpoint estimates for discrete k-spherical maximal functions