Quantitative $$l^p$$-Improving for Discrete Spherical Averages Along the Primes

Abstract

We show quantitative (in terms of the radius) $$l^p$$-improving estimates for the discrete spherical averages along the primes. These averaging operators were defined in [1] and are discrete, prime variants of Stein’s spherical averages. The proof uses a precise decomposition of the Fourier multiplier.