Quantitative $l^p$-improving for discrete spherical averages along the primes

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 by Anderson, Cook, Hughes and Kumchev and are discrete, prime variants of Stein’s spherical averages. The proof uses a precise decomposition of the Fourier multiplier.