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

## TL;DR

This paper establishes quantitative $l^p$-improving bounds for discrete spherical averages along primes, extending Stein's spherical averages to the prime setting with detailed Fourier analysis.

## Contribution

It provides the first quantitative $l^p$-improving estimates for prime-based discrete spherical averages, using a refined Fourier multiplier decomposition.

## Key findings

- Established $l^p$-improving estimates depending on the radius.
- Extended Stein's spherical averages to primes with quantitative bounds.
- Used a precise Fourier multiplier decomposition for the proof.

## 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 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.