Averages Along the Primes: Improving and Sparse Bounds

TL;DR

This paper establishes scale-free $ ext{ell}^p$-improving estimates and sparse bounds for averages along prime numbers, leading to new weighted inequalities for the associated maximal function.

Contribution

It introduces novel scale-free $ ext{ell}^p$-improving and sparse bounds for prime averages, enabling the first weighted inequalities for the maximal function along primes.

Findings

Proves uniform scale-free $ ext{ell}^p$-improving estimates for prime averages.

Establishes $ (p,p)$ sparse bounds for the maximal function for all $ 1< p < 2$.

Demonstrates boundedness of the maximal function on weighted $ ext{ell}^p(w)$ spaces for $ w$ in $ A_p$.

Abstract

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds \begin{equation*} N ^{-1/p' }\lVert \mathcal{A}_N f\rVert_{\ell^{p'}} \leq C_p N ^{- 1/p} \lVert f\rVert_{\ell^p}. \end{equation*} The maximal function $ \mathcal{A}^{\ast} f =\sup_{N} \lvert \mathcal{A}_N f \rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ \mathcal{A}^{\ast} $ is bounded on $ \ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ \mathcal{A}^{\ast} $ were known.