Averages Along the Primes: Improving and Sparse Bounds

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 ${\mathcal{A}}^{\ast }f={\sup }_N|{\mathcal{A}}_{N}f|$ 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.

13 pages