Averages Along the Primes: Improving and Sparse Bounds

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.