  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.
