Averages along the Square Integers: $\ell^p$ improving and Sparse Inequalities

Let $f\in \ell^2(\mathbb Z)$. Define the average of $f$ over the square integers by $A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2)$. We show that $A_N$ satisfies a local scale-free $\ell ^{p}$-improving estimate, for $3/2 < p \leq 2$: \begin{equation*} N ^{-2/p'} \lVert A_N f \rVert _{ p'} \lesssim N ^{-2/p} \lVert f\rVert _{\ell ^{p}}, \end{equation*} provided $f$ is supported in some interval of length $N ^2$, and $p' =\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $1< p < 3/2$. The maximal function $A f = \sup _{N\geq 1} |A_Nf|$ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $A$ follow. A critical step in the proof requires the control of a logarithmic average over $q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$.