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

Abstract

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 $3/2<p\le 2$: \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 $p^{\prime }=\frac{p}{p-1}$ is the conjugate index. The inequality above fails for $1<p<3/2$. The maximal function $Af={\sup }_{N\ge 1}|A_{N}f|$ 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 $x$.

27 pages. To appear in Tunisian Journal of Mathematics