Averages along the square integers ℓp-improving and sparse inequalities

Abstract

Let $f\in {\ell }^2\left(\mathbb{Z}\right)$. Define the average of $f$ over the square integers by

$A_{N}f\left(x\right):=\frac{1}{N}{{\displaystyle {\displaystyle \sum _{k=1}^{N}f\left(x+k^2\right)}}}^{\text{}}.$

We show that $A_N$ satisfies a local scale-free ${\ell }^p$-improving estimate, for :

provided $f$ is supported in some interval of length $N^2$, and $p^{\prime }=p∕\left(p-1\right)$ is the conjugate index. The inequality above fails for . 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 $G\left(q,x\right)$ counting the number of square roots of $xmodq$. One requires an estimate uniform in $x$.