#### Vol. 3, No. 3, 2021

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Averages along the square integers $\ell^p$-improving and sparse inequalities

### Rui Han, Michael T. Lacey and Fan Yang

Vol. 3 (2021), No. 3, 517–550
##### Abstract

Let $f\in {\ell }^{2}\left(ℤ\right)$. Define the average of $f$ over the square integers by ${A}_{N}f\left(x\right):=\frac{1}{N}{\sum }_{k=1}^{N}f\left(x+{k}^{2}\right).$ We show that ${A}_{N}$ satisfies a local scale-free ${\ell }^{p}$-improving estimate, for $\frac{3}{2}:

 ${N}^{-2∕{p}^{\prime }}\parallel {A}_{N}f{\parallel }_{{\ell }^{{p}^{\prime }}}\lesssim {N}^{-2∕p}\parallel f{\parallel }_{{\ell }^{p}},$

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 $1. The maximal function $Af=\underset{N\ge 1}{sup}$|${A}_{N}f$| 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 $x\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}q$. One requires an estimate uniform in $x$.

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