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

 Download this article For screen For printing  Recent Issues Volume 5, Issue 4 Volume 5, Issue 3 Volume 5, Issue 2 Volume 5, Issue 1 Volume 4, Issue 4 Volume 4, Issue 3 Volume 4, Issue 2 Volume 4, Issue 1 Volume 3, Issue 4 Volume 3, Issue 3 Volume 3, Issue 2 Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1 Volume 1, Issue 4 Volume 1, Issue 3 Volume 1, Issue 2 Volume 1, Issue 1  The Journal About the Journal Statement, 2023 Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN (electronic): 2576-7666 ISSN (print): 2576-7658 Author Index To Appear Other MSP Journals  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 $x$.

##### Keywords
improving discrete quadratic residues, sparse bounds, circle method
##### Mathematical Subject Classification 2010
Primary: 11L05, 42A45