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

## TL;DR

This paper establishes $ ext{ell}^p$-improving inequalities for averages over square integers, introduces sparse bounds for the maximal function, and develops weighted and vector-valued inequalities, advancing harmonic analysis on discrete structures.

## Contribution

It proves new $ ext{ell}^p$-improving estimates for square integer averages, including sparse bounds and weighted inequalities, with novel control of quadratic residue counts.

## Key findings

- Established $ ext{ell}^p$-improving estimates for $A_N$ for $3/2 < p extless 2$.
- Proved sparse bounds for the maximal function $A$, leading to weighted inequalities.
- Identified the failure of inequalities for $1< p < 3/2$ and developed uniform estimates for quadratic residue counts.

## 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 \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 $x$.