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Abstract

Let
$f\in {\ell}^{2}\left(\mathbb{Z}\right)$. Define the average
of
$f$ over the square
integers by
${A}_{N}f\left(x\right):=\frac{1}{N}{\sum}_{k=1}^{N}f\left(x+{k}^{2}\right).$ We show
that
${A}_{N}$ satisfies a local
scalefree
${\ell}^{p}$improving
estimate, for
$\frac{3}{2}<p\le 2$:
$${N}^{2\u2215{p}^{\prime}}\parallel {A}_{N}f{\parallel}_{{\ell}^{{p}^{\prime}}}\lesssim {N}^{2\u2215p}\parallel f{\parallel}_{{\ell}^{p}},$$ 
provided
$f$ is supported
in some interval of length
${N}^{2}$,
and
${p}^{\prime}=p\u2215\left(p1\right)$
is the conjugate index. The inequality above fails for
$1<p<\frac{3}{2}$. 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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Keywords
improving discrete quadratic residues, sparse bounds,
circle method

Mathematical Subject Classification 2010
Primary: 11L05, 42A45

Milestones
Received: 3 March 2020
Accepted: 10 August 2020
Published: 13 May 2021

