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 scale-free ${\ell }^p$-improving estimate, for $\frac{3}{2}<p\le 2$:

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 $1<p<\frac{3}{2}$. 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 $xmodq$. One requires an estimate uniform in $x$.

Keywords

Mathematical Subject Classification 2010

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