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2021 Averages along the square integers ${\ell ^p}$-improving and sparse inequalities
Rui Han, Michael T. Lacey, Fan Yang
Tunisian J. Math. 3(3): 517-550 (2021). DOI: 10.2140/tunis.2021.3.517

## Abstract

Let $f \in {\ell ^2}\left(ℤ \right)$. Define the average of $f$ over the square integers by

${A_N}f\left( x \right): = {1 \over N}\mathop \sum \limits_{k = 1}^N f{\left( {x + {k^2}} \right)^}.$

We show that ${A_N}$ satisfies a local scale-free ${\ell ^p}$-improving estimate, for ${3 \over 2} < p \le 2$:

${N^{ - 2 / p'}}\parallel {A_N}f{\parallel _{{\ell ^{p'}}}} \mathbin{\lower.3ex\hbox{\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}}} {N^{ - 2 / p}}\parallel f{\parallel _{{\ell ^p}}},$

provided $f$ is supported in some interval of length ${N^2}$, and $p' = p / \left( {p - 1} \right)$ is the conjugate index. The inequality above fails for $1 < p < {3 \over 2}$. The maximal function $Af = {\sup _{N \ge 1}}$|${A_N}f$| 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 $x\,\bmod \,q$. One requires an estimate uniform in $x$.

## Citation

Rui Han. Michael T. Lacey. Fan Yang. "Averages along the square integers ${\ell ^p}$-improving and sparse inequalities." Tunisian J. Math. 3 (3) 517 - 550, 2021. https://doi.org/10.2140/tunis.2021.3.517

## Information

Received: 3 March 2020; Accepted: 10 August 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/tunis.2021.3.517

Subjects:
Primary: 11L05, 42A45  