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2021 Averages along the square integers 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 f2(). Define the average of f over the square integers by

ANf(x):=1Nk=1Nf(x+k2).

We show that AN satisfies a local scale-free p-improving estimate, for 32<p2:

N2pANfpN2pfp,

provided f is supported in some interval of length N2, and p=p(p1) is the conjugate index. The inequality above fails for 1<p<32. The maximal function Af=supN1|ANf| 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 modq. One requires an estimate uniform in x.

Citation

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Rui Han. Michael T. Lacey. Fan Yang. "Averages along the square integers 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

Rights: Copyright © 2021 Mathematical Sciences Publishers

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