Vol. 3, No. 3, 2021

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Averages along the square integers $\ell^p$-improving and sparse inequalities

Rui Han, Michael T. Lacey and Fan Yang

Vol. 3 (2021), No. 3, 517–550
Abstract

Let f 2(). Define the average of f over the square integers by ANf(x) := 1 N k=1Nf(x + k2). We show that AN satisfies a local scale-free p-improving estimate, for 3 2 < p 2:

N2p ANfp N2pf p,

provided f is supported in some interval of length N2, and p = p(p 1) is the conjugate index. The inequality above fails for 1 < p < 3 2. 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 mod q. One requires an estimate uniform in x.

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
Authors
Rui Han
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
Department of Mathematics
Louisiana State University
Baton Rouge, LA
United States
Michael T. Lacey
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
Fan Yang
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
The Australian National University
Canberra ACT
Australia