Let . Define the average of over the square integers by
We show that satisfies a local scale-free -improving estimate, for :
provided is supported in some interval of length , and is the conjugate index. The inequality above fails for . The maximal function || satisfies a similar sparse bound. Novel weighted and vector valued inequalities for follow. A critical step in the proof requires the control of a logarithmic average over of a function counting the number of square roots of . One requires an estimate uniform in .
"Averages along the square integers -improving and sparse inequalities." Tunisian J. Math. 3 (3) 517 - 550, 2021. https://doi.org/10.2140/tunis.2021.3.517