Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 467-478

MAXIMAL OPERATORS FOR CUBE SKELETONS

Andrea Olivo and Pablo Shmerkin

Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales
Departamento de Matemática, and IMAS-CONICET, Ciudad Universitaria
Pabellón I (C1428EGA), Ciudad de Buenos Aires, Argentina; aolivo 'at' dm.uba.ar

Torcuato Di Tella University, Department of Mathematics and Statistics, and CONICET
Buenos Aires, Argentina; pshmerkin 'at' utdt.edu

Abstract. We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, k-skeletons in Rⁿ. Although these operators are known not to be bounded on any L^p, we obtain nearly sharp L^p bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of Keleti, Nagy and Shmerkin, and of Thornton, on sets that contain a scaled k-sekeleton of the unit cube with center in every point of Rⁿ.

2010 Mathematics Subject Classification: Primary 28A75, 28A80, 42B25.

Key words: Averages over skeletons, maximal functions.

Reference to this article: A. Olivo and P. Shmerkin: Maximal operators for cube skeletons. Ann. Acad. Sci. Fenn. Math. 45 (2020), 467-478.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4513