Annales Academiæ Scientiarum Fennicæ
Volumen 45, 2020, 467-478
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 Lp, we obtain nearly sharp Lp 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.
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