Córdoba–Fefferman collections in harmonic analysis

Abstract

Cordoba-Fefferman collections are defined and used to characterize functions whose corresponding maximal functions are locally integrable. Cordoba-Fefferman collections are also used to show that, if M x and M y respectively denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in R 2 , M HL denotes the standard Hardy-Littlewood maximal operator in R 2 , and f is a measurable function supported in the unit square Q = [0,1] × [0,1], then f Q M HL f ∼ f Q M x f + f Q M y f.