Córdoba-Fefferman collections in harmonic analysis (Q1764363)

The author uses Córdoba-Fefferman collections [\textit{A. Córdoba} and \textit{R. Fefferman}, Ann. Math. (2) 102, 95--100 (1975; Zbl 0324.28004)] to study maximal operators on the unit cube \( Q\subset \mathbb{R}^{2}\). A general integral result for maximal functions defined by Córdoba-Fefferman collections is derived and used to prove that there are constants \(c,C>0\) so that for any measurable function \(f\) on \(Q\), \[ c\int_{Q}M_{HL}f\leq \int_{Q}M_{x}f+\int_{Q}M_{y}f\leq C\int_{Q}M_{HL}f. \]