Properties of the maximal operators associated with bases of rectangles in ℝ³

This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions (t,1/t,s) within a framework of more general Soria bases. The Jessen-Marcinkiewicz-Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps L log² L into L^1,∞. We give a simple geometric condition that guarantees that the L log² L class cannot be enlarged. The proof develops the author's methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fefferman and Pipher.