Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem (Q1085370)

We show that the maximal operator associated to the family of rectangles in \({\mathbb{R}}^ 3\) one of whose sides is parallel to \((1,2^ j,2^ k)\) for some j,k\(\in {\mathbb{H}}Z\) is bounded on \(L^ p\), \(1<p<\infty\). We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.