On the Integrability of Strong Maximal Functions Corresponding to Different Frames

Abstract

Abstract For the frame θ in , let B 2(θ)(𝑥) (𝑥 ∈ ) be a family of all 𝑛-dimensional rectangles containing 𝑥 and having edges parallel to the straight lines of θ, and let MB2(θ) be a maximal operator corresponding to B 2(θ). The main result of the paper is the following Theorem. For any function 𝑓 ∈ 𝐿 (1 + ln+ 𝐿)() (𝑛 ≥ 2) there exists a measure preserving and invertible mapping such that 1. {𝑥 : ω(𝑥) ≠ 𝑥} ⊂ supp 𝑓; 2. This theorem gives a general solution of M. de Guzmán's problem that was previously studied by various authors.