On the Integrability of Strong Maximal Functions Corresponding to Different Frames

Abstract

For the frame θ in Rn, let B2(θ)(x) (x ∈ Rn) 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 B2(θ). The main result of the paper is the followingTheorem. For any function (formula omited)there exists a measure preserving and invertible mapping (formula opmited) such that(formula omited)This theorem gives a general solution of M. de Guzm’n’s problem that was previously studied by various authors. © 1999, Plenum Publishing Corporation. All rights reserved.