Rearrangements and the local integrablility of maximal functions

Paul Alton Hagelstein

Vol. 216 (2004), No. 1, 111–126

DOI: 10.2140/pjm.2004.216.111

Abstract

Let M_HL and M_S respectively denote the Hardy–Littlewood and strong maximal operators, and let M_x and M_y respectively denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in R². It is well known that if f and f are equidistributed functions supported on Q = [0,1] × [0,1], then ∫ _QM_HLf ∼∫ _QM_HLf. This article examines the relationships between ∫ _QM_yf and ∫ _QM_yf, ∫ _QM_yM_xf and ∫ _QM_yM_xf, and ∫ _QM_Sf and ∫ _QM_Sf in the scenario in which f and f are horizontal rearrangements of one another, meaning that f and f are equidistributed on for any value of y.

The rearrangement results provided are not only of intrinsic interest, but also yield tools for more detailed examinations involving the local integrablility of maximal functions. They are used in a companion paper to prove that if f is supported on Q, ∫ _QM_yM_xf < ∞, and ∫ _QM_xM_yf = ∞, then there exists a set A of finite measure in R² such that ∫ _AM_Sf = ∞.

Milestones

Received: 9 June 2000

Published: 1 September 2004