Rearrangements and the Local Integrablility of Maximal Functions

Let MHL and MS respectively denote the Hardy–Littlewood and strong maximal operators, and let Mx and My respectively denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in R. It is well known that if f and f̃ are equidistributed functions supported on Q = [0, 1] × [0, 1], then ∫ Q MHLf ∼ ∫ Q MHLf̃ . This article examines the relationships between ∫ Q Myf̃ and ∫ Q Myf , ∫ Q MyMxf̃ and ∫ Q MyMxf , and ∫ Q MSf̃ and ∫ Q MSf in the scenario in which f̃ and f are horizontal rearrangements of one another, meaning that f̃ (·, y) and f (·, y) are equidistributed on [0, 1] 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, ∫ Q MyMxf < ∞, and ∫ Q MxMyf = ∞, then there exists a set A of finite measure in R such that ∫ A MSf = ∞.