Rearrangements and the local integrablility of maximal functions (Q1764364)

If \(f\) is a measurable function on the unit cube \(Q \subset R^{2}\), one can naturally associate several maximal functions to \(f\), namely, the Hardy-Littlewood maximal function \(M_{HL}f\), defined by cubes, the strong maximal function \(M_{S}f\), defined by oriented rectangles, the one-dimensional horizontal and vertical maximal functions \(M_{x}f\) and \(M_{y}f\), respectively, and their iterates \(M_{x}M_{y}f\) and \(M_{y}M_{x}f\). We say that two functions \(f\) and \(g\) are \textit{horizontal rearrangements} of each other on \(Q\) if \(f(\cdot ,y)\) and \(g(\cdot ,y)\) are equidistributed on \([0,1]\) for any \(y\in [0,1]\). In this paper, the author derives integral relationships involving these operators applied to pairs of functions that are horizontal rearrangements of each other. Several counterexamples demonstrate the sharpness of these results.