Regularity of the Hardy–Littlewood maximal operator on block decreasing functions
Volume 194 / 2009
Studia Mathematica 194 (2009), 253-277
MSC: Primary 42B25, 26B30.
DOI: 10.4064/sm194-3-3
Abstract
We study the Hardy–Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the $\ell _\infty $-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.
