We study the behaviour of the -dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants that appear in the weak type inequalities.
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 -norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily...
Page 1