ShowMe Analysis Meeting 2004 June 3-5, 2004 University of Missouri Columbia, Missouri

Alexander Stokolos (DePaul University)
On Weak Type Inequalities for Rare Maximal Functions in Rⁿ

Abstract: By Rare Maximal Function we understand the maximal function with respect to rectangles whose side length could be any number from a given infinite sparse set of positive real numbers. If this set is dense enough then the Rare Maximal Function is pointwise comparable with the Strong Maximal Function, and thus is of a weak type L(log⁺L)^n-1(Rⁿ), and this is the best possible estimate. Generally, a rarefaction of the set of rectangles could improve the estimate for the corresponding maximal function. In the talk we prove that this is not true for the Rare Maximal Functions in Rⁿ for any n>1, i.e. that the rarefaction of the side-length of the rectangles does not improve the properties of the corresponding maximal functions. Thus, we extend result known for R² only.