Untersuchungen zum verhalten des Hardy-Littlewood-Maximaloperators

Abstract

In this paper we investigate the behavior of the Hardy-Littlewood Maximal Operator. It is well known that for absolutely integrable functions the Hardy-Littlewood Maximal Operator is finite almost everywhere. In this paper it is shown that for each set $E \subset [-\pi,\pi)$ with Lebesgue measure zero there exists a function of vanishing mean oscillation (VMO) such that the Hardy-Littlewood Maximal Operator of this function is infinite for all points of the set $E$. So for VMO-functions the Hardy-Littlewood Maximal Operator has divergence behavior similar to that of absolutely integrable functions. Some applications of these results for the behavior of the Poisson-Integral of VMO-functions are also given.