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.
Citation
Holger Boche. "Untersuchungen zum verhalten des Hardy-Littlewood-Maximaloperators." Illinois J. Math. 44 (2) 221 - 229, Summer 2000. https://doi.org/10.1215/ijm/1255984837
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