## Illinois Journal of Mathematics

### Untersuchungen zum verhalten des Hardy-Littlewood-Maximaloperators

Holger Boche

#### 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.

#### Article information

Source
Illinois J. Math., Volume 44, Issue 2 (2000), 221-229.

Dates
First available in Project Euclid: 19 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1255984837

Digital Object Identifier
doi:10.1215/ijm/1255984837

Mathematical Reviews number (MathSciNet)
MR1775318

Zentralblatt MATH identifier
0951.30028

Subjects
Primary: 30D50
Secondary: 42B35: Function spaces arising in harmonic analysis

#### Citation

Boche, Holger. Untersuchungen zum verhalten des Hardy-Littlewood-Maximaloperators. Illinois J. Math. 44 (2000), no. 2, 221--229. doi:10.1215/ijm/1255984837. https://projecteuclid.org/euclid.ijm/1255984837