A remark on the maximal operator for radial measures

Abstract:

The purpose of this paper is to prove that there exist measures $d\mu (x)=\gamma (x)dx$, with $\gamma (x)=\gamma _{0}(|x|)$ and $\gamma _{0}$ being a decreasing and positive function, such that the Hardy-Littlewood maximal operator, $\mathcal {M}_{\mu }$, associated to the measure $\mu$ does not map $L^{p}_{\mu }(\mathbb {R}^{n})$ into weak $L^{p}_{\mu }(\mathbb {R}^{n})$, for every $p<\infty$. This result answers an open question of P. Sjögren and F. Soria.