  We investigate mapping properties of non-centered Hardy-Littlewood maximal
operators related to the exponential measure $d\mu(x) =
\exp(-|x_1|-\ldots-|x_d|)dx$ in $\mathbb{R}^d$. The mean values are taken over
Euclidean balls or cubes ($\ell^{\infty}$ balls) or diamonds ($\ell^1$ balls).
Assuming that $d \ge 2$, in the cases of cubes and diamonds we prove the
$L^p$-boundedness for $p > 1$ and disprove the weak type $(1,1)$ estimate. The
same is proved in the case of Euclidean balls, under the restriction $d \le 4$
for the positive part.
