On non-centered maximal operators related to a non-doubling and non-radial exponential measure

We investigate mapping properties of non-centered Hardy-Littlewood maximal operators related to the exponential measure $d\mu (x)=\exp (-\mathrm{\mid }{x}_1\mathrm{\mid }-\dots -\mathrm{\mid }{x}_d\mathrm{\mid })dx$ in ${\mathbb{R}}^d$. The mean values are taken over Euclidean balls or cubes (${\mathrm{\ell }}^{\mathrm{\infty }}$ balls) or diamonds (${\mathrm{\ell }}^1$ 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.