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 dμ (x) = exp (-|x_1|-⋯ -|x_d|)dx in ℝ^d . The mean values are taken over Euclidean balls or cubes ( ℓ ^∞ balls) or diamonds ( ℓ ^1 balls). Assuming that d ≥ 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 ≤ 4 for the positive part.