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

Abstract

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