We investigate mapping properties of non-centered Hardy–Littlewood maximal operators related to the exponential measure dμ(x) = exp (- | x1| - ⋯ - | xd|) dx in Rd. The mean values are taken over Euclidean balls or cubes (ℓ∞ balls) or diamonds (ℓ1 balls). Assuming that d≥ 2 , in the cases of cubes and diamonds we prove the Lp-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.
On non-centered maximal operators related to a non-doubling and non-radial exponential measure
Nowak A.;Sasso E.;Sjogren P.;
2023-01-01
Abstract
We investigate mapping properties of non-centered Hardy–Littlewood maximal operators related to the exponential measure dμ(x) = exp (- | x1| - ⋯ - | xd|) dx in Rd. The mean values are taken over Euclidean balls or cubes (ℓ∞ balls) or diamonds (ℓ1 balls). Assuming that d≥ 2 , in the cases of cubes and diamonds we prove the Lp-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.
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