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

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.

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

“…We also point out that recently, Nowak et al. have studied in [25] some properties of the un‐centered maximal function associated with the measure $$\eta _\alpha$$ defined on $$\mathbb {R}^d$$ by $$d\eta _\alpha (x)=\prod _{j=1}^d|x|^{\alpha _j}\text{e}^{-|x_j|}dx_j$$. However, their results cannot help us get ours.…”

“…We remark that endpoint inequalities for Riesz transforms and some spectral multipliers in the Ornstein-Uhlenbeck setting were proved in [19] and [20]. We also point out that recently, Nowak et al have studied in [25] some properties of the un-centered maximal function associated with the measure 𝜂 𝛼 defined on ℝ 𝑑 by 𝑑𝜂 𝛼 (𝑥) = ∏ 𝑑 𝑗=1 |𝑥| 𝛼 𝑗 e −|𝑥 𝑗 | 𝑑𝑥 𝑗 . However, their results cannot help us get ours.…”