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

TL;DR

This paper studies the behavior of non-centered maximal operators associated with a non-doubling exponential measure in multiple dimensions, revealing boundedness properties for certain shapes and dimensions but not others.

Contribution

It establishes $L^p$ boundedness for non-centered maximal operators with exponential measures over specific shapes and dimensions, and shows failure of weak type $(1,1)$ estimates.

Findings

Proves $L^p$ boundedness for $p > 1$ over cubes and diamonds in $ ext{dim} \, d \\ge 2$.

Disproves weak type $(1,1)$ estimates for these operators.

Shows $L^p$ boundedness for Euclidean balls only when $d \\le 4$.

Abstract

We investigate mapping properties of non-centered Hardy-Littlewood maximal operators related to the exponential measure $d\mu(x) = \exp(-|x_1|-\ldots-|x_d|)dx$ in $\mathbb{R}^d$. The mean values are taken over Euclidean balls or cubes ($\ell^{\infty}$ balls) or diamonds ($\ell^1$ 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.