Non-centered maximal operators over ℓ^q balls for exponential and Laguerre-type measures

Abstract

We establish the L^p-boundedness, 1<p<∞, of the non-centered Hardy--Littlewood maximal operator associated with ℓ^q balls, 1<q<∞, for the non-radial exponential measure dμ(x)=e^{−|x₁|−⋯−|x_d|} dx in every dimension d≥1. This settles the higher-dimensional Euclidean case left open by Nowak, Sasso, Sjögren and Stempak, and extends their results to the full range 1<q<∞. We also prove that these operators are not of weak type (1,1) when d≥2. Together with the known results for q=1 and q=∞ for the exponential measure, this gives a complete description of the L^p-boundedness and weak type (1,1) behavior for all 1≤ q≤∞. {We also extend the strong L^p-boundedness to the Laguerre-type measures dμ_α(x)=∏_j=1^d|x_j|^α_je^{−|x₁|−⋯−|x_d|} dx for all α_j≥0 and 1≤ q≤∞, and for the full range α_j>−1 when q=∞.} {The main arguments rely} on geometric estimates for sections of ℓ^q balls and a slicing argument.

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BibTeX RIS

BibTeX

@misc{bui2026noncentered,
  title = {{Non-centered maximal operators over $\ell^q$ balls for exponential and Laguerre-type measures}},
  author = {The Anh Bui and Fu Ken Ly},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04250},
  eprint = {2611.04250},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04250},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Non-centered maximal operators over ℓ^q balls for exponential and Laguerre-type measures
AU  - The Anh Bui
AU  - Fu Ken Ly
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04250
UR  - https://arxiv.org/abs/2611.04250
AB  - We establish the L^p-boundedness, 1<p<∞, of the non-centered Hardy--Littlewood maximal operator associated with ℓ^q balls, 1<q<∞, for the non-radial exponential measure dμ(x)=e^−|x_1|−⋯−|x_d| dx in every dimension d≥1. This settles the higher-dimensional Euclidean case left open by Nowak, Sasso, Sjögren and Stempak, and extends their results to the full range 1<q<∞. We also prove that these operators are not of weak type (1,1) when d≥2. Together with the known results for q=1 and q=∞ for the exponential measure, this gives a complete description of the L^p-boundedness and weak type (1,1) behavior for all 1≤ q≤∞. {We also extend the strong L^p-boundedness to the Laguerre-type measures dμ_α(x)=∏_j=1^d|x_j|^α_je^−|x_1|−⋯−|x_d| dx for all α_j≥0 and 1≤ q≤∞, and for the full range α_j>−1 when q=∞.} {The main arguments rely} on geometric estimates for sections of ℓ^q balls and a slicing argument.
ER  -

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