One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity
Abstract
We study the boundedness of the right-sided Hardy--Littlewood maximal operator on variable Lebesgue spaces over the real line. Motivated by the K0 condition for the classical Hardy--Littlewood maximal operator, we introduce a directional condition K0+ adapted to the geometry of the one-sided maximal operator. We characterize K0+ in terms of the uniform boundedness of a family of one-sided averaging operators and prove that it is a necessary condition for the boundedness of M+ on Lp(·)(ℝ). We also establish one-sided analogues of the modular and reverse Hölder estimates associated with the classical K0 condition. These estimates constitute the main local ingredients in the proof of boundedness of M+ under the combined assumptions K0+ and N∞.
Related
- Weighted boundedness of maximal operator associated with invertible linear transformations on variable Lebesgue spaces
- Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators
- Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators
- The norm of the centered Hardy--Littlewood maximal operator
- The weak-type (1,1) bound for the Hardy--Littlewood maximal function is O(√n log n)
BibTeX
@misc{vallejos2026onesided,
title = {{One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity}},
author = {Lucas A. Vallejos},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03835},
eprint = {2611.03835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03835},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity
AU - Lucas A. Vallejos
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03835
UR - https://arxiv.org/abs/2611.03835
AB - We study the boundedness of the right-sided Hardy--Littlewood maximal operator on variable Lebesgue spaces over the real line. Motivated by the K_0 condition for the classical Hardy--Littlewood maximal operator, we introduce a directional condition K_0^+ adapted to the geometry of the one-sided maximal operator. We characterize K_0^+ in terms of the uniform boundedness of a family of one-sided averaging operators and prove that it is a necessary condition for the boundedness of M^+ on L^p(·)(ℝ). We also establish one-sided analogues of the modular and reverse Hölder estimates associated with the classical K_0 condition. These estimates constitute the main local ingredients in the proof of boundedness of M^+ under the combined assumptions K_0^+ and N_∞.
ER -