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arXiv · 2611.03835

One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity

Lucas A. Vallejos
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Abstract

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 $L^{p(\cdot)}(\mathbb{R})$. 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 $N_\infty$.

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BibTeXRIS

Lucas A. Vallejos. 2026-11-02. One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity. https://arxiv.org/abs/2611.03835

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