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

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 $N_\infty$.