Weighted norm inequalities for the maximal functions associated to a critical radius function on variable Lebesgue spaces

In this work we obtain boundedness on weighted variable Lebesgue spaces of some maximal functions that come from the localized analysis considering a critical radius function. This analysis appears naturally in the context of the Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^d$, where $V$ a non-negative potential satisfying some specific reverse H\"older condition. We consider new classes of weights that locally behave as the Muckenhoupt class for Lebesgue spaces with variable exponents considered in Cruz Uribe et al. (J. Math. Anal. Appl. 394(2):744-760, 2012) and actually include them.