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

TL;DR

This paper establishes weighted norm inequalities for maximal functions associated with a critical radius in variable Lebesgue spaces, relevant for Schrödinger operators with certain potentials, expanding the class of admissible weights.

Contribution

It introduces new weighted inequalities for maximal functions linked to a critical radius, extending previous classes of weights in variable Lebesgue spaces.

Findings

Boundedness of maximal functions on weighted variable Lebesgue spaces.

Introduction of new weight classes including Muckenhoupt-type weights.

Application to Schrödinger operators with specific potential conditions.

Abstract

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.