Boundedness of fractional operators associated with Schrödinger operators on weighted variable Lebesgue spaces via extrapolation

In this work we obtain boundedness results for fractional operators associated with Schr\"odinger operators $\ \mathcal{L}=-\Delta+V$ on weighted variable Lebesgue spaces. These operators include fractional integrals and their respective commutators. Particularly, we obtain weighted inequalities of the type $L^{p(\cdot)}$-$L^{q(\cdot)}$ and estimates of the type $L^{p(\cdot)}$-Lipschitz variable integral spaces. For this purpose, we developed extrapolation results that allow us to obtain boundedness results of the type described above in the variable setting by starting from analogous inequalities in the classical context.