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

Abstract

In this work we obtain boundedness results for fractional operators associated with Schrödinger operators L = −Δ+V on weighted variable Lebesgue spaces. These operators include fractional integrals and their respective commutators. In particular, we obtain weighted inequalities of the type Lp(·)-Lq(·) and estimates of the type Lp(·)-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. Such extrapolation results generalize what was done by Harboure, Macías, and Segovia [Amer. J. Math. 110 no. 3 (1988), 383–397], and by Bongioanni, Cabral, and Harboure [Potential Anal. 38 no. 4 (2013), 1207–1232], for the classic case, that is, V ≡ 0 and p(·) constant, respectively.