Weighted inequalities for Schrödinger type Singular Integrals on variable Lebesgue spaces

In this paper we study the boundedness in weighted variable Lebesgue spaces of operators associated with the semigroup generated by the time-independent Schrödinger operator $\mathcal{L}=-\mathrm{\Delta }+V$ in ${\mathbb{R}}^d$, where $d>2$ and the non-negative potential $V$ belongs to the reverse Hölder class $R{H}_q$ with $q>d\mathrm{/}2$. Each of the operators that we are going to deal with are singular integrals given by a kernel $K(x,y)$, which satisfies certain size and smoothness conditions in relation to a critical radius function $\rho$ which comes appears naturally in the harmonic analysis related to Schrödinger operator $\mathcal{L}$.