Weighted inequalities for variation operators associated with fractional Schr\"{o}dinger semigroups

Let $\{e^{-tL^{\alpha}}\}_{t>0}$ be the fractional Schr\"{o}dinger semigroup associated with $L=-\Delta+V$, where $V$ is a non-negatvie potential belonging to the reverse H\"{o}lder class. In this paper, we establish weighted boundedness properties of the variation operator related to $\{e^{-tL^{\alpha}}\}_{t>0}$, including weighted $L^{p}-L^{q}$ quantitative inequalities and mixed weak-type inequalities.