Weighted inequalities for variation operators associated with fractional Schrödinger semigroups

Abstract: 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.