Let $\mathcal{ L}=-\Delta+V$ be a Schr\"{ o}dinger operator, where the nonnegative potential $V$ belongs to the reverse H\"{ o}lder class $B_{ q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, $\{ e^{ -t\mathcal{ L}^{ \alpha}}\}_{ t>0},$ associated with $\mathcal{ L}$. As an application, we obtain the $BMO^{ \gamma}_{ \mathcal{ L}}$-boundedness of the maximal function, and the Littlewood-Paley $g$-functions associated with $\mathcal{ L}$ via $T1$ theorem, respectively.
Categories: math.CA