Let $\mathcal{ L}$ be the Schr\"odinger operator with potential $V$, that is, $\mathcal L=-\Delta+V$, where it is assumed that $V$ satisfies a reverse H\"older inequality. We consider weighted Morrey-Campanato spaces $BMO_{ \mathcal L,w}^\alpha (\mathbb R^d)$ and $BLO_{ L,w}^\alpha (\mathbb R^d)$ in the Schr\"odinger setting. We prove that the variation operator $V_\sigma (\{ T_t\}_{ t>0})$, $\sigma>2$, and the oscillation operator $O(\{ T_t\}_{ t>0}, \{ t_j\}_{ j\in \mathbb Z})$, where $t_j<t_{ j+1}$, $j\in \mathbb Z$, $\lim_{ j\rightarrow +\infty}t_j=+\infty$ and $\lim_{ j\rightarrow -\infty} t_j=0$, being $T_t=t^k\partial_t^k e^{ -t\mathcal L}$, $t>0$, with $k\in \mathbb N$, are bounded operators from $BMO_{ \mathcal L,w}^\alpha (\mathbb R^d)$ into $BLO_{ \mathcal L,w}^\alpha (\mathbb R^d)$. We also establish the same property for the maximal operators defined by $\{ t^k\partial_t^k e^{ -t\mathcal L}\}_{ t>0}$, $k\in \mathbb N$.
Categories: math.CA