Variation and oscillation operators on weighted Morrey-Campanato spaces in the Schrödinger setting
Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal L=-Δ+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal L,w}^α(\mathbb R^d)$ and $BLO_{L,w}^α(\mathbb R^d)$ in the Schrödinger setting. We prove that the variation operator $V_σ(\{T_t\}_{t>0})$, $σ>2$, and the oscillation operator $O(\{T_t\}_{t>0}, \{t_j\}_{j\in \mathbb Z})$, where $t_j 0$, with $k\in \mathbb N$, are bounded operators from $BMO_{\mathcal L,w}^α(\mathbb R^d)$ into $BLO_{\mathcal L,w}^α(\mathbb R^d)$. We also establish the same property for the maximal operators defined by $\{t^k\partial_t^k e^{-t\mathcal L}\}_{t>0}$, $k\in \mathbb N$.
math.CA↗