Variation and oscillation operators on weighted Morrey-Campanato spaces in the Schrödinger setting

Abstract

Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal{L}=-\Delta +V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BM{O}_{\mathcal{L},w}^{\alpha }({\mathbb{R}}^d)$ and $BL{O}_{L,w}^{\alpha }({\mathbb{R}}^d)$ in the Schrödinger 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 $t_j<t_{j+1}$, $j\in \mathbb{Z}$, ${\lim }_{j\to +\infty }{t}_j=+\infty$ and ${\lim }_{j\to -\infty }{t}_j=0$, being $T_t=t^k{\partial }_t^k{e}^{-t\mathcal{L}}$, $t>0$, with $k\in \mathbb{N}$, are bounded operators from $BM{O}_{\mathcal{L},w}^{\alpha }({\mathbb{R}}^d)$ into $BL{O}_{\mathcal{L},w}^{\alpha }({\mathbb{R}}^d)$. We also establish the same property for the maximal operators defined by $\{t^k{\partial }_t^k{e}^{-t\mathcal{L}}{\}}_{t>0}$, $k\in \mathbb{N}$.