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

Abstract

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