Variation and oscillation operators on weighted Morrey–Campanato spaces in the Schrödinger setting
DOI:
https://doi.org/10.33044/revuma.4327Abstract
We denote by $\mathcal{L}$ 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 ${\mathrm{BMO}}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$ 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 $t_j < t_{j+1}$, $j\in \mathbb{Z}$, $\displaystyle \lim_{j\rightarrow +\infty}t_j=+\infty$ and $\displaystyle\lim_{j\rightarrow -\infty }t_j=0$, being $T_t=t^k\partial _t^ke^{-t\mathcal{L}}$, $t > 0$, with $k\in \mathbb{N}$, are bounded operators from ${\rm BMO}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$ into ${\rm BLO}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$. We also establish the same property for the maximal operators defined by $\{t^k\partial _t^ke^{-t\mathcal L}\}_{t > 0}$, $k\in \mathbb{N}$.
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