Variation and oscillation operators on weighted Morrey-Campanato spaces in the Schr\"odinger setting

TL;DR

This paper studies the boundedness of variation, oscillation, and maximal operators associated with Schr"odinger operators on weighted Morrey-Campanato spaces, extending harmonic analysis tools in the Schr"odinger setting.

Contribution

It establishes boundedness results for variation, oscillation, and maximal operators on weighted Morrey-Campanato spaces in the Schr"odinger context, a novel extension of classical harmonic analysis.

Findings

Boundedness of variation operators $V_\sigma$ on $BMO_{ ext L,w}^ ext{ extalpha}$ to $BLO_{ ext L,w}^ ext{ extalpha}$.

Boundedness of oscillation operators $O$ on $BMO_{ ext L,w}^ ext{ extalpha}$ to $BLO_{ ext L,w}^ ext{ extalpha}$.

Boundedness of maximal operators associated with Schr"odinger semigroup derivatives.

Abstract

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 $\{t^k\partial_t^k e^{-t\mathcal L}\}_{t>0}$, $k\in \mathbb N$.