论文标题
在Schrödinger设置中加权的Morrey-campanato空间上的变异和振荡操作员
Variation and oscillation operators on weighted Morrey-Campanato spaces in the Schrödinger setting
论文作者
论文摘要
令$ \ MATHCAL {l} $为具有潜在$ V $的Schrödinger运营商，即$ \ Mathcal L =-Δ+V $，假定$ V $满足反向Hölder不平等。我们考虑加权的Morrey-campanato空间$ bmo _ {\ Mathcal l，w}^α（\ Mathbb r^d）$和$ blo_ {l，w}^α（\schrödinger设置中的Mathbb r^d）$。我们证明了变体运算符$v_σ（\ {t_t \} _ {t> 0}）$，$σ> 2 $和振荡操作符$ o（\ {t_t \} _ {t> 0} $ t_j <t_ {j+1} $，$ j \ in \ mathbb z $，$ \ lim_ {j \ rightarrow+\ rightarrow+\ infty} t_j =+\ \ \ \ fly $和$ \ lim_ { l} $，$ t> 0 $，带有$ k \ in \ mathbb n $，是$ bmo _ {\ mathcal l，w}^α（\ mathbb r^d）$ to $ bmo _ {\ mathbb r^d）in $ _ {\ blo _ {\ mathcal l，W}^α（\ mathcal l，w}^α（\ mathbb r r^d）$。我们还为由$ \ {t^k \ partial_t^k e^{ - t \ mathcal l} \} _ {t> 0} $，$ k \ in \ mathbb n $建立了相同的属性。
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 $t_j<t_{j+1}$, $j\in \mathbb Z$, $\lim_{j\rightarrow +\infty}t_j=+\infty$ 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 $BMO_{\mathcal L,w}^α(\mathbb R^d)$ into $BLO_{\mathcal L,w}^α(\mathbb R^d)$. We also establish the same property for the maximal operators defined by $\{t^k\partial_t^k e^{-t\mathcal L}\}_{t>0}$, $k\in \mathbb N$.
