半群极大函数交换子与Schr dinger算子的加权Compactness
On weighted Compactness of Commutator of semi-group maximal function associated to Schrödinger operators
摘要：
设$\mathcal{T}^*$是与薛定谔算子$-Δ+V(x)$相关联的半群极大函数，其中$V$满足适当的逆Hölder不等式。本文证明了当$b\in \text{CMO}_θ(ρ)(\mathbb{R}^n)$和$w\in A_p^{ρ,θ}(\mathbb{R}^n)$满足时，$\mathcal{T}^*$的交换子是$L^p(w)$上的紧算子.这里，$\text{ CMO}_θ(ρ)(\mathbb{R}^n)$表示$\text{BMO}_θ(ρ)(\mathbb{R}^n)$（其大于经典的$\text{BMO}(\mathbb{R}^n)$空间）拓扑中的$\mathcal{C}_c^\infty(\mathbb{R}^n)$的闭包。$b$所属的空间和权重类$w$所属的空间分别比通常的$\text{CMO}(\mathbb{R}^n)$空间和Muckenhoupt $A_p$权重类更大。
英文摘要：
Let $\mathcal{T}^*$ be the semi-group maximal function associated to the Schrödinger operator $-Δ+V(x)$ with $V$ satisfying an appropriate reverse Hölder inequality. In this paper, we show that the commutator of $\mathcal{T}^*$ is a compact operator on $L^p(w)$ for $1<p<\infty$ if $b\in \text{CMO}_θ(ρ)(\mathbb{R}^n)$ and $w\in A_p^{ρ,θ}(\mathbb{R}^n)$. Here $\text{ CMO}_θ(ρ)(\mathbb{R}^n)$ denotes the closure of $\mathcal{C}_c^\infty(\mathbb{R}^n)$ in the $\text{BMO}_θ(ρ)(\mathbb{R}^n)$ (which is larger than the classical $\text{BMO}(\mathbb{R}^n)$ space) topology. The space where $b$ belongs and the weighs class $w$ belongs are more larger than the usual $\text{CMO}(\mathbb{R}^n)$ space and the Muckenhoupt $A_p$ weights class, respectively.
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