一类奇异积分算子在加权Hardy空间上的有界性
The boundedness of some singular integral operators on weighted Hardy spaces associated with Schrödinger operators
摘要：
设$L=-Δ+V$是作用在$L^2(\mathbb R^n)$上的薛定谔算子$n\ge1$，其中$V\not\equiv 0$是$\mathbb R^n$上的非负局部可积函数.本文首先定义了与$L$相关的加权Hardy空间$H^p_L(w)$（$0<p\le1$）的分子，并建立了它们的分子特征.然后利用$H^p_L(w)$的原子分解和分子特征，证明了对于$n/{(n+1)}<p\le1$，虚幂$L^{iγ}$在$H^p_L(w)$上有界，分数次积分算子$L^{-α/2}$从$H^p_L(w)$到$H^q_L(w^{q/p})$有界，其中$0<α<\min\{n/2,1\}$，$n/{(n+1)}<p\le n/{(n+α)}$和$1/q=1/p-α/n$ .
英文摘要：
Let $L=-Δ+V$ be a Schrödinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this paper, we first define molecules for weighted Hardy spaces $H^p_L(w)$($0<p\le1$) associated to $L$ and establish their molecular characterizations. Then by using the atomic decomposition and molecular characterization of $H^p_L(w)$, we will show that the imaginary power $L^{iγ}$ is bounded on $H^p_L(w)$ for $n/{(n+1)}<p\le1$, and the fractional integral operator $L^{-α/2}$ is bounded from $H^p_L(w)$ to $H^q_L(w^{q/p})$, where $0<α<\min\{n/2,1\}$, $n/{(n+1)}<p\le n/{(n+α)}$ and $1/q=1/p-α/n$.
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