The boundedness of some singular integral operators on weighted Hardy spaces associated with Schr\"odinger operators

Let $L=-\mathrm{\Delta }+V$ be a Schr"odinger operator acting on $L^2({\mathbb{R}}^n)$, $n\ge 1$, where $V\equiv ̸0$ is a nonnegative locally integrable function on ${\mathbb{R}}^n$. In this paper, we first define molecules for weighted Hardy spaces $H_L^p(w)$($0<p\le 1$) associated to $L$ and establish their molecular characterizations. Then by using the atomic decomposition and molecular characterization of $H_L^p(w)$, we will show that the imaginary power $L^{i\gamma }$ is bounded on $H_L^p(w)$ for $n\mathrm{/}(n+1)<p\le 1$, and the fractional integral operator $L^{-\alpha \mathrm{/}2}$ is bounded from $H_L^p(w)$ to $H_L^q(w^{q\mathrm{/}p})$, where $0<\alpha <\min \{n\mathrm{/}2,1\}$, $n/{(n+1)}<p\le n/{(n+\alpha)}$and$1/q=1/p-\alpha/n$.