The boundedness of some singular integral operators on weighted Hardy spaces associated with Schrödinger operators

Abstract: Let $L=-\Delta+V$ be a Schr\"odinger 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 $L^{i\gamma}$ is bounded on $H^p_L(w)$ for $n/{(n+1)}<p\le1$, and the fractional integral operator $L^{-\alpha/2}$ is bounded from $H^p_L(w)$ to $H^q_L(w^{q/p})$, where $0<\alpha<\min{n/2,1}$, $n/{(n+1)}<p\le n/{(n+\alpha)}$ and $1/q=1/p-\alpha/n$.