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

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Let L=-Δ+V be a Schrödinger operator acting on L²(Rⁿ), n≥1, where V≡ 0 is a nonnegative locally integrable function on Rⁿ. In this paper, we first define molecules for weighted Hardy spaces Hᵖₗ(w)(0<p≤1) associated to L and establish their molecular characterizations. Then by using the atomic decomposition and molecular characterization of Hᵖₗ(w), we will show that the imaginary power Lⁱγ is bounded on Hᵖₗ(w) for n/(n+1)<p≤1, and the fractional integral operator L⁻α/² is bounded from Hᵖₗ(w) to Hqₗ(wq/ᵖ), where 0<α<{n/2,1}, n/(n+1)<p≤ n/(n+α) and 1/q=1/p-α/n.