Remarks on L p -limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems

This paper comprises two parts. We first investigate an {L^{p}} -type of limiting absorption principle for Schrödinger operators {H=-\Delta+V} on {\mathbb{R}^{n}} ( {n\geq 3} ), i.e., we prove the ϵ-uniform {L^{{2(n+1)}/({n+3})}} – {L^{{2(n+1)}/({n-1})}} -estimates of the resolvent {(H-\lambda\pm i\epsilon)^{-1}} for all {\lambda>0} under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce {L^{p}} -bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator {H=(-\Delta)^{\alpha}+V} ( {0<2\alpha<n} ) and prove a uniform Hardy–Littlewood–Sobolev inequality for {(-\Delta)^{\alpha}} , which generalizes the corresponding result of Kenig–Ruiz–Sogge [20].