This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schr\"odinger operators $H=-\Delta+V$, i.e., In $\mathbb{ R}^n$ ($n\ge 3$) we prove the $\epsilon-$uniform $L^{ \frac{ 2(n+1)}{ n+3}}$-$L^{ \frac{ 2(n+1)}{ n-1}}$ estimates of the resolvent $(H-\lambda\pm i\epsilon)^{ -1}$ for all $\lambda>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schr\"odinger operators $H$. Next, we consider the fractional Schr\"odinger 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 \cite{ KRS}.
Categories: math.AP math.CA