  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}.
