Remarks on $L^p$-limiting absorption principle of Schr\"odinger operators and applications to spectral multiplier theorems

This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schr"odinger operators $H=-\mathrm{\Delta }+V$, i.e., In ${\mathbb{R}}^n$ ($n\ge 3$) we prove the $ϵ-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-\lambda \pm iϵ{)}^{-1}$ 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 $H=(-\mathrm{\Delta }{)}^{\alpha }+V$ ($0<2\alpha <n$) and prove a uniform Hardy-Littlewood-Sobolev inequality for $(-\mathrm{\Delta }{)}^{\alpha }$, which generalizes the corresponding result of Kenig-Ruiz-Sogge \cite{KRS}.