Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems (1607.02752v2)
Abstract: This paper comprises two parts. We first investigate a $Lp$ 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 $Lp$ 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}.