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

Abstract

This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schrödinger 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 Schrödinger operators $H$. 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 \cite{KRS}.

15 pages

References in corpus (2)

• Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities
• Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse-square potential