# L^p-MAPPING PROPERTIES FOR SCHRӦDINGER OPERATORS IN OPEN SETS OF R^d

(NO.110) 2015-05-15

Let $H_V = - \Delta +V$ be a Schr\"odinger operator on an arbitrary open set $\Omega \subset \mathbb{R}^d$ ($d \ge 3$), where $\Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $\Omega$. The purpose of this paper is to show $L^p$--boundedness of an operator $\varphi(H_V)$ for any rapidly decreasing function $\varphi$ on $\mathbb{R}$. $\varphi(H_V)$ is defined by the spectral resolution theorem. As a by-product, $L^p$--$L^q$--estimates for $\varphi(H_V)$ are also obtained.

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