$L^p$ estimates for Schrödinger operators with certain potentials

Shen, Zhongwei

Annales de l'Institut Fourier, Tome 45 (1995), p. 513-546 / Harvested from Numdam

Résumé

Nous considérons des opérateurs de Schrödinger $-\Delta +V(x)$ dans ${\Bbb R}^n$ où le facteur non négatif $V(x)$ appartient à la classe de Hölder inversée $B_q$ pour tout $q\ge n/2$. Nous obtenons les estimations optimales $L^p$ pour les opérateurs $(-\Delta +V)^{i\gamma },\nabla ^2 (- \Delta +V)^{-1}, \nabla (- \Delta +V)^{-1/2}$ et $\nabla (- \Delta +V)^{-1}$ où $\gamma \in {\Bbb R}$. En particulier nous montrons que $(- \Delta +V)^{i\gamma }$ est un opérateur de Calderón-Zygmund si $V\in B_{n/2}$ and $\nabla (- \Delta +V)^{-1/2}, \nabla (- \Delta +V)^{-1}\nabla $ sont des opérateurs de Calderón-Zygmund si $V\in B_n$.

Abstract

We consider the Schrödinger operators $-\Delta +V(x)$ in ${\Bbb R}^n$ where the nonnegative potential $V(x)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. We obtain the optimal $L^p$ estimates for the operators $(- \Delta +V)^{i\gamma },\nabla ^2 (- \Delta +V)^{-1}, \nabla (- \Delta +V)^{-1/2}$ and $\nabla (- \Delta +V)^{-1}$ where $\gamma \in {\Bbb R}$. In particular we show that $(- \Delta +V)^{i\gamma }$ is a Calderón-Zygmund operator if $V\in B_{n/2}$ and $\nabla (- \Delta +V)^{-1/2}, \nabla (- \Delta +V)^{-1}\nabla $ are Calderón-Zygmund operators if $V\in B_n$.

Détail

Publié le : 1995-01-01

MR 96h:35037 | Zbl 0818.35021 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1463

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