${L}^{p}$ estimates for Schrödinger operators with certain potentials
Annales de l'Institut Fourier, Volume 45 (1995) no. 2, pp. 513-546.

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

Nous considérons des opérateurs de Schrödinger $-\Delta +V\left(x\right)$ dans ${ℝ}^{n}$ où le facteur non négatif $V\left(x\right)$ 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 $\left(-\Delta +V{\right)}^{i\gamma },{\nabla }^{2}\left(-\Delta +V{\right)}^{-1},\nabla \left(-\Delta +V{\right)}^{-1/2}$ et $\nabla \left(-\Delta +V{\right)}^{-1}$$\gamma \in ℝ$. En particulier nous montrons que $\left(-\Delta +V{\right)}^{i\gamma }$ est un opérateur de Calderón-Zygmund si $V\in {B}_{n/2}$ and $\nabla \left(-\Delta +V{\right)}^{-1/2},\nabla \left(-\Delta +V{\right)}^{-1}\nabla$ sont des opérateurs de Calderón-Zygmund si $V\in {B}_{n}$.

@article{AIF_1995__45_2_513_0,
author = {Shen, Zhongwei},
title = {$L^p$ estimates for {Schr\"odinger} operators with certain potentials},
journal = {Annales de l'Institut Fourier},
pages = {513--546},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {45},
number = {2},
year = {1995},
doi = {10.5802/aif.1463},
zbl = {0818.35021},
mrnumber = {96h:35037},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.1463/}
}
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Shen, Zhongwei. $L^p$ estimates for Schrödinger operators with certain potentials. Annales de l'Institut Fourier, Volume 45 (1995) no. 2, pp. 513-546. doi : 10.5802/aif.1463. http://archive.numdam.org/articles/10.5802/aif.1463/

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