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

Shen, Zhongwei

doi:10.5802/aif.1463

L^{p}

estimates for Schrödinger operators with certain potentials

Shen, Zhongwei

Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 513-546.

Résumé
Abstract

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

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

MR Zbl | 5 citations dans Numdam

DOI : 10.5802/aif.1463

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     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},
     mrnumber = {96h:35037},
     zbl = {0818.35021},
     language = {en},
     url = {http://www.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, Tome 45 (1995) no. 2, pp. 513-546. doi : 10.5802/aif.1463. http://www.numdam.org/articles/10.5802/aif.1463/

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