Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials

Auscher, Pascal; Ben Ali, Besma

doi:10.5802/aif.2320

Maximal inequalities and Riesz transform estimates on

L^{p}

spaces for Schrödinger operators with nonnegative potentials
[Inégalité maximale et de transformée de Riesz dans

L^{p}

pour des opérateurs de Schrödinger avec potentiels positifs]

Auscher, Pascal ¹ ; Ben Ali, Besma ¹

¹ Université de Paris-Sud, UMR du CNRS 8628 91405 Orsay Cedex (France)

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1975-2013.

Résumé
Abstract

On montre des estimations $L^{p}$ pour des opérateurs de Schrödinger $- Δ + V$ sur $ℝ^{n}$ et leurs racines carrées. Le potentiel est dans une classe Hölder inverse améliorant les résultats de Shen. On s’appuie sur une inégalité de type Fefferman-Phong améliorée et des inégalités Hölder inverse pour des solutions faibles de $- Δ + V$ et leurs gradients.

We show various $L^{p}$ estimates for Schrödinger operators $- Δ + V$ on $ℝ^{n}$ and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of $- Δ + V$ and their gradients.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2320

Classification : 35J10, 42B20
Keywords: Schrödinger operators, maximal inequalities, Riesz transforms, Fefferman-Phong inequality, reverse Hölder estimates
Mot clés : opérateurs de Schrödinger, inégalité maximale, transformée de Riesz, inégalité de Fefferman-Phong, inégalités Hölder inverse

Affiliations des auteurs :

Auscher, Pascal ¹ ; Ben Ali, Besma ¹

¹ Université de Paris-Sud, UMR du CNRS 8628 91405 Orsay Cedex (France)

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     title = {Maximal inequalities and {Riesz} transform estimates on $L^p$ spaces for {Schr\"odinger} operators with nonnegative potentials},
     journal = {Annales de l'Institut Fourier},
     pages = {1975--2013},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     zbl = {1161.35003},
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Auscher, Pascal; Ben Ali, Besma. Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials. Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1975-2013. doi : 10.5802/aif.2320. http://www.numdam.org/articles/10.5802/aif.2320/

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