On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators
[Sur les estimations $L^p$ des transformées de Riesz associées aux opérateurs de Schrödinger et préservant la positivité]
Kucharski, Maciej1 ; Wróbel, Błażej2, 1
1Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2, 50-384 Wrocław, Poland
2Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich 8,00-656 Warszawa
Annales de l'Institut Fourier, Online first, 42 p.

We study the $L^{p},$ $1\leqslant p\leqslant \infty ,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}\Delta +V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}\Delta +V)^{-a}$ is bounded on $L^p(\mathbb{R}^d)$ with $1< p\leqslant 2$ whenever $a\leqslant 1/p.$ We demonstrate that the $L^{\infty }(\mathbb{R}^d)$ boundedness holds if $V$ satisfies an $a$-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on $V$ are also obtained on $L^{1}(\mathbb{R}^d).$ In particular our $L^{\infty }$ and $L^1$ results apply to non-negative locally bounded potentials $V$ which globally have a power growth or an exponential growth.

We also discuss a counterexample showing that the $L^{\infty }(\mathbb{R}^d)$ boundedness may fail.

Nous étudions le caractère borné sur $L^p$, $1 \leqslant p \leqslant \infty $, pour les transformées de Riesz de la forme $V^{a}(-\frac{1}{2}\Delta +V)^{-a},$ où $a>0$ et $V$ est un potentiel non-négatif. Nous prouvons que $V^{a}(-\frac{1}{2}\Delta +V)^{-a}$ est bornée sur $L^p(\mathbb{R}^d)$ avec $1< p\leqslant 2$ quand $a\leqslant 1/p.$ Nous démontrons que le caractère borné sur $L^{\infty }(\mathbb{R}^d)$ est valable si $V$ satisfait une condition intégrale dépendante de $a$ et robuste aux petites perturbations. Des résultats similaires avec des hypothèses plus fortes sur $V$ sont également obtenus sur $L^{1}(\mathbb{R}^d).$ En particulier, nos résultats $L^{\infty }$ et $L^1$ s’appliquent aux potentiels non négatifs et localement bornés $V$ qui ont globalement une croissance en puissance ou une croissance exponentielle.

Nous discutons également d’un contre-exemple montrant que le caractère borné sur $L^{\infty }(\mathbb{R}^d)$ peut échouer.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3744
Classification : 47D08, 42B20, 42B37
Keywords: Riesz transform, Schrödinger operator, $L^p$ boundedness
Mots-clés : Transformée de Riesz, Opérateur de Schrödinger, borne $L^p$

Kucharski, Maciej  1   ; Wróbel, Błażej  2 , 1

1 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2, 50-384 Wrocław, Poland
2 Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich 8,00-656 Warszawa 
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Kucharski, Maciej; Wróbel, Błażej. On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators. Annales de l'Institut Fourier, Online first, 42 p.

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