On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators

Kucharski, Maciej; Wróbel, Błażej

doi:10.5802/aif.3744

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, 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

Annales de l'Institut Fourier, Online first, 42 p.

Abstract (VO)
Résumé

We study the $L^{p},$ $1 ⩽ p ⩽ \infty,$ boundedness for Riesz transforms of the form $V^{a} {(- \frac{1}{2} Δ + V)}^{- a},$ where $a > 0$ and $V$ is a non-negative potential. We prove that $V^{a} {(- \frac{1}{2} Δ + V)}^{- a}$ is bounded on $L^{p} (ℝ^{d})$ with $1 < p ⩽ 2$ whenever $a ⩽ 1 / p .$ We demonstrate that the $L^{\infty} (ℝ^{d})$ 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 $L^{1} (ℝ^{d}) .$ 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 $L^{\infty} (ℝ^{d})$ boundedness may fail.

Nous étudions le caractère borné sur $L^{p}$ , $1 ⩽ p ⩽ \infty$ , pour les transformées de Riesz de la forme $V^{a} {(- \frac{1}{2} Δ + V)}^{- a},$ où $a > 0$ et $V$ est un potentiel non-négatif. Nous prouvons que $V^{a} {(- \frac{1}{2} Δ + V)}^{- a}$ est bornée sur $L^{p} (ℝ^{d})$ avec $1 < p ⩽ 2$ quand $a ⩽ 1 / p .$ Nous démontrons que le caractère borné sur $L^{\infty} (ℝ^{d})$ 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 $L^{1} (ℝ^{d}) .$ 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 $L^{\infty} (ℝ^{d})$ peut échouer.

Reçu le : 2023-06-29
Révisé le : 2024-02-08
Accepté le : 2024-03-21
Première publication : 2025-09-22

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 $L^p$

Affiliations des auteurs :

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

@unpublished{AIF_0__0_0_A32_0,
     author = {Kucharski, Maciej and Wr\'obel, B{\l}a\.zej},
     title = {On $L^p$ estimates for positivity-preserving {Riesz} transforms related to {Schr\"odinger} operators},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3744},
     language = {en},
     note = {Online first},
}

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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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