ANNALES DE L'INSTITUT FOURIER

Heat kernel and Riesz transform of Schrödinger operators
[Noyau de la chaleur et transformée de Riesz des opérateurs de Schrödinger]
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 457-513.

Le but de cet article est double : dans une première partie, nous donnons une preuve analytique des estimées Gaussiennes pour un opérateur de Schrödinger $\Delta +𝒱$ dont le potential $𝒱$ est « petit à l’infini » en un sens (faible) intégral. Nos résultats améliorent des résultats connus, qui avaient été prouvés précédemment par des techniques probabilistes, et éclairent les hypothèses qui doivent être faites sur le potentiel $𝒱$. Dans une seconde partie, nous prouvons des résultats optimaux concernant l’action de la transformée de Riesz avec potentiel $\mathrm{d}{\left(\Delta +𝒱\right)}^{-1/2}$ sur les espaces ${L}^{p}$. Une charactérisation particulièrement simple de la non-$p$-parabolicité en terme de borne inférieure de la croissance du volume, qui a un intérêt en tant que tel, est aussi obtenue.

The goal of this article is two-fold: in the first part, we give a purely analytic proof of the Gaussian estimates for the heat kernel of Schrödinger operators $\Delta +𝒱$ whose potential $𝒱$ is “small at infinity” in a (weak) integral sense. Our results improve known results that have been proved by probabilistic techniques, and shed light on the hypotheses that are required on the potential $𝒱$. In a second part, we prove sharp boundedness results for the Riesz transform with potential $\mathrm{d}{\left(\Delta +𝒱\right)}^{-1/2}$. A very simple characterization of $p$-non-parabolicity in terms of lower bounds for the volume growth, which is of independent interest, is also presented.

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DOI : https://doi.org/10.5802/aif.3249
Classification : 35Kxx,  31Exx,  58Jxx
Mots clés : Noyau de la chaleur, opérateurs de Schrödinger, transformée de Riesz, $p$-non-parabolicité
@article{AIF_2019__69_2_457_0,
author = {Devyver, Baptiste},
title = {Heat kernel and {Riesz} transform of {Schr\"odinger} operators},
journal = {Annales de l'Institut Fourier},
pages = {457--513},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {69},
number = {2},
year = {2019},
doi = {10.5802/aif.3249},
zbl = {07067410},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3249/}
}
Devyver, Baptiste. Heat kernel and Riesz transform of Schrödinger operators. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 457-513. doi : 10.5802/aif.3249. https://aif.centre-mersenne.org/articles/10.5802/aif.3249/

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