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 Δ+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 d(Δ+𝒱) -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 Δ+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 d(Δ+𝒱) -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 = {http://archive.numdam.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. http://archive.numdam.org/articles/10.5802/aif.3249/

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