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 whose potential 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 . In a second part, we prove sharp boundedness results for the Riesz transform with potential . A very simple characterization of -non-parabolicity in terms of lower bounds for the volume growth, which is of independent interest, is also presented.
Le but de cet article est double : dans une première partie, nous donnons une preuve analytique des estimées Gaussiennes pour un opérateur de Schrödinger dont le potential 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 . Dans une seconde partie, nous prouvons des résultats optimaux concernant l’action de la transformée de Riesz avec potentiel sur les espaces . Une charactérisation particulièrement simple de la non--parabolicité en terme de borne inférieure de la croissance du volume, qui a un intérêt en tant que tel, est aussi obtenue.
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3249
Keywords: Heat kernel, Schrödinger operators, Riesz transform, $p$-non-parabolicity.
Mot clés : Noyau de la chaleur, opérateurs de Schrödinger, transformée de Riesz, $p$-non-parabolicité
Devyver, Baptiste 1
@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/} }
TY - JOUR AU - Devyver, Baptiste TI - Heat kernel and Riesz transform of Schrödinger operators JO - Annales de l'Institut Fourier PY - 2019 SP - 457 EP - 513 VL - 69 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3249/ DO - 10.5802/aif.3249 LA - en ID - AIF_2019__69_2_457_0 ER -
%0 Journal Article %A Devyver, Baptiste %T Heat kernel and Riesz transform of Schrödinger operators %J Annales de l'Institut Fourier %D 2019 %P 457-513 %V 69 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3249/ %R 10.5802/aif.3249 %G en %F AIF_2019__69_2_457_0
Devyver, Baptiste. Heat kernel and Riesz transform of Schrödinger operators. Annales de l'Institut Fourier, Volume 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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