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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp Gaussian bounds and $ L^p$-growth of semigroups associated with elliptic and Schrödinger operators


Author: El Maati Ouhabaz
Journal: Proc. Amer. Math. Soc. 134 (2006), 3567-3575
MSC (2000): Primary 47D08, 47D06, 35P15
Published electronically: May 31, 2006
MathSciNet review: 2240669
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Abstract: We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrödinger operators, including Schrödinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrödinger operators $ A=-\sum_{k = 1}^d (\tfrac{\partial}{\partial x_k}-i b_k)^2 + V $, we have the $ L^\infty$-estimate (for large $ t$):

$\displaystyle \Vert e^{-tA} \Vert_{{\mathcal L}(L^\infty(\mathbb{R}^d))} \le C e^{-s(A)t} ( t\ln t)^{d/4}$

where $ s(A) := \inf \sigma(A)$ is the spectral bound of $ A.$ The same estimate holds for elliptic and Schrödinger operators on general domains.


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Additional Information

El Maati Ouhabaz
Affiliation: Institut de Mathématiques de Bordeaux, Laboratoire d’Analyse et Géométrie, C.N.R.S. UMR 5467, Université Bordeaux 1-351, Cours de la Libération, 33405 Talence, France
Email: Elmaati.Ouhabaz@math.u-bordeaux1.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08430-9
PII: S 0002-9939(06)08430-9
Received by editor(s): March 15, 2005
Received by editor(s) in revised form: June 24, 2005
Published electronically: May 31, 2006
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.