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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds


Authors: David Cruz-Uribe, SFO and Cristian Rios
Journal: Trans. Amer. Math. Soc. 364 (2012), 3449-3478
MSC (2010): Primary 35J70, 35K45, 35K65, 35C15; Secondary 47D06, 47N20
Published electronically: February 20, 2012
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Abstract: We prove the Kato conjecture for degenerate elliptic operators on $ {\mathbb{R}^n}$. More precisely, we consider the divergence form operator $ {\mathcal {L}}_w=-w^{-1} {\mathrm {div}}\mathbf {A}{\nabla } $, where $ w$ is a Muckenhoupt $ A_{2}$ weight and $ \mathbf {A}$ is a complex-valued $ n\times n$ matrix such that $ w^{-1}\mathbf {A}$ is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup $ e^{-t{\mathcal {L}_w}}$ satisfies Gaussian bounds, then the weighted Kato square root estimate, $ \Vert{\mathcal {L}}_w^{1/2} f\Vert _{L^{2}\left ( w\right ) }\approx \Vert {\nabla } f\Vert _{L^{2}\left ( w\right ) }$, holds.


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Additional Information

David Cruz-Uribe, SFO
Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106
Email: david.cruzuribe@trincoll.edu

Cristian Rios
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
Email: crios@math.ucalgary.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05380-3
PII: S 0002-9947(2012)05380-3
Received by editor(s): July 23, 2009
Received by editor(s) in revised form: May 18, 2010
Published electronically: February 20, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.