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The Kato problem for operators with weighted ellipticity


Authors: David Cruz-Uribe SFO and Cristian Rios
Journal: Trans. Amer. Math. Soc. 367 (2015), 4727-4756
MSC (2010): Primary 35J15, 35J25, 35J70, 35D30, 47D06, 35B30, 31B10, 35B45
DOI: https://doi.org/10.1090/S0002-9947-2015-06131-5
Published electronically: March 2, 2015
MathSciNet review: 3335399
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Abstract: We consider second order operators $ \mathcal {L}_{w}=-w^{-1}\operatorname {div}\mathbf {A}_{w}\nabla $ with ellipticity controlled by a Muckemphout $ A_{2}$ weight $ w $. We prove that the Kato square root estimate $ \left \Vert \mathcal {L} _{w}^{1/2}f\right \Vert _{L^{2}\left ( w\right ) }\approx \left \Vert \nabla f\right \Vert _{L^{2}\left ( w\right ) }$ holds in the weighted space $ L^{2}\left ( w\right ) $.


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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, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: crios@ucalgary.ca

DOI: https://doi.org/10.1090/S0002-9947-2015-06131-5
Keywords: Elliptic operators, Kato problem, weighted norm inequalities, functional calculus
Received by editor(s): September 24, 2012
Received by editor(s) in revised form: January 6, 2013, March 9, 2013, March 10, 2013, March 11, 2013, and March 13, 2013
Published electronically: March 2, 2015
Additional Notes: The first author was partially supported by the Stewart-Dorwart faculty development fund at Trinity College
The second author was supported by the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.