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Kato square root problem with unbounded leading coefficients


Authors: Luis Escauriaza and Steve Hofmann
Journal: Proc. Amer. Math. Soc. 146 (2018), 5295-5310
MSC (2010): Primary 35B45, 35J15, 35J25, 42B20, 42B37, 47B44
DOI: https://doi.org/10.1090/proc/14224
Published electronically: September 17, 2018
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Abstract: We prove the Kato conjecture for elliptic operators, $ L=-\nabla \cdot \left ((\mathbf A+\mathbf D)\nabla \ \right )$, with $ \mathbf A$ a complex measurable bounded coercive matrix and $ \mathbf D$ a measurable real-valued skew-symmetric matrix in $ \mathbb{R}^n$ with entries in $ BMO(\mathbb{R}^n)$; i.e., the domain of $ \sqrt {L}\,$ is the Sobolev space $ \dot H^1(\mathbb{R}^n)$ in any dimension, with the estimate $ \Vert\sqrt {L}\, f\Vert _2 \lesssim \Vert \nabla f\Vert _2$.


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Additional Information

Luis Escauriaza
Affiliation: UPV/EHU, Departamento de Matemáticas, Barrio Sarriena s/n, 48940 Leioa, Spain
Email: luis.escauriaza@ehu.eus

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: hofmann@math.missouri.edu

DOI: https://doi.org/10.1090/proc/14224
Keywords: Kato's conjecture
Received by editor(s): December 28, 2017
Received by editor(s) in revised form: April 24, 2018
Published electronically: September 17, 2018
Additional Notes: The first author was supported by grants MTM2014-53145-P and IT641-13 (GIC12/96).
The second author was supported by NSF grant no. DMS-1664047.
This material is based upon work supported by the National Science Foundation under Grant No. DMS- 1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2018 American Mathematical Society

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