Kato square root problem with unbounded leading coefficients

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)\...

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Bibliographic Details
Published in:arXiv.org 2017-12
Main Authors: Escauriaza, Luis, Hofmann, Steve
Format: Article
Language:English
Subjects:
Coercivity
Mathematical analysis
Matrix methods
Sobolev space
Online Access:Get full text
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Summary: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 \(\|\sqrt{L}\, f\|_2\lesssim \| \nabla f\|_2\).
ISSN:2331-8422