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The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on Rn

Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh and Ph. Tchamitchian
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 633-654
Published by: Annals of Mathematics
DOI: 10.2307/3597201
Stable URL: http://www.jstor.org/stable/3597201
Page Count: 22
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The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on  Rn
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Abstract

We prove the Kato conjecture for elliptic operators on Rn. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator $L=-\text{div}(A\nabla)$ with bounded measurable coefficients in Rn is the Sobolev space $H^{1}({\Bbb R}^{n})$ in any dimension with the estimate $\parallel \sqrt{L}f\parallel _{2}\sim \parallel \nabla f\parallel _{2}$.

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