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The Solution of the Kato Problem for Divergence Form Elliptic Operators with Gaussian Heat Kernel Bounds

Steve Hofmann, Michael Lacey and Alan McIntosh
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 623-631
Published by: Annals of Mathematics
DOI: 10.2307/3597200
Stable URL: http://www.jstor.org/stable/3597200
Page Count: 9
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The Solution of the Kato Problem for Divergence Form Elliptic Operators with Gaussian Heat Kernel Bounds
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Abstract

We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly 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 $\parallel \sqrt{L}f\parallel _{2}\sim \parallel \nabla f\parallel _{2}$. We note, in particular, that for such operators, the Gaussian hypothesis holds always in two dimensions.

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