The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds
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Abstract:
We prove the Kato conjecture for degenerate elliptic operators on ${\mathbb {R}^n}$. More precisely, we consider the divergence form operator ${\mathcal {L}}_w=-w^{-1} {\mathrm {div}}\mathbf {A}{\nabla }$, where $w$ is a Muckenhoupt $A_{2}$ weight and $\mathbf {A}$ is a complex-valued $n\times n$ matrix such that $w^{-1}\mathbf {A}$ is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup $e^{-t{\mathcal {L}_w}}$ satisfies Gaussian bounds, then the weighted Kato square root estimate, $\|{\mathcal {L}}_w^{1/2} f\| _{L^{2}\left ( w\right ) }\approx \| {\nabla } f\| _{L^{2}\left ( w\right ) }$, holds.References
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Additional Information
- SFO David Cruz-Uribe
- Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106
- Email: david.cruzuribe@trincoll.edu
- Cristian Rios
- Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada
- Email: crios@math.ucalgary.ca
- Received by editor(s): July 23, 2009
- Received by editor(s) in revised form: May 18, 2010
- Published electronically: February 20, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3449-3478
- MSC (2010): Primary 35J70, 35K45, 35K65, 35C15; Secondary 47D06, 47N20
- DOI: https://doi.org/10.1090/S0002-9947-2012-05380-3
- MathSciNet review: 2901220