The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds

David Cruz-Uribe, SFO; Rios, Cristian

doi:10.1090/S0002-9947-2012-05380-3

The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds
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by SFO David Cruz-Uribe and Cristian Rios PDF

Trans. Amer. Math. Soc. 364 (2012), 3449-3478 Request permission

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.

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