Kato square root problem with unbounded leading coefficients

Escauriaza, Luis; Hofmann, Steve

doi:10.1090/proc/14224

Kato square root problem with unbounded leading coefficients
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by Luis Escauriaza and Steve Hofmann PDF

Proc. Amer. Math. Soc. 146 (2018), 5295-5310 Request permission

Abstract:

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 $\|\sqrt {L} f\|_2 \lesssim \| \nabla f\|_2$.

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