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The solution of the Kato square root problem for second order elliptic operators on $$\mathbb R^n$$. (English) Zbl 1128.35316
Summary: We prove the Kato conjecture for elliptic operators on $$\mathbb R^n$$. 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 $$\mathbb R^n$$ is the Sobolev space $$H^1(\mathbb R^n)$$ in any dimension with the estimate $$\|\sqrt L f\|_2\sim \| \nabla f\|_2$$.

##### MSC:
 35J15 Second-order elliptic equations 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47B44 Linear accretive operators, dissipative operators, etc. 47F05 General theory of partial differential operators
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