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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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