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

35J15Second order elliptic equations, general
42B20Singular and oscillatory integrals, several variables
47B44Accretive operators, dissipative operators, etc. (linear)
47F05Partial differential operators
Full Text: DOI Euclid