# The solution of the Kato square root problem for second order elliptic operators on $\mathbb{R}^n$

### Abstract

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 $L = -\mathrm{div} (A\nabla)$ with bounded measurable coefficients in $\mathbb{R}^n$ is the Sobolev space $H^1(\mathbb{R}^n)$ in any dimensio with the estimate in $\Vert \sqrt{L} f\Vert_2 \sim \Vert \nabla f\Vert_2$.

## Authors

Pascal Auscher

Steve Hofmann

Michael Lacey

Alan McIntosh

Philippe Tchamitchian