# The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds

### Abstract

We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly 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 dimension with the estimate $\Vert \sqrt{L}f\Vert _2 \sim \Vert \nabla f\Vert_2$. We note, in particular, that for such operators, the Gaussian hypothesis holds always in two dimensions.

Steve Hofmann

Michael Lacey

Alan McIntosh