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.

DOI: 10.2307/3597200

Authors

Steve Hofmann

Michael Lacey

Alan McIntosh