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

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 = -div(A∇) with bounded measurable coefficients in ℝⁿ is the Sobolev space H¹(ℝⁿ) in any dimension with the estimate ||√ L f || ₂ ∼ ||nabla;f||₂. We note, in particular, that for such operators, the Gaussian hypothesis holds always in two dimensions.