The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds (Q2839940)

The authors give a positive answer to the Kato square root conjecture for a class of degenerate elliptic operators on \(\mathbb R^{n}\), under the assumption that the associated heat kernel satisfies classic Gaussian upper bounds. More precisely, they consider the divergence form operator \(\mathcal L_{w}=-w^{-1}\text{div}{\mathbf A}\nabla\), where \(w\) is a Muckenhoupt \(A_{2}\) weight and \(\mathbf A\) is a complex-valued \(n\times n\) matrix such that \(w^{-1}\mathbf A\) is bounded and uniformly elliptic. It is shown that if the heat kernel of the associated semigroup \(e^{-t\mathcal L_{w}}\) satisfies Gaussian bounds, then the weighted Kato square root estimate, \(\|\mathcal L_{w}^{-1/2}f\|_{L^{2}(w)}\approx \|\nabla f\|_{L^{2}(w)}\), holds.