The Kato square root problem follows from an extrapolation property of the Laplacian
Moritz Egert, Robert Haller‐Dintelmann, Patrick Tolksdorf · Project Euclid (Cornell University) · 2016
On a domain $\\Omega \\subseteq \\mathbb{R}^d$ we consider second-order elliptic systems in divergence-form with bounded complex coefficients, realized via a sesquilinear form with domain $\\mathrm{H}_0^1(\\Omega) \\subseteq \\mathcal{V} \\subseteq \\mathrm{H}^1(\\Omega)$. Under very mild assumptions on~$\\Omega$ and $\\mathcal{V}$ we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends earlier results of McIntosh [25] and Axelsson-Keith-McIntosh [6] to non-smooth coefficients and domains.