The Kato Square Root Problem follows from an Extrapolation Property of the Laplacian

Abstract

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 $V \subseteq H^1(\Omega)$. Under very mild assumptions on $\Omega$ and $V$ we show that the Kato Square Root Problem for such systems can be reduced to a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of McIntosh to non-smooth coefficients.