The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

Abstract

We prove the Kato conjecture for elliptic operators and N×N-systems in divergence form of arbitrary order 2m on $ \Bbb R^n $ . More precisely, we assume the coefficients to be bounded measurable and the ellipticity is taken in the sense of a Garding inequality. We identify the domain of their square roots as the natural Sobolev space $ H^m(\Bbb R^n,\Bbb C^N) $ . We also make some remarks on the relation between various ellipticity conditions and Garding inequality.