The aim of the paper under review is to prove the Kato conjecture for elliptic \(N\times N\)-systems in divergence form of arbitrary order \(2m\) in \({\mathbb R}^n\). More precisely, the considered operators have bounded measurable coefficients and the ellipticity is defined in the sense of a Gårding inequality. The considered operators define maximal-accretive operators in \(L^2({\mathbb R}^n; {\mathbb C}^N)\); thus their square roots may be defined using the holomorphic functional calculus. The main result of the paper identifies the domains of these square roots as being equal to the Sobolev space \(H^m({\mathbb R}^n; {\mathbb C}^N)\), with equivalent norms.
The argument for the proof of the main result has two parts. First, the semigroup kernel of the operator is assumed to satisfy a pointwise upper bound; the required result is a consequence of a Carleson measure estimate, which is proved using a “T(b)” argument. In the second part of the proof, the assumption on the semigroup kernel is removed by noticing that this assumption is fulfilled by operators with high enough order.
The relation between various ellipticity conditions and the Gårding inequality is also investigated.