The solution of Kato's conjectures (Q2738864)

The present paper announces a series of results including the solution of Kato's conjectures. The six authors of this note contributed to the proof of one or several of the theorems with full details presented in [Ann. Math. (2) 156, 633-654 (2002; Zbl 1128.35316)]. The results are summarized as follows:NEWLINENEWLINENEWLINETheorem 1. For any second order, divergence-form operator, \(Lf=-\text{div} A\nabla\) satisfying \(\lambda|\xi |^2\leq \text{Re} A\xi \cdot\xi^*\) and \(|A\xi\cdot\xi^* |\leq\Lambda |\xi ||\zeta|\) for \(\xi,\zeta\in \mathbb{C}^n\) and \(0<\lambda \leq\Lambda< \infty\), the domain of \(\sqrt L\) coincides with \(H^1(\mathbb{R}^n)\) and \(\|\sqrt L f\|_2 \approx\|\nabla f\|_2\) holds with constants depending only on \(n,\lambda\), and \(\Lambda\).NEWLINENEWLINENEWLINETheorem 2. If \(n\) is greater or equal to 2, the domain of the square root of an elliptic second order operator subject to Dirichlet or Neumann boundary conditions on a strongly Lipschitz domain is equal to a closed subspace \(V\) and one has the estimates \(\|\sqrt L f \|_{L^2(\Omega)} \leq C\|f\|_{H^1 (\Omega)}\) for all \(f\in V\) with \(C\) depending only on \(n,\lambda,\Lambda\) and \(\Omega\).NEWLINENEWLINENEWLINETheorem 3 and Theorem 4 also have some very interesting estimates. The announcement is very timely for the mathematical community.