On a quadratic estimate related to the Kato conjecture and boundary value problems

Abstract

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.Comment: Text of the lectures given at the El Escorial 2008 conference. Revised after the suggestions of the referee. Some historical material added. A short proof of the main result added under a further assumption. To appear in the Proceeding