On a quadratic estimate related to the Kato conjecture and boundary   value problems
  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. 
Pascal Auscher (LM-Orsay), Andreas Axelsson, Alan McIntosh (CMA)