2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230455We 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.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 ProceedingsClassical Analysis and ODEsAnalysis of PDEsFunctional Analysis35J25, 35J55, 47N20, 47F05, 42B25text