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

Pascal Auscher (1) , Andreas Axelsson (2) , Alan Mcintosh (3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Matematiska institutionen
3 CMA - Center for Mathematics and its Applications

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.

Mots clés: Littlewood-Paley estimate; functional calculus; boundary value problems; second order elliptic equations and systems; square root problem

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Pascal Auscher, Andreas Axelsson, Alan Mcintosh. On a quadratic estimate related to the Kato conjecture and boundary value problems. Harmonic Analysis and Partial Differential Equations. Harmonic Analysis and Partial Differential Equations, pp.105-129, 2010, Contemp. Math., 505, Amer. Math. Soc., Providence, RI, 2010. ⟨hal-00331494v2⟩