Quadratic estimates and functional calculi of perturbed Dirac operators

Abstract

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge-Dirac operator on compact manifolds depend analytically on L ∞ changes in the metric. We also recover a unified proof of many results in the Calderón program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces. Show more