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 $L_{\infty }$ 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.

To appear in Inventiones Mathematicae. Minor final changes added 4/7 2005

Cited by in corpus (21)

• Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
• Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi
• Lp-estimates for the square root of elliptic systems with mixed boundary conditions
• The Kato Square Root Problem on locally uniform domains
• Density problems on vector bundles and manifolds
• Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems
• The abstract Hodge-Dirac operator and its stable discretization
• The Kato square root problem on vector bundles with generalised bounded geometry
• On uniqueness results for Dirichlet problems of elliptic systems without DeGiorgi-Nash-Moser regularity
• Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
• Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric
• Functional calculus for $C_0$-groups using (co)type
• Perturbation of Sectorial Projections of Elliptic Pseudo-differential Operators
• The Kato Square Root Problem for Divergence Form Operators with Potential
• Rough metrics on manifolds and quadratic estimates
• Basic Functional Analysis Puzzles of Spectral Flow
• Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data
• First-Kind Boundary Integral Equations for the Dirac Operator in 3D Lipschitz Domains
• Functional calculus and harmonic analysis in geometry
• Elliptic Pre-Complexes, Hodge-like Decompositions and Overdetermined Boundary-Value Problems
• Hardy spaces for boundary value problems of elliptic systems with block structure