Quadratic estimates and functional calculi of perturbed Dirac operators

2004/12/31 by Andreas Axelsson, Stephen Keith, Alan McIntosh · 2 citations
Mathematics · #Advanced Operator Algebra Research #Bounded function #Cauchy distribution #Dirac (video compression format) #Dirac operator #Holomorphic and Operator Theory #Lipschitz continuity #Operator (biology) #Quadratic equation #Spectral Theory in Mathematical Physics #Square root #math.CA #math.SP

paper · pdf · doi:10.1007/s00222-005-0464-x

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

arxiv created 2005/07/04 · openalex publication_date 2005/10/24 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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.

Citations

• Multiple Integrals in the Calculus of Variations
• Elliptic Partial Differential Equations of Second Order
• Cauchy integrals on Lipschitz curves and related operators

Cited by

• The S-resolvent estimates for the Dirac operator on hyperbolic and spherical spaces
• The Whitney method of fundamental solutions with Lusin wavelets