Quadratic estimates and functional calculi of perturbed Dirac operators

By A. Axelsson, S. Keith, A. Mcintosh. 2004

163 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6756466cedb79e4cb2496a2cd2b049936655f90a

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In Applied Mathematics

1. P. Auscher
2. S. Hofmann
3. A. Mcintosh
4. A. Axelsson
5. Mihalis Mourgoglou
6. Tuomas Hytönen
7. Dachun Yang
8. J. V. Neerven

In Mathematical Physics

1. P. Auscher
2. A. Axelsson
3. A. Mcintosh
4. S. Hofmann
5. Tuomas Hytönen
6. Mihalis Mourgoglou
7. J. V. Neerven
8. Pierre Portal

Citations in the corpus

• Hardy Spaces of Differential Forms on Riemannian Manifolds (2006, methodology): “The proof is done by induction on k and relies on a commutator argument, as in [9], Proposition 5.”
• Conical square function estimates in UMD Banach spaces and applications to H∞-functional calculi (2007, methodology): “We know from [4] that (I + zDB) −1 satisfies off-diagonal estimates of arbitrary order and that DB has an H ∞(Sθ)-calculus on L(2) ⊕ (L(2)).”
• Kato's square root problem in Banach spaces (2007, methodology): “The Kato estimates follow from the functional calculus of ΠB applied to the sign function z 7→ z/ √ z2, as in [9] Corollary 2.”
• Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems (2007, methodology): “10 we construct a new set of test functions f Q which also can be used in the Carleson measure estimate in [9] to simplify the proof there.”
• Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I (2009, background): “When σ = 0, each of the operators D0 and D̃0 is of the type considered in [14, 4], and each has two complementary invariant subspaces.”
• Representation and uniqueness for boundary value elliptic problems via first order systems (2014, methodology): “ perturbations under the impetus of Steve Hofmann (see [HK, AAAHK, HMiMo, BM]). The solution of the Kato problem for second order operators and systems [CMcM, HMc, AHLMcT, AHMcT] and its extension in [AKMc] to DBoperators brought a wealth of new estimates and techniques. The estimates for square roo”
• On a quadratic estimate related to the Kato conjecture and boundary value problems (2008, methodology): “This theorem is proved in [8] for first order D, i.”
• Solvability of elliptic systems with square integrable boundary data (2008, methodology): “This is a non-trivial result and is in fact equivalent to the Kato square root problem, proved by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian [5].”