L'integrale de Cauchy Definit un Operateur Borne sur L 2 Pour Les Courbes Lipschitziennes

Coifman, Ronald R.; McIntosh, Alan; Meyer, Yves

doi:10.2307/2007065

Cited by 569 publications

(455 citation statements)

References 5 publications

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“…As such, this extends results from the Euclidean context and for standard constant coefficient Dirac operators in [9], [24], [17] and [35].…”

Section: Inverting Generalized Dirac Operators and Laplacianssupporting

confidence: 82%

“…One intriguing aspect is that there are actually two Cauchy operators naturally associated with D: one which has a "holomorphic" kernel and one which reproduces "holomorphic" functions. As we shall see momentarily, they satisfy similar properties (such as L p boundedness and jump relations) to the ordinary Cauchy operator on Lipschitz curves of the complex plane as discussed in [9]. For now, if f : ∂Ω → E is an arbitrary section set:…”

Section: Inverting Generalized Dirac Operators and Laplaciansmentioning

confidence: 73%

“…The point (1) follows directly from Theorem 2.9 in [34] (where some of the main results of [9] are extended to manifolds). To see (2), we first note the readily verified identity…”

Section: Inverting Generalized Dirac Operators and Laplaciansmentioning

confidence: 99%

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Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Mitrea

2001

The Journal of Fourier Analysis and Applications

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We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study PlemeljCalderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of 'inner' and 'outer' monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L p -decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allows one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a 'half' Dirichlet problem for a suitable Dirac operator.

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“…As such, this extends results from the Euclidean context and for standard constant coefficient Dirac operators in [9], [24], [17] and [35].…”

Section: Inverting Generalized Dirac Operators and Laplacianssupporting

confidence: 82%

Section: Inverting Generalized Dirac Operators and Laplaciansmentioning

confidence: 73%

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Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Mitrea

2001

The Journal of Fourier Analysis and Applications

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“…The boundedness of the Cauchy integral C γ was first proved for small L by Calderón [7], and in the general case by Coifman-M c Intosh-Meyer [10]. Boundedness of other operators in the functional calculus of iD γ have been proved by Coifman-Meyer [9], Kenig-Meyer [18] and M c Intosh-Qian [27].…”

Section: Consequencesmentioning

confidence: 98%

Quadratic estimates and functional calculi of perturbed Dirac operators

2005

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Abstract. We prove quadratic estimates for complex perturbations of Diractype 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.

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“…We point out that the use of layer potentials in the periodic setting relies on two crucial developments. The first one is the proof of Coifman-McIntosh-Meyer [7] of the L p boundedness of the Cauchy integrals on Lipschitz curves. By the method of rotation this gives the L p boundedness of layer potentials on Lipschitz surfaces for elliptic systems with constants coefficients.…”

Section: Introductionmentioning

confidence: 99%

Layer potential methods for elliptic homogenization problems

Kenig

Shen

2010

Comm Pure Appl Math

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In this paper we use the method of layer potentials to study L 2 boundary value problems in a bounded Lipschitz domain Ω for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the theory of homogenization. Let L ε = −div A(ε −1 X)∇ . Under the assumption that A(X) is elliptic, symmetric, periodic and Hölder continuous, we establish the solvability of the L 2 Dirichlet, regularity, and Neumann problems for L ε (u ε ) = 0 in Ω with optimal estimates uniform in ε > 0.

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L'integrale de Cauchy Definit un Operateur Borne sur L 2 Pour Les Courbes Lipschitziennes

Cited by 569 publications

References 5 publications

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Quadratic estimates and functional calculi of perturbed Dirac operators

Layer potential methods for elliptic homogenization problems

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