Clifford Cauchy type integrals on Ahlfors-David regular surfaces in
$$\mathbb{R}^{m + 1} $$

Blaya, Ricardo Abreu; Peña, Dixan Peña; Reyes, Juan Bory

doi:10.1007/s00006-003-0008-7

Cited by 27 publications

(9 citation statements)

References 43 publications

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“…Moreover, neither the hypotheses nor the conclusions of Theorems 2.1, 2.2 are affected if the Lipschitz condition on Γ is replaced by the so-called Ahlfors David regular (AD-regular for short) assumption. In fact, we have only used the existence of continuous limit values of the Cauchy transform with a given Hölder continuous data, which remains valid for any Jordan domain Ω with an AD-regular boundary Γ (see [1]) after understanding the unit normal vector in Federer's sense [18].…”

Section: Theorem 22 (Aronov and Kytmanov) Letmentioning

confidence: 99%

Holomorphic Extension Theorems in Lipschitz Domains of $${\mathbb{C}}^{2}$$

Blaya

Reyes²,

Peña

et al. 2008

AACA

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The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions f : R 2n → Cn of the so-The aim of this paper is to bring together these two areas which are intended as a good generalization of the classical one-dimensional complex analysis.In particular, it is of interest to study how far some classical holomorphic extension theorems can be stretched when the regularity of the boundary is reduced from C 1 -smooth to Lipschitz. As an illustration, we give a complete viewpoint on simplified proofs of Kytmanov-Aronov-Aȋzenberg type theorems for the case n = 2.We will denote by {e 1 , . . . , e m } an orthonormal basis of the Euclidean space R m . Let C m be the complex Clifford algebra constructed over R m . The noncommutative multiplication in C m is governed by the rules:The Clifford algebra C m is generated additively by elements of the formwhere A = {j 1 , . . . , j k } ⊂ {1, . . . , m} is such that j 1 < · · · < j k , and so the dimension of C m is 2 m . For A = ∅, e ∅ = 1 is the identity element.

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Section: Theorem 22 (Aronov and Kytmanov) Letmentioning

confidence: 99%

Holomorphic Extension Theorems in Lipschitz Domains of $${\mathbb{C}}^{2}$$

Blaya

Reyes²,

Peña

et al. 2008

AACA

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“…Finally, the Clifford-Dirac case was later generalized to the full Dirac operator case on a Riemannian manifold by Mitrea [69,83,84]. See also [1][2][3][4][5][6]9,49,65,112] and the books [17,41,55,81,82,93] for related results and more references. The methods used to prove boundedness on these Banach spaces work for Lipschitz domains and use, amongst other methods, singular harmonic measures (e.g.…”

Section: Manifolds With Cornersmentioning

confidence: 99%

Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners

Loya

2007

Journal of Differential Equations

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In this paper we give a new perspective on the Cauchy integral and transform and Hardy spaces for Diractype operators on manifolds with corners of codimension two. Instead of considering Banach or Hilbert spaces, we use polyhomogeneous functions on a geometrically "blown-up" version of the manifold called the total boundary blow-up introduced by Mazzeo and Melrose [R.R. Mazzeo, R.B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1) (1995) 14-75]. These polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a "Taylor series" (with possible log terms) in polar coordinates. The main application of our analysis is a complete Fredholm theory for boundary value problems of Dirac operators on manifolds with corners of codimension two.

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“…For instance there is a very general notion of the unit normal n(y) introduced by Federer [21] such that the Stokes's Theorem still holds for boundaries with H n (Γ) < +∞. It is exactly this version of Stokes's Theorem we need to establish basic formulas in Clifford analysis such as the following Borel-Pompeiu formula: Here we refer the reader to [1,12] for a detailed presentation of the easiest way to make the desired formula true.…”

Section: Clifford Algebras and Monogenic Functionsmentioning

confidence: 99%

“…It is a continuation of a series of papers by the Cuban Clifford research group [1][2][3][4]11] where we have been expressed our vision of the role of the Cauchy transform in generalizations of the one-dimensional complex analysis to a great extent. In particular, properties of the boundary values of the Cliffordian-Cauchy transform under weaker assumptions imposed on the boundary of the domains than almost all reference until nowadays (to the authors's knowledge) were examined.…”

Section: Introductionmentioning

confidence: 99%

Minkowski Dimension and Cauchy Transform in Clifford Analysis

Blaya

Reyes

García

2007

Complex anal.oper. theory

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In the present paper we provide some conditions of a geometrical character for continuous extendibility of the Clifford-Cauchy transform to the boundary of a domain in the Euclidean space of higher dimensions if its density satisfies a Hölder condition. The criterion obtained in this work is an extension to a very general class of domains of a result, which has already become classical, obtained by Viorel Iftimie, who proved in 1965, for the case of a domain with compact Liapunov boundary, that the Clifford-Cauchy transform has Hölder-continuous limit values for any Hölder-continuous density. Mathematics Subject Classification (2000). 30E20, 30E25, 30G35.

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Clifford Cauchy type integrals on Ahlfors-David regular surfaces in $$\mathbb{R}^{m + 1} $$

Cited by 27 publications

References 43 publications

Holomorphic Extension Theorems in Lipschitz Domains of $${\mathbb{C}}^{2}$$

Holomorphic Extension Theorems in Lipschitz Domains of $${\mathbb{C}}^{2}$$

Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners

Minkowski Dimension and Cauchy Transform in Clifford Analysis

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