Iterated Integral Operators in Clifford Analysis

Begehr, Heinrich

doi:10.4171/zaa/887

Cited by 36 publications

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“…A hierarchy of analog integral operators and related higher order representations of Cauchy-Pompeiu type are attained, compare [4,11]. The same observation is made in Clifford analysis, see [6], where the Dirac operator is the adequate differential operator. In the case of several complex variables the situation is more involved.…”

Section: Introductionmentioning

confidence: 62%

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Second Order Cauchy-Pompeiu Representations

Begehr

1999

Complex Methods for Partial Differential Equations

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Second order Cauchy Pompeiu formulas are given in the case of one and of several complex variables, for Douglis-algebra-valued functions of one complex variable and for Clifford-algebra-valued functions of several real variables. The kernels of the integral operators attained provide fundamental solutions for the respective differential operators. INTRODUCTIONIn classical real analysis integral representation formulas are deduced from the GauE theorem. An example is the well-known Green representation for solutions to the Poisson equation. In complex analysis the Cauchy representation of analytic functions is also a consequence of the GauE theorem. But it thus is just a special case of the Cauchy-Pompeiu formula. This formula expresses any differentiable function through its boundary values and its first-order derivatives. Iterating the Cauchy-Pompeiu formula leads to higher order representations for functions n-times differentiable. They express the function through a combination of boundary integrals of the lower order derivatives and an area integral of the n-th order derivative. This was done in [9,10] where a hierarchy of integral operators was created by consecutive iteration of the Pompeiu operator and its complex conjugate and by differentiating the operators attained. In the same manner one can proceed for Douglis-algebra-valued functions. Here the Cauchy-Riemann operator from complex analysis is replaced by the Douglis operator. A hierarchy of analog integral operators and related higher order

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Section: Introductionmentioning

confidence: 62%

“…(11.31) are available from the second and fourth Cauchy-Pompeiu formulas. Continuing iteration leads to higher order representation formulas, see [6].…”

Section: Second Order Representations In Clifford Analysismentioning

confidence: 99%

Second Order Cauchy-Pompeiu Representations

Begehr

1999

Complex Methods for Partial Differential Equations

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“…The theory of functions valued in a Clifford algebra has important recent developments (see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]), it generalizes, to higher dimensions, the theory of holomorphic functions in the plane and refines the theory of harmonic functions. Including some on regular functions in Euclidean space which are generalizations of well-known classical results on analytic functions of one complex variables (see for example [4-6, 7-10, 14, 16]).…”

Section: Introductionmentioning

confidence: 99%

“…Including some on regular functions in Euclidean space which are generalizations of well-known classical results on analytic functions of one complex variables (see for example [4-6, 7-10, 14, 16]). In [5,16,14] Delanghe, Brackx, Begehr and Ryan studied k-regular functions in Euclidean space with values in Clifford algebra C(V 0,n ), the corresponding Cauchy integral formula and Taylor series were obtained. In [7][8][9], the author discusses the Cauchy integral formula of k-regular functions in Euclidean space with values in Clifford algebra C(V n,n ).…”

Section: Introductionmentioning

confidence: 99%

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

2008

AACA

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We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator D (k) s on locally defined polynomial forms on the unit sphere of R n ; functions valued in the universal Clifford algebra C(Vn,n), here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations D (k) s g = f . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation ∆sg = f . (2000). 30G35, 35C15, 35J05. Mathematics Subject Classification

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“…Theorem 3.2 is a convolution identity on C ∞ 0 (R n+1 ) involving the iterated Cauchy kernel and Dirac operator. This is a special case of results in [4]. See also [5] and [6].…”

Section: Introductionmentioning

confidence: 99%

Fourier Multipliers and Dirac Operators

Nolder

Wang

2017

Adv. Appl. Clifford Algebras

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Abstract. We use Fourier multipliers of the Dirac operator and Cauchy transform to obtain composition theorems and integral representations. In particular we calculate the multiplier of the Π-operator. This operator is the hypercomplex version of the Beurling Ahlfors transform in the plane. The hypercomplex Beuling Ahlfors transform is a direct generalization of the Beurling Ahlfors transform and reduces to this operator in the plane. We give an integral representation for iterations of the hypercomplex Beurling Ahlfors transform and we present here a bound for the L p -norm. Such L p -bounds are essential for applications of the Beurling Ahlfors transformation in the plane. The upper bound presented here is m(p * − 1) where m is the dimension of the Euclidean space on which the functions are defined, 1 < p < ∞ and p * = max(p, p/(p − 1)). We use recent estimates on second order Riesz transforms to obtain this result. Using the Fourier multiplier of the Π operator we express this operator as a hypercomplex linear combination of second order Riesz transforms.

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Iterated Integral Operators in Clifford Analysis

Cited by 36 publications

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Second Order Cauchy-Pompeiu Representations

Second Order Cauchy-Pompeiu Representations

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

Fourier Multipliers and Dirac Operators

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