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Deconstructing Dirac Operators. II: Integral Representation Formulas

  • Mircea MartinEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We set up generalized Cauchy-Pompeiu and Bochner-Martinelli- Koppelman representation formulas for arbitrary pairs (\(\mathfrak{D}, \Phi\)), where \(\mathfrak{D}\) is a first-order homogeneous differential operator on ℝn with coefficients in a Banach algebra \(\mathfrak{A} \), and ф is a smooth \(\mathfrak{A} \)-valued function on ℝn \ { 0} homogeneous of degree 1 –n, n ≥2. Within our general framework we prove that the integral representation formulas include the expected components, as well as some remainders that are explicitly computed in terms of \(\mathfrak{A} \) and ф. As a consequence, we obtain necessary and sufficient conditions that ensure the existence of genuine Cauchy-Pompeiu or Bochner-Martinelli-Koppelman formulas for such operator-kernel pairs (\(\mathfrak{D}, \Phi\)). Properly interpreted in a Clifford algebra setting these conditions prove valuable in investigating Dirac and Cauchy-Riemann operators.

Keywords

Dirac operators Cauchy-Riemann operators first-order differential operators Cauchy-Pompeiu formula Bochner-Martinelli-Koppelman formula integral representation formulas 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsBaker UniversityBaldwin CityUSA

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