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.
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Mathematics Subject Classification (2010). 32A26, 35F05, 47B34, 47F05.
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© 2011 Springer Basel AG
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Martin, M. (2011). Deconstructing Dirac Operators. II: Integral Representation Formulas. In: Sabadini, I., Sommen, F. (eds) Hypercomplex Analysis and Applications. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0346-0246-4_14
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DOI: https://doi.org/10.1007/978-3-0346-0246-4_14
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Online ISBN: 978-3-0346-0246-4
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