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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cauchy-Green type formulae in Clifford analysis

Author: John Ryan
Journal: Trans. Amer. Math. Soc. 347 (1995), 1331-1341
MSC: Primary 30G35; Secondary 58G99
MathSciNet review: 1249888
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Abstract: A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation $ ({D^k} + \sum\nolimits_{m = 0}^{k - 1} {{b_m}{D^m})f = 0} $, where each $ {b_m}$ is a complex number, $ D$ is the Dirac operator in $ {R^n}$, and $ f$ is defined on a domain in $ ^{{R^n}}$ and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximation theorem. We also introduce a Bergman kernel for square integrable solutions to $ (D + \lambda )f = 0$ over bounded domains with piecewise $ {C^1}$, or Lipschitz, boundary.

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