Deconstructing Dirac Operators. II: Integral Representation Formulas

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.

Mathematics Subject Classification (2010). 32A26, 35F05, 47B34, 47F05.