Several complex variables and analytic spaces Holomorphic functions of several variables Theorem

Cauchy–Fantappiè formula

Builds on Bochner–Martinelli formula Dbar operator Holomorphic function of several variables

Statement

Let $n \ge 1$ and let $D \subseteq \C^n$ be a bounded domain with $C^1$ boundary $\partial D$ , oriented as the boundary of $D$ . Fix $z \in D$ . Let $\eta = (\eta_1, \dots, \eta_n)$ be a map of class $C^1$ from a neighbourhood of $\partial D$ to $\C^n$ such that $\sum_{j=1}^n \eta_j(\zeta) (\zeta_j - z_j) = 1 \quad \text{for all } \zeta \in \partial D.$ Put $\omega_\eta(\zeta) = \frac{(n-1)!}{(2\pi i)^n} \sum_{j=1}^n \eta_j(\zeta) \, d\zeta_j \wedge \bigwedge_{k \ne j} \left( \bar\partial \eta_k \wedge d\zeta_k \right),$ where $\bar\partial$ is the $\bar\partial$ operator in $\zeta$ . Let $f$ be a holomorphic function on a neighbourhood of $\overline{D}$ . Then $f(z) = \int_{\partial D} f(\zeta) \, \omega_\eta(\zeta).$

Remarks

The forms $\bar\partial \eta_k \wedge d\zeta_k$ have degree $2$ , so the order of the factors in the product does not matter. With $\eta_j(\zeta) = (\bar\zeta_j - \bar z_j)/\abs{\zeta - z}^2$ , the form $\omega_\eta$ is the kernel of the Bochner–Martinelli formula, and the statement reduces to that formula for holomorphic $f$ . For a general $\eta$ , a homotopy argument joining $\eta$ to the Bochner–Martinelli choice shows that the boundary integral does not depend on $\eta$ . For $n = 1$ the condition forces $\eta = 1/(\zeta - z)$ , and the formula is the Cauchy integral formula. On a convex domain with $C^2$ boundary, one can choose $\eta$ depending holomorphically on $z$ , which gives integral formulas with kernels holomorphic in $z$ .

Going deeper

For a student

The Bochner–Martinelli kernel is universal but not holomorphic in $z$ , which limits its use. The freedom in $\eta$ allows kernels that are holomorphic in $z$ on convex or strongly pseudoconvex domains. Such kernels, built by Henkin, Ramírez and others, give solutions of the $\bar\partial$ equation with estimates and results on approximation by holomorphic functions.

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(p,q)-form concept
Dbar operator concept
Holomorphic function of several variables concept
Bochner–Martinelli formula statement
Cauchy–Fantappiè formula statement

Tag	`R9JJ44TMQB`
URI	`m2://R9JJ44TMQB`
Kind	statement, theorem
Writing	W0 stub
MSC	32A26
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About	Holomorphic function of several variables
Uses	Bochner–Martinelli formula, Dbar operator, Holomorphic function of several variables