Complex Analysis and Operator Theory

, Volume 13, Issue 2, pp 431–478

# Cauchy–Fantappiè Integral Formula for Holomorphic Functions on Special Tube Domains in $$\mathbb {C}^2$$

• N. Alexandrou
• A. Vidras
Article

## Abstract

Let $$T_{B}=\mathbb {R}^2\times i\{(y_1,y_2)\in \mathbb {R}^2 :y_1^2+y_2^2 <1\}$$ be the tube in $$\mathbb {C}^2$$ with base the imaginary disk $$B=i\{(y_1,y_2)\in \mathbb {R}^2 :y_1^2+y_2^2 <1\}$$. The defining function of the tube is $$\Phi (\zeta ,\bar{\zeta } )=(\frac{\zeta _1-\bar{\zeta } _1}{ 2i})^2+(\frac{\zeta _2-\bar{\zeta } _2}{ 2i})^2-1$$, $$\zeta \in \mathbb {C}^2$$. Let $$H^2(T_{B})$$ be the space of holomorphic functions $$f(z)=\int \limits _{\mathbb {R}^2}h(t)e^{2\pi i z\cdot t}dt$$, $$z\in T_{B}$$, whenever the function h satisfies the estimate
\begin{aligned} \sup _{y\in B}\int \limits _{\mathbb {R}^2}\vert h(t)\vert ^2e^{-4\pi y\cdot t}dt \le A^2<+\infty . \end{aligned}
The main result of the present paper asserts that every function $$f\in H^2(T_{B})$$ is represented by $$Cauchy\text {--}Fantappi\grave{e}$$ formula supported on the boundary $$\partial T_{B}$$. That is, for every $$z\in T_{B}$$ one has that
\begin{aligned} f(z)=\frac{1}{(2\pi )^2}\int \limits _{\mathbb {R}^2\times iS^1} \frac{f(\zeta )\partial \Phi (\zeta , \bar{\zeta })\wedge \bar{\partial } \partial \Phi (\zeta , \bar{\zeta })}{ \langle \nabla \Phi (\zeta ,\bar{\zeta }), \zeta -z\rangle ^2}. \end{aligned}
The absence of Stokes’ theorem for unbounded domain is superseded by a separation of singularities type theorem allowing to express a function $$f\in H^2(T_{B})$$ as a difference of two holomorphic functions $$f_1\in H^2(T_{(S_H^-)^{int}})$$ and $$f_2\in H^2(T_{(S_H^+)^{int}})$$, defined on suitable tubes $$T_{(S_H^-)^{int}}$$ and $$T_{(S_H^+)^{int}}$$, whose base contains a cone, and satisfying $$T_{B}=T_{(S_H^-)^{int}}\cap T_{(S_H^+)^{int}}$$. It is proved that every function $$f_1\in H^2(T_{(S_H^-)^{int}})$$ or $$f_2\in H^2(T_{(S_H^+)^{int}})$$ is representable by $$Cauchy\text {--}Fantappi\grave{e}$$ formula (and conversely). The main result then follows as a direct consequence of a duality argument.

## Keywords

Generalized dual Integral representation

## Mathematics Subject Classification

32A35 32A40 32E99

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