Complex Analysis and Operator Theory

, Volume 13, Issue 2, pp 431–478 | Cite as

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

  • N. Alexandrou
  • A. VidrasEmail author


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The absence of Stokes’ theorem for unbounded domain is superseded by a separation of singularities type theorem allowing to express a function \(f\in H^2(T_{B})\) as a difference of two holomorphic functions \(f_1\in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.


Generalized dual Integral representation 

Mathematics Subject Classification

32A35 32A40 32E99 


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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