Several complex variables and analytic spaces Extension phenomena and analytic continuation Theorem

Edge-of-the-wedge theorem

Builds on Holomorphic function of several variables Convex cone Open set

Statement

Let $C \subseteq \R^n$ be a nonempty open convex cone with vertex $0$ , let $E \subseteq \R^n$ be a nonempty open set, and let $\delta > 0$ . Write $\R^n \subseteq \C^n$ as the set of points with zero imaginary part, and define the two wedges $W_\pm = \set{x + iy \in \C^n : x \in E, \ \pm y \in C, \ \abs{y} < \delta}.$ Let $f_+$ and $f_-$ be holomorphic functions on $W_+$ and $W_-$ . Suppose that $f_+$ extends continuously to $W_+ \cup E$ , that $f_-$ extends continuously to $W_- \cup E$ , and that the two extensions agree on $E$ . Then there are an open set $G \subseteq \C^n$ with $E \subseteq G$ and a holomorphic function $F$ on $G$ such that $F = f_+$ on $G \cap W_+$ and $F = f_-$ on $G \cap W_-$ .

Remarks

For $n = 1$ the wedges are a rectangle above and a rectangle below the interval $E$ . Then the theorem follows from Morera's theorem: the glued function is continuous, and its integrals over small triangles vanish. In several variables the edge $E$ has only real dimension $n$ inside $\C^n$ , and a direct Morera argument is not available. Proofs use Cauchy integrals over suitable families of analytic discs, or Fourier analysis. The theorem also holds when the continuity hypothesis is replaced by the condition that $f_+$ and $f_-$ have the same boundary value on $E$ in the sense of distributions.

The agreement on $E$ is necessary: $f_+ = 0$ and $f_- = 1$ satisfy all other hypotheses but cannot be glued to one holomorphic function.

Going deeper

For a student

In one variable this is the familiar gluing of holomorphic functions across a real interval. In several variables it is surprising, because the edge has only half the real dimension of the space, yet it still suffices for gluing. The theorem arose in quantum field theory, where it is used to prove analyticity properties of correlation functions. It is also a basic tool in the theory of hyperfunctions, where generalised functions on $\R^n$ are represented by boundary values of holomorphic functions on wedges.

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Convex cone concept
Holomorphic function of several variables concept
Open set concept
Edge-of-the-wedge theorem statement

Tag	`ATZEZ428CY`
URI	`m2://ATZEZ428CY`
Kind	statement, theorem
Writing	W0 stub
MSC	32D15, 32A40
Status	This copy of the site has no build data. Run `m2 status ATZEZ428CY`.
About	Holomorphic function of several variables
Uses	Holomorphic function of several variables, Convex cone, Open set
Same as	Wikidata Q4454925 Wikipedia Edge-of-the-wedge theorem

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