Tube domain

Definition

Let Rn denote real coordinate space of dimension n and Cn denote complex coordinate space. Then any element of Cn can be decomposed into real and imaginary parts: :a=(z_1,\dots,z_n) = (x_1+iy_1, \dots, x_n+iy_n) = (x_1,\dots,x_n) + i(y_1,\dots,y_n)=x+iy.

Let A be an open subset of Rn. The tube over *A''''', denoted *TA*, is the subset of **C'''n consisting of all elements whose real parts lie in A: :T_A = { z=x+iy\in\mathbb{C}^n\mid x\in A}.

Tubes as domains of holomorphy

Main article: Bochner's tube theorem

Suppose that A is a connected open set. Then any complex-valued function that is holomorphic in a tube T**A can be extended uniquely to a holomorphic function on the convex hull of the tube ch T**A, which is also a tube, and in fact

:\operatorname{ch} , T_A = T_{\operatorname{ch}, A}.

Since any convex open set is a domain of holomorphy (holomorphically convex), a convex tube is also a domain of holomorphy. So the holomorphic envelope of any tube is equal to its convex hull.

Hardy spaces

Let A be an open set in Rn. The Hardy space H p(T**A) is the set of all holomorphic functions F in T**A such that

:\int_{\mathbb{R}^n} |F(x+iy)|^p, dy

for all x in A.

In the special case of p = 2, functions in H2(T**A) can be characterized as follows. Let ƒ be a complex-valued function on Rn satisfying :\sup_{x\in A}\int_{\mathbb{R}^n}|f(t)|^2e^{-4\pi x\cdot t},dt

The Fourier–Laplace transform of ƒ is defined by :F(x+iy) = \int_{\mathbb{R}^n}e^{2\pi z\cdot t}f(t),dt.

Then F is well-defined and belongs to H2(T**A). Conversely, every element of H2(T**A) has this form.

A corollary of this characterization is that H2(T**A) contains a nonzero function if and only if A contains no straight line.

Tubes over cones

Let A be an open convex cone in Rn. This means that A is an open convex set such that, whenever x lies in A, so does the entire ray from the origin to x. Symbolically, :x\in A \implies tx\in A\ \ \ \text{for all}\ t0.

If A is a cone, then the elements of H2(T**A) have L2 boundary limits in the sense that :\lim_{y\to 0} F(x+iy) exists in L2(B). There is an analogous result for H**p(T**A), but it requires additional regularity of the cone (specifically, the dual cone A* needs to have nonempty interior).

Notes

Citations

Sources

• {{Citation| contribution = Tube domain | orig-year = First published 1994 | contribution-url = https://encyclopediaofmath.org/index.php?title=Tube_domain
• {{citation| title = Holography and the future tube | author-link = Gary Gibbons
• {{citation| title = Introduction to complex analysis in several variables | author-link = Lars Hörmander
• {{citation| title = Introduction to Fourier Analysis on Euclidean Spaces | author1-link = Elias M. Stein | author2-link = Guido Weiss
• .