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{{other uses of|tube|topic=mathematics|Tube (disambiguation)#Mathematics}}
In [[mathematics]], a '''tube domain''' is a generalization of the notion of a vertical strip (or [[half-plane]]) in the [[complex plane]] to [[several complex variables]].  A strip can be thought of as the collection of complex numbers whose [[real part]] lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained.

Tube domains are [[domain of a function|domains]] of the [[Laplace transform]] of a function of several [[real number|real]] variables (see [[multidimensional Laplace transform]]).  [[Hardy space]]s on tubes can be defined in a manner in which a version of the [[Paley–Wiener theorem]] from one variable continues to hold, and characterizes the elements of Hardy spaces as the Laplace transforms of functions with appropriate integrability properties.  Tubes over [[convex set]]s are [[domain of holomorphy|domains of holomorphy]].  The Hardy spaces on tubes over convex [[Monge cone|cones]] have an especially rich structure, so that precise results are known concerning the boundary values of ''H''<sup>''p''</sup> functions.  In mathematical physics, the [[future tube]] is the tube domain associated to the interior of the past [[null cone]] in [[Minkowski space]], and has applications in [[relativity theory]] and [[quantum gravity]].{{sfn|Gibbons|2000}} Certain tubes over cones support a [[Bergman metric]] in terms of which they become [[bounded symmetric domain]]s.  One of these is the [[Siegel half-space]] which is fundamental in [[arithmetic]].

==Definition==
Let '''R'''<sup>''n''</sup> denote [[real coordinate space]] of dimension ''n'' and '''C'''<sup>''n''</sup> denote [[complex number|complex]] coordinate space.  Then any element of '''C'''<sup>''n''</sup> can be decomposed into real and imaginary parts:
:<math>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.</math>

Let ''A'' be an [[open set|open]] subset of '''R'''<sup>''n''</sup>.  The '''tube over ''A''''', denoted ''T''<sub>''A''</sub>, is the subset of '''C'''<sup>''n''</sup> consisting of all elements whose real parts lie in ''A'':{{sfn|Hörmander|1990}}{{efn|Some conventions instead define a tube to be a domain such that the imaginary part lies in ''A'' {{harv|Stein|Weiss|1971}}.}}
:<math>T_A = \{ z=x+iy\in\mathbb{C}^n\mid x\in A\}.</math>

==Tubes as domains of holomorphy==
{{Main|Bochner's tube theorem}}
Suppose that ''A'' is a connected open set.  Then any complex-valued function that is [[Holomorphic function|holomorphic]] in a tube ''T''<sub>''A''</sub> can be extended uniquely to a holomorphic function on the [[convex hull]] of the tube {{nowrap|ch ''T''<sub>''A''</sub>}},{{sfn|Hörmander|1990}} which is also a tube, and in fact

:<math>\operatorname{ch} \, T_A = T_{\operatorname{ch}\, A}.</math>

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.{{sfn|Chirka|2001}}{{sfn|Carmignani|1973}}

==Hardy spaces==
Let ''A'' be an [[open set]] in '''R'''<sup>''n''</sup>.  The [[Hardy space]] ''H''<sup>&nbsp;''p''</sup>(''T''<sub>''A''</sub>) is the set of all [[holomorphic function]]s ''F'' in ''T''<sub>''A''</sub> such that

:<math>\int_{\mathbb{R}^n} |F(x+iy)|^p\, dy < \infty</math>

for all ''x'' in ''A''.

In the special case of ''p''&nbsp;=&nbsp;2, functions in ''H''<sup>2</sup>(''T''<sub>''A''</sub>) can be characterized as follows.{{sfn|Stein|Weiss|1971}} Let ''&fnof;'' be a complex-valued function on '''R'''<sup>''n''</sup> satisfying
:<math>\sup_{x\in A}\int_{\mathbb{R}^n}|f(t)|^2e^{-4\pi x\cdot t}\,dt < \infty.</math>

The Fourier–Laplace transform of ''&fnof;'' is defined by
:<math>F(x+iy) = \int_{\mathbb{R}^n}e^{2\pi z\cdot t}f(t)\,dt.</math>

Then ''F'' is well-defined and belongs to ''H''<sup>2</sup>(''T''<sub>''A''</sub>).  Conversely, every element of ''H''<sup>2</sup>(''T''<sub>''A''</sub>) has this form.

A corollary of this characterization is that ''H''<sup>2</sup>(''T''<sub>''A''</sub>) contains a nonzero function if and only if ''A'' contains no straight line.

===Tubes over cones===
Let ''A'' be an open convex cone in '''R'''<sup>''n''</sup>.  This means that ''A'' is an [[open set|open]] [[convex set]] such that, whenever ''x'' lies in ''A'', so does the entire ray from the origin to ''x''.  Symbolically,
:<math>x\in A \implies tx\in A\ \ \ \text{for all}\ t>0.</math>

If ''A'' is a cone, then the elements of ''H''<sub>2</sub>(''T''<sub>''A''</sub>) have ''L''<sup>2</sup> boundary limits in the sense that{{sfn|Stein|Weiss|1971}}
:<math>\lim_{y\to 0} F(x+iy)</math>
exists in ''L''<sup>2</sup>(''B'').  There is an analogous result for ''H''<sup>''p''</sup>(''T''<sub>''A''</sub>), but it requires additional regularity of the cone (specifically, the [[dual cone]] ''A''* needs to have nonempty interior).

==See also==
* [[Reinhardt domain]]
* [[Siegel domain]]

==Notes==
{{notelist}}

===Citations===
{{Reflist|22em}}

==Sources==
{{refbegin}}
*{{Citation| contribution = Tube domain
 | last = Chirka | first = E.M. | year = 2001
 | title = [[Encyclopedia of Mathematics]]
 | orig-year = First published 1994
 | publisher = [[European Mathematical Society|EMS Press]]
 | contribution-url = https://encyclopediaofmath.org/index.php?title=Tube_domain
}}.
*{{citation| title = Holography and the future tube
 | last = Gibbons | first = G.W. | year = 2000
 | author-link = Gary Gibbons
 | journal = Classical and Quantum Gravity
 | volume = 17 | issue = 5 | pages = 1071–1079
 | arxiv = hep-th/9911027 | doi = 10.1088/0264-9381/17/5/316
| bibcode = 2000CQGra..17.1071G | s2cid = 14045117 }}.
*{{citation| title = Introduction to complex analysis in several variables
 | last = Hörmander | first = Lars | year = 1990
 | author-link = Lars Hörmander
 | publisher = North-Holland | location = New York
 | isbn = 0-444-88446-7
}}.
*{{citation| title = Introduction to Fourier Analysis on Euclidean Spaces
 | last1 = Stein | first1 = Elias
 | last2 = Weiss | first2 = Guido
 | author1-link = Elias M. Stein
 | author2-link = Guido Weiss
 | year = 1971
 | publisher = Princeton University Press | location = Princeton, N.J.
 | url = https://archive.org/details/introductiontofo0000stei | url-access = registration | via = [[Internet Archive]]
 | isbn = 978-0-691-08078-9
}}.
*{{cite journal |doi=10.1090/S0002-9947-1973-0316748-1|jstor=1996512|title=Envelopes of Holomorphy and Holomorphic Convexity|last1=Carmignani|first1=Robert|journal=Transactions of the American Mathematical Society|year=1973|volume=179|pages=415–431|hdl=1911/14576|hdl-access=free}}.
{{refend}}

[[Category:Fourier analysis]]
[[Category:Harmonic analysis]]
[[Category:Several complex variables]]