Hartogs's extension theorem: Difference between revisions

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{{Short description | Singularities of holomorphic functions extend infinitely outward}}

In mathematics, precisely in the theory of functions of [[several complex variables]], '''Hartogs's extension theorem''' is a statement about the [[Singularity (mathematics)|singularities]] of [[holomorphic function]]s of several variables. Informally, it states that the [[Support (mathematics)|support]] of the singularities of such functions cannot be [[compact space|compact]], therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an [[isolated singularity]] is always a [[removable singularity]] for any [[analytic function]] of {{math|''n'' > 1}} complex variables. A first version of this theorem was proved by [[Friedrich Hartogs]],<ref name="hartogs">See the original paper of {{Harvtxt|Hartogs|1906}} and its description in various historical surveys by {{harvtxt|Osgood|1963|pp=56–59}}, {{harvtxt|Severi|1958|pp=111–115}} and {{harvtxt|Struppa|1988|pp=132–134}}. In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of {{harv|Hartogs|1906}}, and as the reader shall soon see, the key tool in the proof is the [[Cauchy integral formula]]''".</ref> and as such it is known also as '''Hartogs's lemma''' and '''Hartogs's principle''': in earlier [[Soviet Union|Soviet]] literature,<ref>See for example {{harvtxt|Vladimirov|1966|p=153}}, which refers the reader to the book of {{harvtxt|Fuks|1963|p=284}} for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).</ref> it is also called '''Osgood–Brown theorem''', acknowledging later work by [[Arthur Barton Brown]] and [[William Fogg Osgood]].<ref>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref> This property of holomorphic functions of several variables is also called '''[[#Hartogs's phenomenon|Hartogs's phenomenon]]''': however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of [[System of equations|systems]] of [[partial differential equation|partial differential]] or [[convolution operator|convolution equation]]s satisfying Hartogs type theorems.<ref>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>

{{Redirect|Hartogs' lemma|the lemma on infinite ordinals|Hartogs number}}

In mathematics, precisely in the theory of functions of [[Function of several complex variables|several complex variables]], '''Hartogs's extension theorem''' is a statement about the [[Singularity (mathematics)|singularities]] of [[holomorphic function]]s of several variables. Informally, it states that the [[Support (mathematics)|support]] of the singularities of such functions cannot be [[compact space|compact]], therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an [[isolated singularity]] is always a [[removable singularity]] for any [[analytic function]] of {{math|''n'' > 1}} complex variables. A first version of this theorem was proved by [[Friedrich Hartogs]],<ref name="hartogs">See the original paper of {{Harvtxt|Hartogs|1906}} and its description in various historical surveys by {{harvtxt|Osgood|19631966|pp=56–59}}, {{harvtxt|Severi|1958|pp=111–115}} and {{harvtxt|Struppa|1988|pp=132–134}}. In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of {{harv|Hartogs|1906}}, and as the reader shall soon see, the key tool in the proof is the [[Cauchy integral formula]]''".</ref> and as such it is known also as '''Hartogs's lemma''' and '''Hartogs's principle''': in earlier [[Soviet Union|Soviet]] literature,<ref>See for example {{harvtxt|Vladimirov|1966|p=153}}, which refers the reader to the book of {{harvtxt|Fuks|1963|p=284}} for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).</ref> it is also called the '''Osgood–Brown theorem''', acknowledging later work by [[Arthur Barton Brown]] and [[William Fogg Osgood]].<ref>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref> This property of holomorphic functions of several variables is also called '''[[#Hartogs's phenomenon|Hartogs's phenomenon]]''': however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of [[System of equations|systems]] of [[partial differential equation|partial differential]] or [[convolution operator|convolution equation]]s satisfying Hartogs -type theorems.<ref>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>

==Historical note==

The original proof was given by [[Friedrich Hartogs]] in 1906, using [[Cauchy's integral formula]] for [[functions of [[several complex variables]].<ref name="hartogs"/> Today, usual proofs rely on either the [[Bochner–Martinelli–Koppelman formula]] or the solution of the inhomogeneous [[Cauchy–Riemann equations]] with compact support. The latter approach is due to [[Leon Ehrenpreis]] who initiated it in the paper {{Harv|Ehrenpreis|1961}}. Yet another very simple proof of this result was given by [[Gaetano Fichera]] in the paper {{Harv|Fichera|1957}}, by using his solution of the [[Dirichlet problem]] for [[holomorphic function]]s of several variables and the related concept of [[CR-function]]:<ref>Fichera's profproof as well as his epoch making paper {{Harv|Fichera|1957}} seem to have been overlooked by many specialists of the [[SeveralFunction of several complex variables|theory of functions of several complex variables]]: see {{Harvtxt|Range|2002}} for the correct attribution of many important theorems in this field.</ref> later he extended the theorem to a certain class of [[partial differential operator]]s in the paper {{Harv|Fichera|1983}}, and his ideas were later further explored by Giuliano Bratti.<ref>See {{Harvtxt|Bratti|1986a}} {{Harv|Bratti|1986b}}.</ref> Also the Japanese school of the theory of [[partial differential operator]]s worked much on this topic, with notable contributions by Akira Kaneko.<ref>See his paper {{Harv|Kaneko|1973}} and the references therein.</ref> Their approach is to use [[Ehrenpreis's fundamental principle]].

==Hartogs's phenomenon==

A phenomenon that holds in several variables but does not hold in one variable is called '''Hartogs's phenomenon''', which lead to the notion of this Hartogs's extension theorem and the [[___domain of holomorphy]], hence the [[Several complex variables|theory of several complex variables]].

For example, in two variables, consider the interior ___domain

:<math>H_\varepsilon = \{z=(z_1,z_2)\in\Delta^2:|z_1|<\varepsilon\ \ \text{or}\ \ 1-\varepsilon< |z_2|\}</math>

in the two-dimensional polydisk <math>\Delta^2=\{z\in\mathbb{C}^2;|z_1|<1,|z_2|<1\}</math> where <math>0 < \varepsilon < 1.</math> .

'''Theorem''' {{harvtxt|Hartogs|1906}}: anyAny holomorphic functionsfunction <math>f</math> on <math>H_\varepsilon</math> arecan be analytically continued to <math>\Delta^2.</math> . Namely, there is a holomorphic function <math>F</math> on <math>\Delta^2</math> such that <math>F=f</math> on <math>H_\varepsilon.</math> .

ASuch a phenomenon that holds in several variables but does not hold in one variable is called '''Hartogs's phenomenon''', which lead to the notion of this Hartogs's extension theorem and the [[___domain of holomorphy]], hence the [[Several complex variables|theory of several complex variables]].

==Formal statement and proof==

:Let {{mvar|f}} be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G''&nbsp;\&nbsp;''K''}}, where {{mvar|G}} is an open subset of {{math|'''C'''^''n''}} ({{math|''n'' ≥ 2}}) and {{mvar|K}} is a compact subset of {{mvar|G}}. If the [[Complement (set theory)|complement]] {{math|''G''&nbsp;\&nbsp;''K''}} is connected, then {{mvar|f}} can be extended to a unique holomorphic function {{mvar|F}} on {{mvar|G}}.{{sfnm|1a1=Hörmander|1y=1990|1loc=Theorem 2.3.2}}

Ehrenpreis' proof is based on the existence of smooth [[bump function]]s, unique continuation of holomorphic functions, and the [[Poincaré lemma]] &mdash; the last in the form that for any smooth and compactly supported differential (0,1)-form {{mvar|&omega;}} on {{math|'''C'''^''n''}} with {{math|{{overline|∂}}''&omega;'' {{=}} 0}}, there exists a smooth and compactly supported function {{mvar|&eta;}} on {{math|'''C'''^''n''}} with {{math|{{overline|∂}}''&eta;'' {{=}} ''&omega;''}}. The crucial assumption {{math|''n'' ≥ 2}} is required for the validity of this Poincaré lemma; if {{math|''n'' {{=}} 1}} then it is generally impossible for {{mvar|&eta;}} to be compactly supported.{{sfnm|1a1=Hörmander|1y=1990|1p=30}}

The [[ansatz]] for {{mvar|F}} is {{math|''&phi; f'' − ''v''}} for smooth functions {{mvar|&phi;}} and {{mvar|v}} on {{mvar|G}}; such an expression is meaningful provided that {{mvar|&phi;}} is identically equal to zero where {{mvar|f}} is undefined (namely on {{mvar|K}}). Furthermore, given any holomorphic function on {{mvar|G}} which is equal to {{mvar|f}} on ''some'' [[open set]], unique continuation (based on connectedness of {{math|''G''&nbsp;\&nbsp;''K''}}) shows that it is equal to {{mvar|f}} on ''all'' of {{math|''G''&nbsp;\&nbsp;''K''}}.

'''Theorem''' {{harvtxt|Hartogs|1906}}: any holomorphic functions <math>f</math> on <math>H_\varepsilon</math> are analytically continued to <math>\Delta^2</math> . Namely, there is a holomorphic function <math>F</math> on <math>\Delta^2</math> such that <math>F=f</math> on <math>H_\varepsilon</math> .

The holomorphicity of this function is identical to the condition {{math|{{overline|∂}}''v'' {{=}} ''f'' {{overline|∂}}''&phi;''}}. For any smooth function {{mvar|&phi;}}, the differential (0,1)-form {{math|''f'' {{overline|∂}}''&phi;''}} is {{math|{{overline|∂}}}}-closed. Choosing {{mvar|&phi;}} to be a [[Smoothness|smooth function]] which is identically equal to zero on {{mvar|K}} and identically equal to one on the complement of some compact subset {{mvar|L}} of {{mvar|G}}, this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate {{mvar|v}} of compact support. This defines {{mvar|F}} as a holomorphic function on {{mvar|G}}; it only remains to show (following the above comments) that it coincides with {{mvar|f}} on some open set.

In fact, using the [[Cauchy integral formula]] we obtain the extended function <math>F</math> . All holomorphic functions are analytically continued to the polydisk, which is strictly larger than the ___domain on which the original holomorphic function is defined. Such phenomena never happen in the case of one variable.

On the set {{math|'''C'''^''n''&nbsp;\&nbsp;''L''}}, {{mvar|v}} is holomorphic since {{mvar|&phi;}} is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of {{math|''G''&nbsp;\&nbsp;''L''}}.<ref>Any connected component of {{math|'''C'''^''n''&nbsp;\&nbsp;''L''}} must intersect {{math|''G''&nbsp;\&nbsp;''L''}} in a nonempty open set. To see the nonemptiness, connect an arbitrary point {{mvar|p}} of {{math|'''C'''^''n''&nbsp;\&nbsp;''L''}} to some point of {{mvar|L}} via a line. The intersection of the line with {{math|'''C'''^''n''&nbsp;\&nbsp;''L''}} may have many connected components, but the component containing {{mvar|p}} gives a continuous path from {{mvar|p}} into {{math|''G''&nbsp;\&nbsp;''L''}}.</ref> Thus, on this open subset, {{mvar|F}} equals {{mvar|f}} and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of {{mvar|G}}.

==Formal statement==

:Let {{mvar|f}} be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G''&nbsp;\&nbsp;''K''}}, where {{mvar|G}} is an open subset of {{math|'''C'''^''n''}} ({{math|''n'' ≥ 2}}) and {{mvar|K}} is a compact subset of {{mvar|G}}. If the [[Complement (set theory)|complement]] {{math|''G''&nbsp;\&nbsp;''K''}} is connected, then {{mvar|f}} can be extended to a unique holomorphic function on {{mvar|G}}.

==Counterexamples in dimension one==

The theorem does not hold when {{math|''n'' {{=}} 1}}. To see this, it suffices to consider the function {{math|''f''(''z'') {{=}} ''z''⁻¹}}, which is clearly holomorphic in {{math|'''C'''&nbsp;\&nbsp;{0},}} but cannot be continued as a holomorphic function on the whole of {{math|'''C'''}}. Therefore, the Hartogs's phenomenon is an elementary phenomenon that highlights the difference between the theory of functions of one and several complex variables.

== Notes ==

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==References==

{{refbegin}}

===Historical references===

*{{Citation

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| doi = 10.1007/BF03024609

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*{{Citation

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| mr = 0201669

| zbl = 0125.31904| title-link = Nikolay Bogolyubov

}} ([[Zentralblatt]] review of the original [[Russian language|Russian]] edition). One of the first modern monographs on the theory of [[Function of several complex variables|several complex variables]], being different from other ones of the same period due to the extensive use of [[generalized function]]s.

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*{{Citation

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}}

*{{planetmath reference|idurlname=10242FailureOfHartogsTheoremInOneDimension|title=Failurefailure of Hartogs'Hartogs’ theorem in one dimension (counterexample)}}

*{{PlanetMath|urlname=HartogsTheorem|title=Hartogs' theorem}}

*{{planetmath referencePlanetMath|idurlname=10238ProofOfHartogsTheorem|title=Proof of Hartogs' theorem}}

[[Category:Several complex variables]]