Hartogs's extension theorem

In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be 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 $n > 1$ complex variables. A first version of this theorem was proved by Friedrich Hartogs,^[1] and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlier Soviet literature,^[2] it is also called the Osgood–Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood.^[3] This property of holomorphic functions of several variables is also called Hartogs's phenomenon: however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs-type theorems.^[4]

Historical note

The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables.^[1] 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 (Ehrenpreis 1961). Yet another very simple proof of this result was given by Gaetano Fichera in the paper (Fichera 1957), by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept of CR-function:^[5] later he extended the theorem to a certain class of partial differential operators in the paper (Fichera 1983), and his ideas were later further explored by Giuliano Bratti.^[6] Also the Japanese school of the theory of partial differential operators worked much on this topic, with notable contributions by Akira Kaneko.^[7] Their approach is to use Ehrenpreis's fundamental principle.

Hartogs's phenomenon

For example, in two variables, consider the interior domain

H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2}:|z_{1}|<\varepsilon \ \ {\text{or}}\ \ 1-\varepsilon <|z_{2}|\}

in the two-dimensional polydisk $\Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}$ where $0<\varepsilon <1.$

Theorem Hartogs (1906): Any holomorphic function $f$ on $H_{\varepsilon }$ can be analytically continued to $\Delta ^{2}.$ Namely, there is a holomorphic function $F$ on $\Delta ^{2}$ such that $F=f$ on $H_{\varepsilon }.$

Such a phenomenon is called Hartogs's phenomenon, which lead to the notion of this Hartogs's extension theorem and the domain of holomorphy.

Formal statement and proof

Let

f

be a holomorphic function on a set

G \ K

, where

G

is an open subset of

C n

(

n \geq 2

) and

K

is a compact subset of

G

. If the complement

G \ K

is connected, then

f

can be extended to a unique holomorphic function

F

on

G

.^[8]

Ehrenpreis' proof is based on the existence of smooth bump functions, unique continuation of holomorphic functions, and the Poincaré lemma — the last in the form that for any smooth and compactly supported differential (0,1)-form $ω$ on $C n$ with $\partial ω = 0$ , there exists a smooth and compactly supported function $η$ on $C n$ with $\partial η = ω$ . The crucial assumption $n \geq 2$ is required for the validity of this Poincaré lemma; if $n = 1$ then it is generally impossible for $η$ to be compactly supported.^[9]

The ansatz for $F$ is $φ f - v$ for smooth functions $φ$ and $v$ on $G$ ; such an expression is meaningful provided that $φ$ is identically equal to zero where $f$ is undefined (namely on $K$ ). Furthermore, given any holomorphic function on $G$ which is equal to $f$ on some open set, unique continuation (based on connectedness of $G \ K$ ) shows that it is equal to $f$ on all of $G \ K$ .

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

On the set $Cn \ L$ , $v$ is holomorphic since $φ$ is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of $G \ L$ .^[10] Thus, on this open subset, $F$ equals $f$ and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of $G$ .

Counterexamples in dimension one

The theorem does not hold when $n = 1$ . To see this, it suffices to consider the function $f (z) = z -1$ , which is clearly holomorphic in $C \ {0},$ but cannot be continued as a holomorphic function on the whole of $C$ . 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

^ ^a ^b See the original paper of Hartogs (1906) and its description in various historical surveys by Osgood (1966, pp. 56–59), Severi (1958, pp. 111–115) and 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 (Hartogs 1906), and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula".
^ See for example Vladimirov (1966, p. 153), which refers the reader to the book of Fuks (1963, p. 284) for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).
^ See Brown (1936) and Osgood (1929).
^ See Fichera (1983) and Bratti (1986a) (Bratti 1986b).
^ Fichera's proof as well as his epoch making paper (Fichera 1957) seem to have been overlooked by many specialists of the theory of functions of several complex variables: see Range (2002) for the correct attribution of many important theorems in this field.
^ See Bratti (1986a) (Bratti 1986b).
^ See his paper (Kaneko 1973) and the references therein.
^ Hörmander 1990, Theorem 2.3.2.
^ Hörmander 1990, p. 30.
^ Any connected component of $Cn \ L$ must intersect $G \ L$ in a nonempty open set. To see the nonemptiness, connect an arbitrary point $p$ of $Cn \ L$ to some point of $L$ via a line. The intersection of the line with $Cn \ L$ may have many connected components, but the component containing $p$ gives a continuous path from $p$ into $G \ L$ .

References

Historical references

Fuks, B. A. (1963), Introduction to the Theory of Analytic Functions of Several Complex Variables, Translations of Mathematical Monographs, vol. 8, Providence, RI: American Mathematical Society, pp. vi+374, ISBN 9780821886441, MR 0168793, Zbl 0138.30902.
Osgood, William Fogg (1966) [1913], Topics in the theory of functions of several complex variables (unabridged and corrected ed.), New York: Dover, pp. IV+120, JFM 45.0661.02, MR 0201668, Zbl 0138.30901.
Range, R. Michael (2002), "Extension phenomena in multidimensional complex analysis: correction of the historical record", The Mathematical Intelligencer, 24 (2): 4–12, doi:10.1007/BF03024609, MR 1907191, S2CID 120531925. A historical paper correcting some inexact historical statements in the theory of holomorphic functions of several variables, particularly concerning contributions of Gaetano Fichera and Francesco Severi.
Severi, Francesco (1931), "Risoluzione del problema generale di Dirichlet per le funzioni biarmoniche", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 6 (in Italian), 13: 795–804, JFM 57.0393.01, Zbl 0002.34202. This is the first paper where a general solution to the Dirichlet problem for pluriharmonic functions is given for general real analytic data on a real analytic hypersurface. A translation of the title reads as:-"Solution of the general Dirichlet problem for biharmonic functions".
Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, Zbl 0094.28002. A translation of the title is:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome". This book consist of lecture notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), and includes appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty.
Struppa, Daniele C. (1988), "The first eighty years of Hartogs' theorem", Seminari di Geometria 1987–1988, Bologna: Università degli Studi di Bologna – Dipartimento di Matematica, pp. 127–209, MR 0973699, Zbl 0657.35018.
Vladimirov, V. S. (1966), Ehrenpreis, L. (ed.), Methods of the theory of functions of several complex variables. With a foreword of N.N. Bogolyubov, Cambridge-London: The M.I.T. Press, pp. XII+353, MR 0201669, Zbl 0125.31904 (Zentralblatt review of the original Russian edition). One of the first modern monographs on the theory of several complex variables, being different from other ones of the same period due to the extensive use of generalized functions.

Scientific references

Bochner, Salomon (October 1943), "Analytic and meromorphic continuation by means of Green's formula", Annals of Mathematics, Second Series, 44 (4): 652–673, doi:10.2307/1969103, JSTOR 1969103, MR 0009206, Zbl 0060.24206.
Bochner, Salomon (March 1, 1952), "Partial Differential Equations and Analytic Continuations", PNAS, 38 (3): 227–230, Bibcode:1952PNAS...38..227B, doi:10.1073/pnas.38.3.227, MR 0050119, PMC 1063536, PMID 16589083, Zbl 0046.09902.
Bratti, Giuliano (1986a), "A proposito di un esempio di Fichera relativo al fenomeno di Hartogs" [About an example of Fichera concerning Hartogs's phenomenon], Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, serie 5 (in Italian and English), X (1): 241–246, MR 0879111, Zbl 0646.35007, archived from the original on 2011-07-26
Bratti, Giuliano (1986b), "Estensione di un teorema di Fichera relativo al fenomeno di Hartogs per sistemi differenziali a coefficenti costanti" [Extension of a theorem of Fichera for systems of P.D.E. with constant coefficients, concerning Hartogs's phenomenon], Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, serie 5 (in Italian and English), X (1): 255–259, MR 0879114, Zbl 0646.35008, archived from the original on 2011-07-26
Bratti, Giuliano (1988), "Su di un teorema di Hartogs" [On a theorem of Hartogs], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 79: 59–70, MR 0964020, Zbl 0657.46033
Brown, Arthur B. (1936), "On certain analytic continuations and analytic homeomorphisms", Duke Mathematical Journal, 2: 20–28, doi:10.1215/S0012-7094-36-00203-X, JFM 62.0396.02, MR 1545903, Zbl 0013.40701
Ehrenpreis, Leon (1961), "A new proof and an extension of Hartog's theorem", Bulletin of the American Mathematical Society, 67 (5): 507–509, doi:10.1090/S0002-9904-1961-10661-7, MR 0131663, Zbl 0099.07801. A fundamental paper in the theory of Hartogs's phenomenon. The typographical error in the title is reproduced as it appears in the original version of the paper.
Fichera, Gaetano (1957), "Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analitica di più variabili complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 8 (in Italian), 22 (6): 706–715, MR 0093597, Zbl 0106.05202. An epoch-making paper in the theory of CR-functions, where the Dirichlet problem for analytic functions of several complex variables is solved for general data. A translation of the title reads as:-"Characterization of the trace, on the boundary of a domain, of an analytic function of several complex variables".
Fichera, Gaetano (1983), "Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali", Rendiconti Dell' Istituto Lombardo di Scienze e Lettere. Scienze Matemàtiche e Applicazioni, Series A. (in Italian), 117: 199–211, MR 0848259, Zbl 0603.35013. An English translation of the title reads as:-"Hartogs phenomenon for certain linear partial differential operators".
Fueter, Rudolf (1939–1940), "Über einen Hartogs'schen Satz" [On a theorem of Hartogs], Commentarii Mathematici Helvetici (in German), 12 (1): 75–80, doi:10.1007/bf01620640, JFM 65.0363.03, S2CID 120266425, Zbl 0022.05802, archived from the original on 2011-10-02, retrieved 2011-01-16. Available at the SEALS Portal Archived 2012-11-10 at the Wayback Machine.
Fueter, Rudolf (1941–1942), "Über einen Hartogs'schen Satz in der Theorie der analytischen Funktionen von $n$ komplexen Variablen" [On a theorem of Hartogs in the theory of analytic functions of $n$ complex variables], Commentarii Mathematici Helvetici (in German), 14 (1): 394–400, doi:10.1007/bf02565627, JFM 68.0175.02, MR 0007445, S2CID 122750611, Zbl 0027.05703, archived from the original on 2011-10-02, retrieved 2011-01-16 (see also Zbl 0060.24505, the cumulative review of several papers by E. Trost). Available at the SEALS Portal Archived 2012-11-10 at the Wayback Machine.
Hartogs, Fritz (1906), "Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.", Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse (in German), 36: 223–242, JFM 37.0443.01.
Hartogs, Fritz (1906a), "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselber durch Reihen welche nach Potentzen einer Veränderlichen fortschreiten", Mathematische Annalen (in German), 62: 1–88, doi:10.1007/BF01448415, JFM 37.0444.01, S2CID 122134517. Available at the DigiZeitschriften.
Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001.
Kaneko, Akira (January 12, 1973), "On continuation of regular solutions of partial differential equations with constant coefficients", Proceedings of the Japan Academy, 49 (1): 17–19, doi:10.3792/pja/1195519488, MR 0412578, Zbl 0265.35008, available at Project Euclid.
Martinelli, Enzo (1942–1943), "Sopra una dimostrazione di R. Fueter per un teorema di Hartogs" [On a proof by R. Fueter of a theorem of Hartogs], Commentarii Mathematici Helvetici (in Italian), 15 (1): 340–349, doi:10.1007/bf02565649, MR 0010729, S2CID 119960691, Zbl 0028.15201, archived from the original on 2011-10-02, retrieved 2011-01-16. Available at the SEALS Portal Archived 2012-11-10 at the Wayback Machine.
Osgood, W. F. (1929), Lehrbuch der Funktionentheorie. II, Teubners Sammlung von Lehrbüchern auf dem Gebiet der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (in German), vol. Bd. XX - 1 (2nd ed.), Leipzig: B. G. Teubner, pp. VIII+307, ISBN 9780828401821, JFM 55.0171.02.
Severi, Francesco (1932), "Una proprietà fondamentale dei campi di olomorfismo di una funzione analitica di una variabile reale e di una variabile complessa", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 6 (in Italian), 15: 487–490, JFM 58.0352.05, Zbl 0004.40702. An English translation of the title reads as:-"A fundamental property of the domain of holomorphy of an analytic function of one real variable and one complex variable".
Severi, Francesco (1942–1943), "A proposito d'un teorema di Hartogs" [About a theorem of Hartogs], Commentarii Mathematici Helvetici (in Italian), 15 (1): 350–352, doi:10.1007/bf02565650, MR 0010730, S2CID 120514642, Zbl 0028.15301, archived from the original on 2011-10-02, retrieved 2011-06-25. Available at the SEALS Portal Archived 2012-11-10 at the Wayback Machine.

External links

Chirka, E. M. (2001) [1994], "Hartogs theorem", Encyclopedia of Mathematics, EMS Press
"failure of Hartogs' theorem in one dimension". PlanetMath.
Hartogs' theorem at PlanetMath.
Proof of Hartogs' theorem at PlanetMath.

[hartogs-1] See the original paper of Hartogs (1906) and its description in various historical surveys by Osgood (1966, pp. 56–59), Severi (1958, pp. 111–115) and 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 (Hartogs 1906), and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula".

[2] See for example Vladimirov (1966, p. 153), which refers the reader to the book of Fuks (1963, p. 284) for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).

[3] See Brown (1936) and Osgood (1929).

[4] See Fichera (1983) and Bratti (1986a) (Bratti 1986b).

[5] Fichera's proof as well as his epoch making paper (Fichera 1957) seem to have been overlooked by many specialists of the theory of functions of several complex variables: see Range (2002) for the correct attribution of many important theorems in this field.

[6] See Bratti (1986a) (Bratti 1986b).

[7] See his paper (Kaneko 1973) and the references therein.

[FOOTNOTEHörmander1990Theorem_2.3.2-8] Hörmander 1990, Theorem 2.3.2.

[FOOTNOTEHörmander199030-9] Hörmander 1990, p. 30.

[10] Any connected component of $Cn \ L$ must intersect $G \ L$ in a nonempty open set. To see the nonemptiness, connect an arbitrary point $p$ of $Cn \ L$ to some point of $L$ via a line. The intersection of the line with $Cn \ L$ may have many connected components, but the component containing $p$ gives a continuous path from $p$ into $G \ L$ .

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