Hartogs's extension theorem

In mathematics, precisely 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 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

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 theory of several complex variables.

For example, in two variables, consider the interior domain

in the two-dimensional polydisk where .

Theorem Hartogs (1906): any holomorphic functions on are analytically continued to . Namely, there is a holomorphic function on such that on .

In fact, using the Cauchy integral formula we obtain the extended function . 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.

Formal statement
Let f be a holomorphic function on a set G \ K, where G is an open subset of Cn (n ≥ 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 on 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 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
  1. See the original paper of Hartogs (1906) and its description in various historical surveys by Osgood (1963, 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 prof 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.

References

Historical references

  • Fuks, B. A. (1963), Introduction to the Theory of Analytic Functions of Several Complex Variables, Translations of Mathematical Monographs, 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. 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

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