Andreotti–Norguet formula

The Andreotti–Norguet formula, first introduced by Aldo Andreotti and François Norguet (1964, 1966),[1] is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function. Precisely, this formula express the value of the partial derivative of any multiindex order of a holomorphic function of several variables,[2] in any interior point of a given bounded domain, as a hypersurface integral of the values of the function on the boundary of the domain itself. In this respect, it is analogous and generalizes the Bochner–Martinelli formula,[3] reducing to it when the absolute value of the multiindex order of differentiation is $0$ .[4] When considered for functions of $n = 1$ complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function:[5] however, when $n > 1$ , its integral kernel is not obtainable by simple differentiation of the Bochner–Martinelli kernel.[6]

Historical note

The Andreotti–Norguet formula was first published in the research announcement (Andreotti & Norguet 1964, p. 780):[7] however, its full proof was only published later in the paper (Andreotti & Norguet 1966, pp. 207–208).[8] Another, different proof of the formula was given by Martinelli (1975).[9] In 1977 and 1978, Lev Aizenberg gave still another proof and a generalization of the formula based on the Cauchy–Fantappiè–Leray kernel instead on the Bochner–Martinelli kernel.[10]

The Andreotti–Norguet integral representation formula

Notation

The notation adopted in the following description of the integral representation formula is the one used by Kytmanov (1995, p. 9) and by Kytmanov & Myslivets (2010, p. 20): the notations used in the original works and in other references, though equivalent, are significantly different.[11] Precisely, it is assumed that

$n > 1$ is a fixed natural number,
$ζ, z \in ℂ n$ are complex vectors,
$α = (α 1,..., α n) \in ℕ n$ is a multiindex whose absolute value is $| α |$ ,
$D \subset ℂ n$ is a bounded domain whose closure is $D$ ,
$A (D)$ is the function space of functions holomorphic on the interior of $D$ and continuous on its boundary $\partialD$ .
the iterated Wirtinger derivatives of order $α$ of a given complex valued function $f \in A (D)$ are expressed using the following simplified notation:

\partial ^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial z_{1}^{\alpha _{1}}\cdots \partial z_{n}^{\alpha _{n}}}}.

The Andreotti–Norguet kernel

Definition 1. For every multiindex $α$ , the Andreotti–Norguet kernel $ω α (ζ, z)$ is the following differential form in $ζ$ of bidegree $(n, n - 1)$ :

\omega _{\alpha }(\zeta ,z)={\frac {(n-1)!\alpha _{1}!\cdots \alpha _{n}!}{(2\pi i)^{n}}}\sum _{j=1}^{n}{\frac {(-1)^{j-1}({\bar {\zeta }}_{j}-{\overline {z}}_{j})^{\alpha _{j}+1}\,d{\bar {\zeta }}^{\alpha +I}[j]\land d\zeta }{\left(|z_{1}-\zeta _{1}|^{2(\alpha _{1}+1)}+\cdots +|z_{n}-\zeta _{n}|^{2(\alpha _{n}+1)}\right)^{n}}},

where $I = (1,...,1) \in ℕ n$ and

d{\bar {\zeta }}^{\alpha +I}[j]=d{\bar {\zeta }}_{1}^{\alpha _{1}+1}\land \cdots \land d{\bar {\zeta }}_{j-1}^{\alpha _{j+1}+1}\land d{\bar {\zeta }}_{j+1}^{\alpha _{j-1}+1}\land \cdots \land d{\bar {\zeta }}_{n}^{\alpha _{n}+1}

The integral formula

Theorem 1 (Andreotti and Norguet). For every function $f \in A (D)$ , every point $z \in D$ and every multiindex $α$ , the following integral representation formula holds

\partial ^{\alpha }f(z)=\int _{\partial D}f(\zeta )\omega _{\alpha }(\zeta ,z).

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Notes

For a brief historical sketch, see the "historical section" of the present entry.
Partial derivatives of a holomorphic function of several complex variables are defined as partial derivatives respect to its complex arguments, i.e. as Wirtinger derivatives.
See (Aizenberg & Yuzhakov 1983, p. 38), Kytmanov (1995, p. 9), Kytmanov & Myslivets (2010, p. 20) and (Martinelli 1984, pp. 152–153).
As remarked in (Kytmanov 1995, p. 9) and (Kytmanov & Myslivets 2010, p. 20).
As remarked by Aizenberg & Yuzhakov (1983, p. 38).
See the remarks by Aizenberg & Yuzhakov (1983, p. 38) and Martinelli (1984, p. 153, footnote (1)).
As correctly stated by Aizenberg & Yuzhakov (1983, p. 250, §5) and Kytmanov (1995, p. 9). Martinelli (1984, p. 153, footnote (1)) cites only the later work (Andreotti & Norguet 1966) which, however, contains the full proof of the formula.
See (Martinelli 1984, p. 153, footnote (1)).
According to Aizenberg & Yuzhakov (1983, p. 250, §5), Kytmanov (1995, p. 9), Kytmanov & Myslivets (2010, p. 20) and Martinelli (1984, p. 153, footnote (1)), who does not describe his results in this reference, but merely mentions them.
See (Aizenberg 1993, p.289, §13), (Aizenberg & Yuzhakov 1983, p. 250, §5), the references cited in those sources and the brief remarks by Kytmanov (1995, p. 9) and by Kytmanov & Myslivets (2010, p. 20): each of these works gives Aizenberg's proof.
Compare, for example, the original ones by Andreotti and Norguet (1964, p. 780, 1966, pp. 207–208) and those used by Aizenberg & Yuzhakov (1983, p. 38), also briefly described in reference (Aizenberg 1993, p. 58).

References

Aizenberg, Lev (1993) [1990], Carleman's Formulas in Complex Analysis. Theory and applications, Mathematics and Its Applications, 244 (2nd ed.), Dordrecht–Boston–London: Kluwer Academic Publishers, pp. xx+299, doi:10.1007/978-94-011-1596-4, ISBN 0-7923-2121-9, MR 1256735, Zbl 0783.32002, revised translation of the 1990 Russian original.
Aizenberg, L. A.; Yuzhakov, A. P. (1983) [1979], Integral Representations and Residues in Multidimensional Complex Analysis, Translations of Mathematical Monographs, 58, Providence R.I.: American Mathematical Society, pp. x+283, ISBN 0-8218-4511-X, MR 0735793, Zbl 0537.32002.
Andreotti, Aldo; Norguet, François (20 January 1964), "Problème de Levi pour les classes de cohomologie" [The Levi problem for cohomology classes], Comptes rendus hebdomadaires des séances de l'Académie des Sciences (in French), 258 (Première partie): 778–781, MR 0159960, Zbl 0124.38803.
Andreotti, Aldo; Norguet, François (1966), "Problème de Levi et convexité holomorphe pour les classes de cohomologie" [The Levi problem and holomorphic convexity for cohomology classes], Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie III (in French), 20 (2): 197–241, MR 0199439, Zbl 0154.33504.
Berenstein, Carlos A.; Gay, Roger; Vidras, Alekos; Yger, Alain (1993), Residue currents and Bezout identities, Progress in Mathematics, 114, Basel–Berlin–Boston: Birkhäuser Verlag, pp. xi+158, doi:10.1007/978-3-0348-8560-7, ISBN 3-7643-2945-9, MR 1249478, Zbl 0802.32001 ISBN 0-8176-2945-9, ISBN 978-3-0348-8560-7.
Kytmanov, Alexander M. (1995) [1992], The Bochner–Martinelli integral and its applications, Birkhäuser Verlag, pp. xii+305, ISBN 978-3-7643-5240-0, MR 1409816, Zbl 0834.32001.
Kytmanov, Alexander M.; Myslivets, Simona G. (2010), Интегральные представления и их приложения в многомерном комплексном анализе [Integral representations and their application in multidimensional complex analysis], Красноярск: СФУ, p. 389, ISBN 978-5-7638-1990-8, archived from the original on 2014-03-23.
Kytmanov, Alexander M.; Myslivets, Simona G. (2015), Multidimensional integral representations. Problems of analytic continuation, Cham–Heidelberg–New York–Dordrecht–London: Springer Verlag, pp. xiii+225, doi:10.1007/978-3-319-21659-1, ISBN 978-3-319-21658-4, MR 3381727, Zbl 1341.32001, ISBN 978-3-319-21659-1 (ebook).
Martinelli, Enzo (1975), "Sopra una formula di Andreotti–Norguet" [On a formula of Andreotti–Norguet], Bollettino dell'Unione Matematica Italiana, IV Serie (in Italian), 11 (3, Supplemento): 455–457, MR 0390270, Zbl 0317.32006. Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday.
Martinelli, Enzo (1984), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali [Elementary introduction to the theory of functions of complex variables with particular regard to integral representations], Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II, archived from the original on 2011-09-27, retrieved 2014-03-22. The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "Professore Linceo".

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[1] For a brief historical sketch, see the "historical section" of the present entry.

[2] Partial derivatives of a holomorphic function of several complex variables are defined as partial derivatives respect to its complex arguments, i.e. as Wirtinger derivatives.

[3] See (Aizenberg & Yuzhakov 1983, p. 38), Kytmanov (1995, p. 9), Kytmanov & Myslivets (2010, p. 20) and (Martinelli 1984, pp. 152–153).

[4] As remarked in (Kytmanov 1995, p. 9) and (Kytmanov & Myslivets 2010, p. 20).

[5] As remarked by Aizenberg & Yuzhakov (1983, p. 38).

[6] See the remarks by Aizenberg & Yuzhakov (1983, p. 38) and Martinelli (1984, p. 153, footnote (1)).

[7] As correctly stated by Aizenberg & Yuzhakov (1983, p. 250, §5) and Kytmanov (1995, p. 9). Martinelli (1984, p. 153, footnote (1)) cites only the later work (Andreotti & Norguet 1966) which, however, contains the full proof of the formula.

[8] See (Martinelli 1984, p. 153, footnote (1)).

[9] According to Aizenberg & Yuzhakov (1983, p. 250, §5), Kytmanov (1995, p. 9), Kytmanov & Myslivets (2010, p. 20) and Martinelli (1984, p. 153, footnote (1)), who does not describe his results in this reference, but merely mentions them.

[10] See (Aizenberg 1993, p.289, §13), (Aizenberg & Yuzhakov 1983, p. 250, §5), the references cited in those sources and the brief remarks by Kytmanov (1995, p. 9) and by Kytmanov & Myslivets (2010, p. 20): each of these works gives Aizenberg's proof.

[11] Compare, for example, the original ones by Andreotti and Norguet (1964, p. 780, 1966, pp. 207–208) and those used by Aizenberg & Yuzhakov (1983, p. 38), also briefly described in reference (Aizenberg 1993, p. 58).