Andreotti–Norguet formula explained

The Andreotti–Norguet formula, first introduced by,^[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 .^[4] When considered for functions of complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function:^[5] however, when, 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 :^[7] however, its full proof was only published later in the paper .^[8] Another, different proof of the formula was given by .^[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 and by : the notations used in the original works and in other references, though equivalent, are significantly different.^[11] Precisely, it is assumed that

• is a fixed natural number,

\zeta, z \in \Complex ^{n} are complex vectors, \alpha = (\alpha _{1,} ... , \alpha _{n)} \in N ^{n} is a multiindex whose absolute value is,D \subset \Complex ^{n} is a bounded domain whose closure is,

• is the function space of functions holomorphic on the interior of and continuous on its boundary .
• the iterated Wirtinger derivatives of order of a given complex valued function are expressed using the following simplified notation: $$\backslash partial^\backslash alpha\; f\; =\; \backslash frac.$$

The Andreotti–Norguet kernel

For every multiindex, the Andreotti–Norguet kernel is the following differential form in of bidegree :$$\backslash omega\_\backslash alpha(\backslash zeta,z)\; =\; \backslash frac\backslash sum\_^n\; \backslash frac,$$where

I = ( 1 , ... , 1 ) \in \N ^{n} and$$d\backslash bar\backslash zeta^[j]\; =\; d\backslash bar\backslash zeta\_1^\; \backslash land\; \backslash cdots\; \backslash land\; d\backslash bar\backslash zeta\_^\; \backslash land\; d\backslash bar\backslash zeta\_^\; \backslash land\; \backslash cdots\; \backslash land\; d\backslash bar\backslash zeta\_n^$$

The integral formula

For every function, every point and every multiindex, the following integral representation formula holds$$\backslash partial^\backslash alpha\; f(z)\; =\; \backslash int\_\; f(\backslash zeta)\backslash omega\_\backslash alpha(\backslash zeta,z).$$

See also

• Bergman–Weil formula

References

• , revised translation of the 1990 Russian original.
• .
• .
• .
• , .
• .
• .
• , (ebook).
• . Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday.
• . The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "Professore Linceo".
• , :

Notes and References

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,, and .
4. As remarked in and .
5. As remarked by .
6. See the remarks by and .
7. As correctly stated by and . cites only the later work which, however, contains the full proof of the formula.
8. See .
9. According to,, and, who does not describe his results in this reference, but merely mentions them.
10. See,, the references cited in those sources and the brief remarks by and by : each of these works gives Aizenberg's proof.
11. Compare, for example, the original ones by and those used by, also briefly described in reference .