Bochner-Martinelli representation formula

Bochner–Martinelli representation, Bochner–Martinelli formula

An integral representation for holomorphic functions, which is defined as follows [1], [2]. Let the function $ f $ be holomorphic in a domain $ D \subset {\mathbf C ^ {n} } $ with piecewise-smooth boundary $ \partial D $, and let $ f $ be continuous in its closure $ \overline{D}\; $. Then the expression

$$ \tag{1 } \frac{(n-1)! }{(2 \pi i) ^ {n} } \int\limits _ {\partial D } \frac{f( \zeta ) }{| \zeta -z | ^ {2n} } \times $$

$$ \times \sum _ { j=1 } ^ { n } ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} ) \ d \overline \zeta \; _ {1} \wedge d \zeta _ {1} \wedge \dots \wedge \ [d \overline \zeta \; _ {j} ] \wedge d \zeta _ {j} \wedge \dots \wedge d \zeta _ {n\ } = $$

$$ = \ \left \{ \begin{array}{cl} f(z), & \textrm{ if } z \in D, \\ 0, & \textrm{ if } z \notin \overline{D}\; , \\ \end{array} \right .$$

here $ [d \overline \zeta \; _ {j} ] $ means that the term $ d \overline \zeta \; _ {j} $ is to be omitted, is called the Bochner–Martinelli representation. For $ n = 1 $ this representation is identical with the Cauchy integral formula (cf. Cauchy integral), but for $ n > 1 $ its kernel is not holomorphic in $ z $, which is the reason for the limited applicability of the Bochner–Martinelli representation in the theory of functions of several complex variables. The kernel of the Bochner–Martinelli representation is the $ \zeta $- differential form of bidegree $ (n, n - 1) $:

$$ \omega ( \zeta , z ) = \ (n-1)! over {(2 \pi i) ^ {n} } \frac{1}{| \zeta -z | ^ {2n} } \times $$

$$ \times \sum _ { j=1 } ^ { n } ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} ) d \overline \zeta \; _ {1} \wedge d \zeta _ {1} \wedge \dots \wedge [ d \overline \zeta \; _ {j} ] \wedge d \zeta _ {j} \wedge \dots \wedge d \overline \zeta \; _ {n} \wedge d \zeta _ {n} , $$

which is defined in $ \mathbf C ^ {n} $, has a singular point at $ \zeta = z $, and is $ \overline \partial \; $- closed (i.e. $ \overline \partial \; \omega = 0 $) outside the singular point. If $ n > 1 $, the form $ \omega $ is equal to $ \partial \omega ^ \prime ( \zeta , z) $, where

$$ \omega ^ \prime ( \zeta , z) = - (n-2)! over {(2 \pi i) ^ {n} } \cdot \frac{1}{| \zeta -z | ^ {2n-2} } \sum _ { j=1 } ^ { n } \left ( \prod _ { k\neq j } d \overline \zeta \; _ {k} \wedge d \zeta _ {k} \right ) $$

is a form of bidegree $ (n - 1, n - 1) $, the coefficient of which is a fundamental solution of the Laplace equation; here

$$ \partial \phi = \ \sum dz _ {k} \frac{\partial \phi }{\partial z _ {k} } \ \textrm{ and } \ \ \overline \partial \; \phi = \ \sum d \overline{z}\; _ {k} \frac{\partial \phi }{\partial z _ {k} } . $$

The following integral representation, which generalizes formula (1), is the analogue of the Cauchy–Green formula (cf. Cauchy integral): If the function $ f $ is continuously differentiable in the closure of a domain $ D \subset \mathbf C ^ {n} $ with piecewise-smooth boundary $ \partial D $, then, for any point $ z \in D $,

$$ \tag{2 } f(z) = \int\limits _ {\partial D } f( \zeta ) \omega ( \zeta , z) - \int\limits _ { D } \overline \partial \; f( \zeta ) \wedge \omega ( \zeta , z). $$

The function

$$ \widehat{f} (z) = \ \int\limits _ \Gamma f( \zeta ) \omega ( \zeta , z), $$

where $ \Gamma $ is a smooth hypersurface in $ \mathbf R ^ {2n} = \mathbf C ^ {n} $ and $ f $ is a function on $ \Gamma $ which is Lebesgue-integrable, is said to be an integral of Bochner–Martinelli type. As for Cauchy-type integrals, Sokhotskii's formula, with the usual restrictions on $ \Gamma $ and $ f $, is applicable to Bochner–Martinelli-type integrals. A Bochner–Martinelli-type integral is a complex function which is harmonic everywhere outside $ \Gamma $; in the general case this function is holomorphic only for $ n = 1 $. If $ \Gamma = \partial D $, then if $ n \geq 1 $, the condition $ \widehat{f} (z) \equiv 0 $ outside $ \overline{D}\; $ is equivalent to the holomorphy of $ \widehat{f} $ in $ D $.

The Bochner–Martinelli representation is employed to demonstrate other integral representations (e.g. the Bergman–Weil representation), in holomorphic continuation from the boundary, and also in the theory of boundary values of holomorphic functions of several complex variables. It was introduced by S. Bochner [1] and by E. Martinelli [2].

References

[1]	S. Bochner, "Analytic and meromorphic continuation by means of Green's formula" Ann. of Math. (2) , 44 : 4 (1943) pp. 652–673
[2]	E. Martinelli, Rend. Accad. Ital. , 9 (1938) pp. 269–283
[3]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

Sokhotskii's formula is also known as Plemelj's formula. The Cauchy–Green formula is also known as the Cauchy–Pompeiu formula.

The Bochner–Martinelli kernel is a special case of a Cauchy–Fantappié kernel. The integral representation (2) solves the $ \overline \partial \; $- equation:

$$ \overline \partial \; f = \ v \ \textrm{ with } \ \ \overline \partial \; v = 0, $$

for compactly-supported $ (0, 1) $- forms $ v $, by substituting $ v $ for $ \partial f $ in the right-hand side of (2) and omitting the integral over the boundary. When $ v $ is not compactly supported, the boundary integral causes difficulties. These can be solved for strictly pseudo-convex domains and the Bochner–Martinelli kernel then occurs in an explicit solution operator for the $ \overline \partial \; $- equation.

References

[a1]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
[a2]	G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984)
[a3]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3