Bergman-Weil representation

Bergman–Weil formula, Weil formula

An integral representation of holomorphic functions, obtained by S. Bergman [1] and A. Weil [2] and defined as follows. Let $ D $ be a domain of holomorphy in $ \mathbf C ^ {n} $, let the functions $ W _ {1} \dots W _ {j} $ be holomorphic in $ D $ and let $ V = \{ {z \in D } : {| W _ {k} (z) | < 1, k = 1 \dots N } \} $ compactly belong to $ D $. It is then possible to represent any function $ f $ holomorphic in $ V $ and continuous on $ \overline{V}\; $ at any point $ z \in V $ by the formula:

$$ \tag{* } f(z) = { \frac{1}{(2 \pi i) ^ {n} } } \sum \int\limits \frac{f ( \zeta ) \mathop{\rm det} (P _ {ij _ {k} } ) }{\prod _ { k=1 } ^ { n } (W _ {j _ {k} } ( \zeta ) - W _ {j _ {k} } (z)) } \ d \zeta , $$

where the summation is performed over all $ j _ {1} < \dots < j _ {n} $, while the integration is carried out over suitably-oriented $ n $- dimensional surfaces $ \sigma _ {j _ {1} \dots j _ {n} } $, forming the skeleton of the domain $ V $( cf. Analytic polyhedron), $ d \zeta = d \zeta _ {1} \wedge \dots \wedge d \zeta _ {n} $. Here the functions $ P _ {ij } ( \zeta , z) $ are holomorphic in the domain $ D \times D $ and are defined, in accordance with Hefer's lemma [3], by the equations

$$ W _ {j} ( \zeta ) - W _ {j} (z) = \ \sum _ { i=1 } ^ { n } ( \zeta _ {i} - z _ {i} ) P _ {ij} ( \zeta , z). $$

The integral representation (*) is called the Bergman–Weil representation.

The domains $ V $ appearing in the Bergman–Weil representation are called Weil domains; an additional condition must usually be imposed, viz. that the ranks of the matrices $ ( \partial W _ {j _ \nu } / \partial z _ \mu ) $, $ \nu = 1 \dots k $, $ \mu = 1 \dots n $, $ k \leq n $, on the corresponding sets

$$ \{ {z \in \overline{V}\; } : {| W _ {j _ {1} } | = \dots = | W _ {j _ {k} } | = 1 } \} $$

are maximal $ (=k) $ for all $ j _ {1} < \dots < j _ {k} $( such Weil domains are called regular). The Weil domains in the Bergman–Weil representations may be replaced by analytic polyhedra $ U $ compactly belonging to D,

$$ U = \{ {z \in D } : {W _ {j} (z) \in D _ {j} ,\ j =1 \dots N } \} , $$

where the $ D _ {j} $ are bounded domains with piecewise-smooth boundaries $ \partial D _ {j} $ in the plane $ \mathbf C $. The Bergman–Weil representation defines the value of a holomorphic function $ f $ inside the analytic polyhedron $ U $ from the values of $ f $ on the skeleton $ \sigma $; for $ n > 1 $ the dimension of $ \sigma $ is strictly lower than that of $ \partial U $. If $ n = 1 $, analytic polyhedra become degenerate in a domain with piecewise-smooth boundary, the skeleton and the boundary become identical, and if, moreover, $ N = 1 $ and $ W(z) = z $, then the Bergman–Weil representation becomes identical with Cauchy's integral formula.

An important property of the Bergman–Weil representation is that its kernel is holomorphic in $ z $. Accordingly, if the holomorphic function $ f $ is replaced by an arbitrary function which is integrable over $ \sigma $, then the right-hand side of the Weil representation gives a function which is holomorphic everywhere in $ U $ and almost-everywhere in $ D \setminus \partial U $; such functions are called integrals of Bergman–Weil type. If $ f $ is holomorphic in $ U $ and continuous on $ \overline{U}\; $, then its integral of Bergman–Weil type is zero almost-everywhere on $ D \setminus \overline{U}\; $.

Bergman–Weil representations in a Weil domain $ V $ yield, after the substitution

$$ (W _ {j _ {k} } ( \zeta ) - W _ {j _ {k} } (z)) ^ {-1} = \ \sum _ { v=0 } ^ \infty \frac{W _ {j _ {k} } ^ { v } (z) }{W _ {j _ {k} } ^ { v+1 } ( \zeta ) } $$

the Weil decomposition

$$ f (z) = $$

$$ = \ \sum _ {s _ {k} \geq 0 } \sum _ {j _ {1} < \dots < j _ {k} } Q _ {j _ {1} \dots j _ {n} s _ {1} \dots s _ {n} } (z) (W _ {j _ {1} } ^ { s _ {1} } (z) \dots W _ {j _ {k} } ^ { s _ {k} } (z)) $$

into a series of functions, holomorphic in $ D $, and this series is uniformly convergent on compact subsets of $ V $.

References

[1]	S.B. Bergman, Mat. Sb. , 1 (43) (1936) pp. 242–257
[2]	A. Weil, "L'intégrale de Cauchy et les fonctions de plusieurs variables" Math. Ann. , 111 (1935) pp. 178–182
[3]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

References

[a1]	G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1983)