Bergman kernel function

Bergman kernel

A function of complex variables with the reproducing kernel property, defined for any domain $ D \subset \mathbf C ^ {n} $ in which there exist holomorphic functions $ f \neq 0 $ of class $ L _ {2} (D) $ with respect to the Lebesgue measure $ dV $. The function was introduced by S. Bergman [1]. The set of these functions $ f $ forms the Hilbert space $ L _ {2,h} (D) \subset L _ {2} (D) $ with orthonormal basis $ \{ \phi _ {1} , \phi _ {2} ,\dots \} $; $ L _ {2,h} (D) = L _ {2} (D) \cap O(D) $, where $ O(D) $ is the space of holomorphic functions. The function

$$ K _ {D} (z, \zeta ) = \ K (z, \zeta ) = \ \sum _ { j=1 } ^ \infty \phi _ {j} (z) \overline{ {\phi _ {j} ( \zeta ) }}\; , $$

$$ z = (z _ {1} \dots z _ {n} ),\ \zeta = ( \zeta _ {1} \dots \zeta _ {n} ), $$

is called the Bergman kernel function (or simply the kernel function) of $ D $. The series on the right-hand side converges uniformly on compact subsets of $ D $, and belongs to $ L _ {2,h} (D) $ for each given $ \zeta \in D $, the sum does not depend on the choice of the orthonormal basis $ \{ \phi _ {j} \} $. The Bergman kernel function depends on $ 2n $ complex variables, and is defined in the domain $ D \times D \subset \mathbf C ^ {2n } $; it has the symmetry property $ K ( \zeta , z ) = {K(z, \zeta ) } bar $, it is holomorphic with respect to the variable $ z $ and anti-holomorphic with respect to $ \zeta $. If $ D = D ^ \prime \times D ^ {\prime\prime} $, $ D ^ \prime \subset \mathbf C ^ {m} $, $ D ^ {\prime\prime} \subset \mathbf C ^ {n - m } $, then

$$ K _ {D} (z, \zeta ) = \ K _ {D ^ \prime } (z ^ \prime , \zeta ^ \prime ) K _ {D ^ {\prime\prime} } (z ^ {\prime\prime} , \zeta ^ \prime ), $$

where $ z ^ \prime = ( z _ {1} \dots z _ {m} ), z ^ {\prime\prime } = ( z _ {m+1 } \dots z _ {n} ) $.

The most important characteristic of the Bergman kernel function is its reproducing property: For any function $ f \in L _ {2,h} (D) $ and for any point $ z \in D $ the following integral representation is valid:

$$ f(z) = \int\limits _ { D } f ( \zeta ) K (z, \zeta ) dV ( \zeta ). $$

Extremal properties of the Bergman kernel function are:

1) For any point $ z \in D $

$$ K (z, z) = \sup \ \{ {| f (z) | ^ {2} } : { f \in L _ {2,h} (D),\ \| f \| _ {L _ {2} (D) } \leq 1 } \} . $$

2) Let a point $ \zeta \in D $ be such that the class $ L _ {2,h} (D) $ contains functions satisfying the condition $ f ( \zeta ) = 1 $. The function $ K(z, \zeta )/K( \zeta , \zeta ) $ then satisfies this condition and has norm $ K( \zeta , \zeta ) ^ {-1/2} $, which is minimal for all such $ f $. The function $ K(z, \zeta )/K( \zeta , \zeta ) $ is called the extremal function of $ D $.

Changes in the Bergman kernel function under biholomorphic mappings are expressed in the following theorem: If $ \phi $ is a biholomorphic mapping of a domain $ D $ onto a domain $ D ^ {*} $, $ \phi (z) = w $, $ \phi ( \zeta ) = \eta $, then

$$ K _ {D ^ {*} } ( w , \eta ) = \ K _ {D} (z, \zeta ) \ \frac{dz}{d w } \ \frac{\overline{ {d \zeta }}\; }{d \eta } , $$

where $ dz / dw $ is the Jacobian of the inverse mapping. Owing to this property the Hermitian quadratic form

$$ ds ^ {2} = \ \sum _ { j,k=1 } ^ { n } \frac{\partial ^ {2} \mathop{\rm log} K(z, z) }{\partial z _ {j} \partial \overline{z}\; _ {k} } \ dz _ {j} d \overline{z}\; _ {k} $$

is invariant under biholomorphic mappings.

The function $ K(z) = K(z, z) $, which is also called a kernel function, plays an important role in the intrinsic geometry of domains. In the general case it is non-negative, while the function $ \mathop{\rm log} K (z) $ is plurisubharmonic. In domains $ D $ where $ K(z) $ is positive (e.g. in bounded domains), the functions $ K(z) $ and $ \mathop{\rm log} K(z) $ are strictly plurisubharmonic. The latter is tantamount to saying that in such domains $ D $ the form $ ds ^ {2} $ is positive definite and, consequently, gives a Hermitian Riemannian metric in $ D $. This metric remains unchanged under biholomorphic mappings and is called the Bergman metric. It may be considered as a special case of a Kähler metric. It follows from extremal property 1) that the coefficients of the Bergman metric increase to infinity on approaching certain boundary points. If $ D \subset \mathbf C ^ {n} $ is a strictly pseudo-convex domain or an analytic polyhedron, then $ K(z) $ increases to infinity for any approach of $ z $ to the boundary of the domain. Every domain which has this property of the Bergman kernel function is a domain of holomorphy.

For domains of the simplest type, the Bergman kernel function can be explicitly calculated. Thus, for the ball $ B = \{ {z } : {| z | < R } \} $ in $ \mathbf C ^ {n} $, the Bergman function has the following form:

$$ K _ {B} (z , \zeta ) = \ \frac{n ! R ^ {n} }{\pi ^ {n} } \left ( R ^ {2} - \sum _ { j=1 } ^ { n } z _ {j} \overline \zeta \; _ {j} \right ) ^ {-n-1} , $$

and for the polydisc $ U = \{ {z } : {| z _ {j} | < R _ {j} , j = 1 \dots n } \} $, in $ \mathbf C ^ {n} $:

$$ K _ {U} (z, \zeta ) = \ \frac{1}{\pi ^ {n} } \prod _ { j=1 } ^ { n } \frac{R _ {j} ^ {2} }{(R _ {j} ^ {2} -z _ {j} \overline \zeta \; _ {j} ) ^ {2} } . $$

In the special case when $ n=1 $ and $ U = B $ is the disc $ \{ {z } : {| z | < R } \} $ in the complex $ z $- plane, the Bergman metric becomes the classical hyperbolic metric

$$ ds ^ {2} = \ \frac{2R ^ {2} }{(R ^ {2} - | z | ^ {2} ) ^ {2} } \ | dz | ^ {2} , $$

which is invariant under conformal mappings and which defines the Lobachevskii geometry in $ U $.

References

[1]	S. Bergman, "The kernel function and conformal mapping" , Amer. Math. Soc. (1950)
[2]	B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)
[3]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)

Comments

Recently, additional properties of the Bergman kernel have been discovered. For a large class of pseudo-convex domains with $ C ^ \infty $ boundary, which includes strictly pseudo-convex domains, and domains of finite type, the kernel function $ K (z, w) $ on $ D \times D $ is smooth up to the boundary in $ z $ if $ w $ remains fixed in $ D $. This is a consequence of the compactness of the Neumann operator $ N $ for the complex Laplacian on $ D $ and the identity

$$ P = I - \overline \partial \; * N \overline \partial \; . $$

Here $ P $ is the Bergman projection, that is, the orthogonal projection of $ L _ {2} (D) $ onto $ L _ {2,h} (D) $ given by integration against $ K $; $ \overline \partial \; $ is the Cauchy–Riemann operator and $ \overline \partial \; ^ {*} $ its Hilbert space adjoint. In fact, for these domains $ P $ satisfies the so-called "condition R for the Bergman projectioncondition R" , that is $ P $ maps $ L _ {2, s + 2 } (D) $ continuously into $ L _ {2,s} (D) $, where $ L _ {2,k} (D) $ denotes the Sobolev space of order $ k $. This property of the Bergman kernel function is employed in the study of proper holomorphic and biholomorphic mappings. (See [a2], [a4], [a5].) Moreover, the asymptotic behaviour of $ K (z, w) $ has been studied; for strictly pseudo-convex domains $ D $ one has

$$ K (z, w) = \ F (z, w) (i \psi (z, w)) ^ {- n - 1 } + G (z, w) \mathop{\rm log} \ (i \psi (z, w)), $$

where $ F, G $ and $ \psi $ are $ C ^ \infty $ functions on $ \overline{D}\; \times \overline{D}\; $ and $ \psi $ satisfies

a) $ \psi (z, z) = \rho (z)/i $, where $ \rho $ is a strictly-plurisubharmonic defining function for $ D $;

b) $ \overline \partial \; _ {z} \psi $ and $ \overline \partial \; _ {w} \psi $ vanish to infinite order at $ z = w $; and

c) $ \psi (z, w) = {- \psi (w, z) } bar $.

Similar results have been obtained for certain weakly pseudo-convex domains, see [a1], [a3], [a4].

The Bergman kernel has also been studied for other domains, e.g. Cartan domains (cf. [a6]) and Siegel domains (cf. [a7], Siegel domain).

References

[a1]	L. Boutet de Monvel, J. Sjöstrand, "Sur la singularité des noyaux de Bergman et de Szegö" Astérisque , 34–35 (1976) pp. 123–164
[a2]	D. Catlin, "Global regularity of the -Neumann problem" , Proc. Symp. Pure Math. , 41 , Amer. Math. Soc. (1984) pp. 39–49
[a3]	K. Diederich, G. Herbort, T. Ohsawa, "The Bergman kernel on uniformly extendable pseudo-convex domains" Math. Ann. , 273 (1986) pp. 471–478
[a4]	C. Fefferman, "The Bergman kernel and biholomorphic mappings of pseudo-convex domains" Invent. Math. , 37 (1974) pp. 1–65
[a5]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 7
[a6]	L.K. Hua, "Harmonic analysis of functions of several complex variables in the classical domains" , Amer. Math. Soc. (1963)
[a7]	S.G. Gindikin, "Analysis on homogeneous domains" Russian Math. Surveys , 19 (1964) pp. 1–89 Uspekhi Mat. Nauk , 19 (1964) pp. 3–92