# Bergman kernel function

*Bergman kernel*

A function of complex variables with the reproducing kernel property, defined for any domain in which there exist holomorphic functions of class with respect to the Lebesgue measure . The function was introduced by S. Bergman [1]. The set of these functions forms the Hilbert space with orthonormal basis ; , where is the space of holomorphic functions. The function

is called the Bergman kernel function (or simply the kernel function) of . The series on the right-hand side converges uniformly on compact subsets of , and belongs to for each given , the sum does not depend on the choice of the orthonormal basis . The Bergman kernel function depends on complex variables, and is defined in the domain ; it has the symmetry property , it is holomorphic with respect to the variable and anti-holomorphic with respect to . If , , , then

where .

The most important characteristic of the Bergman kernel function is its reproducing property: For any function and for any point the following integral representation is valid:

Extremal properties of the Bergman kernel function are:

1) For any point

2) Let a point be such that the class contains functions satisfying the condition . The function then satisfies this condition and has norm , which is minimal for all such . The function is called the extremal function of .

Changes in the Bergman kernel function under biholomorphic mappings are expressed in the following theorem: If is a biholomorphic mapping of a domain onto a domain , , , then

where is the Jacobian of the inverse mapping. Owing to this property the Hermitian quadratic form

is invariant under biholomorphic mappings.

The function , 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 is plurisubharmonic. In domains where is positive (e.g. in bounded domains), the functions and are strictly plurisubharmonic. The latter is tantamount to saying that in such domains the form is positive definite and, consequently, gives a Hermitian Riemannian metric in . 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 is a strictly pseudo-convex domain or an analytic polyhedron, then increases to infinity for any approach of 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 in , the Bergman function has the following form:

and for the polydisc , in :

In the special case when and is the disc in the complex -plane, the Bergman metric becomes the classical hyperbolic metric

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

#### 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 boundary, which includes strictly pseudo-convex domains, and domains of finite type, the kernel function on is smooth up to the boundary in if remains fixed in . This is a consequence of the compactness of the Neumann operator for the complex Laplacian on and the identity

Here is the Bergman projection, that is, the orthogonal projection of onto given by integration against ; is the Cauchy–Riemann operator and its Hilbert space adjoint. In fact, for these domains satisfies the so-called "condition R for the Bergman projectioncondition R" , that is maps continuously into , where denotes the Sobolev space of order . 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 has been studied; for strictly pseudo-convex domains one has

where and are functions on and satisfies

a) , where is a strictly-plurisubharmonic defining function for ;

b) and vanish to infinite order at ; and

c) .

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 |

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Bergman kernel function.

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