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
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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
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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:
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Extremal properties of the Bergman kernel function are:
1) For any point
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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
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where is the Jacobian of the inverse mapping. Owing to this property the Hermitian quadratic form
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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:
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and for the polydisc , in
:
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In the special case when and
is the disc
in the complex
-plane, the Bergman metric becomes the classical hyperbolic metric
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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
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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
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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 ![]() |
[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 |
Bergman kernel function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bergman_kernel_function&oldid=11473