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 $K ( \zeta , z ) = \overline{ {K(z, \zeta ) }}\;$, 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) $\psi (z, w) = \overline{ {- \psi (w, z) }}\;$.

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 $\partial$-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