# 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 . 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 $U$.

How to Cite This Entry:
Bergman kernel function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bergman_kernel_function&oldid=46211
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article