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''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 [[#References|[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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Bergman,   "The kernel function and conformal mapping" , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Fuks,   "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>

====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|Sobolev space]] of order  $  k $. 
This property of the Bergman kernel function is employed in the study of proper holomorphic and biholomorphic mappings. (See [[#References|[a2]]], [[#References|[a4]]], [[#References|[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 [[#References|[a1]]], [[#References|[a3]]], [[#References|[a4]]].

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

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Boutet de Monvel,   J. Sjöstrand,   "Sur la singularité des noyaux de Bergman et de Szegö"  ''Astérisque'' , '''34–35'''  (1976)  pp. 123–164</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Catlin,   "Global regularity of the $\partial$-Neumann problem" , ''Proc. Symp. Pure Math.'' , '''41''' , Amer. Math. Soc.  (1984)  pp. 39–49</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Diederich,   G. Herbort,   T. Ohsawa,   "The Bergman kernel on uniformly extendable pseudo-convex domains"  ''Math. Ann.'' , '''273'''  (1986)  pp. 471–478</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Fefferman,   "The Bergman kernel and biholomorphic mappings of pseudo-convex domains"  ''Invent. Math.'' , '''37'''  (1974)  pp. 1–65</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.K. Hua,   "Harmonic analysis of functions of several complex variables in the classical domains" , Amer. Math. Soc.  (1963)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.G. Gindikin,   "Analysis on homogeneous domains"  ''Russian Math. Surveys'' , '''19'''  (1964)  pp. 1–89  ''Uspekhi Mat. Nauk'' , '''19'''  (1964)  pp. 3–92</TD></TR>
</table>