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''Bergman kernel''
 
''Bergman kernel''
  
A function of complex variables with the reproducing kernel property, defined for any domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155601.png" /> in which there exist holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155602.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155603.png" /> with respect to the Lebesgue measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155604.png" />. The function was introduced by S. Bergman [[#References|[1]]]. The set of these functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155605.png" /> forms the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155606.png" /> with orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155607.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155608.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b0155609.png" /> is the space of holomorphic functions. The function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556010.png" /></td> </tr></table>
+
$$
 +
K _ {D} (z, \zeta )  = \
 +
K (z, \zeta )  = \
 +
\sum _ { j=1 } ^  \infty 
 +
\phi _ {j} (z)
 +
\overline{ {\phi _ {j} ( \zeta ) }}\; ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556011.png" /></td> </tr></table>
+
$$
 +
= (z _ {1} \dots z _ {n} ),\  \zeta  = ( \zeta _ {1} \dots \zeta _ {n} ),
 +
$$
  
is called the Bergman kernel function (or simply the kernel function) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556012.png" />. The series on the right-hand side converges uniformly on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556013.png" />, and belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556014.png" /> for each given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556015.png" />, the sum does not depend on the choice of the orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556016.png" />. The Bergman kernel function depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556017.png" /> complex variables, and is defined in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556018.png" />; it has the symmetry property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556019.png" />, it is holomorphic with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556020.png" /> and anti-holomorphic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556024.png" />, then
+
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 ) = {K(z, \zeta ) } bar $,  
 +
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556025.png" /></td> </tr></table>
+
$$
 +
K _ {D} (z, \zeta )  = \
 +
K _ {D  ^  \prime  }
 +
(z  ^  \prime  , \zeta  ^  \prime  )
 +
K _ {D  ^ {\prime\prime}  }
 +
(z  ^ {\prime\prime} , \zeta  ^  \prime  ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556026.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556027.png" /> and for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556028.png" /> the following integral representation is valid:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556029.png" /></td> </tr></table>
+
$$
 +
f(z)  = \int\limits _ { D } f ( \zeta ) K (z, \zeta )  dV ( \zeta ).
 +
$$
  
 
Extremal properties of the Bergman kernel function are:
 
Extremal properties of the Bergman kernel function are:
  
1) For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556030.png" />
+
1) For any point $  z \in D $
 +
 
 +
$$
 +
K (z, z)  = \sup \
 +
\{ {| f (z) |  ^ {2} } : {
 +
f \in L _ {2,h} (D),\
 +
\| f \| _ {L _ {2}  (D) }
 +
\leq  1 } \}
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556031.png" /></td> </tr></table>
+
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 $.
  
2) Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556032.png" /> be such that the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556033.png" /> contains functions satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556034.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556035.png" /> then satisfies this condition and has norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556036.png" />, which is minimal for all such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556037.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556038.png" /> is called the extremal function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556039.png" />.
+
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
  
Changes in the Bergman kernel function under biholomorphic mappings are expressed in the following theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556040.png" /> is a biholomorphic mapping of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556041.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556044.png" />, then
+
$$
 +
K _ {D  ^ {*}  }
 +
( w , \eta )  = \
 +
K _ {D} (z, \zeta ) \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556045.png" /></td> </tr></table>
+
\frac{dz}{d w }
 +
\
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556046.png" /> is the Jacobian of the inverse mapping. Owing to this property the Hermitian quadratic form
+
\frac{\overline{ {d \zeta }}\; }{d \eta }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556047.png" /></td> </tr></table>
+
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.
 
is invariant under biholomorphic mappings.
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556048.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556049.png" /> is plurisubharmonic. In domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556050.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556051.png" /> is positive (e.g. in bounded domains), the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556053.png" /> are strictly plurisubharmonic. The latter is tantamount to saying that in such domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556054.png" /> the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556055.png" /> is positive definite and, consequently, gives a Hermitian Riemannian metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556056.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556057.png" /> is a strictly pseudo-convex domain or an analytic polyhedron, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556058.png" /> increases to infinity for any approach of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556059.png" /> to the boundary of the domain. Every domain which has this property of the Bergman kernel function is a domain of holomorphy.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556061.png" />, the Bergman function has the following form:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556062.png" /></td> </tr></table>
+
$$
 +
K _ {B} (z , \zeta )  = \
  
and for the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556063.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556064.png" />:
+
\frac{n ! R  ^ {n} }{\pi  ^ {n} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556065.png" /></td> </tr></table>
+
\left ( R  ^ {2} - \sum _ { j=1 } ^ { n }
 +
z _ {j} \overline \zeta \; _ {j} \right )  ^ {-n-1} ,
 +
$$
  
In the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556067.png" /> is the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556068.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556069.png" />-plane, the Bergman metric becomes the classical hyperbolic metric
+
and for the polydisc  $  U = \{ {z } : {| z _ {j} | < R _ {j} ,  j = 1 \dots n } \} $,
 +
in $  \mathbf C  ^ {n} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556070.png" /></td> </tr></table>
+
$$
 +
K _ {U} (z, \zeta )  = \
  
which is invariant under conformal mappings and which defines the Lobachevskii geometry in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556071.png" />.
+
\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====
 
====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>
 
<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====
 
====Comments====
Recently, additional properties of the Bergman kernel have been discovered. For a large class of pseudo-convex domains with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556072.png" /> boundary, which includes strictly pseudo-convex domains, and domains of finite type, the kernel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556074.png" /> is smooth up to the boundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556075.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556076.png" /> remains fixed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556077.png" />. This is a consequence of the compactness of the Neumann operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556078.png" /> for the complex Laplacian on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556079.png" /> and the identity
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556080.png" /></td> </tr></table>
+
$$
 +
= I - \overline \partial \; * N \overline \partial \; .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556081.png" /> is the Bergman projection, that is, the orthogonal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556082.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556083.png" /> given by integration against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556084.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556085.png" /> is the Cauchy–Riemann operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556086.png" /> its Hilbert space adjoint. In fact, for these domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556087.png" /> satisfies the so-called  "condition R for the Bergman projectioncondition R" , that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556088.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556089.png" /> continuously into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556091.png" /> denotes the [[Sobolev space|Sobolev space]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556092.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556093.png" /> has been studied; for strictly pseudo-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556094.png" /> one has
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556095.png" /></td> </tr></table>
+
$$
 +
K (z, w)  = \
 +
F (z, w)
 +
(i \psi (z, w)) ^ {- n - 1 } +
 +
G (z, w)  \mathop{\rm log} \
 +
(i \psi (z, w)),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556097.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556098.png" /> functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560100.png" /> satisfies
+
where $  F, G $
 +
and $  \psi $
 +
are $  C  ^  \infty  $
 +
functions on $  \overline{D}\; \times \overline{D}\; $
 +
and $  \psi $
 +
satisfies
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560102.png" /> is a strictly-plurisubharmonic defining function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560103.png" />;
+
a) $  \psi (z, z) = \rho (z)/i $,  
 +
where $  \rho $
 +
is a strictly-plurisubharmonic defining function for $  D $;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560105.png" /> vanish to infinite order at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560106.png" />; and
+
b) $  \overline \partial \; _ {z} \psi $
 +
and $  \overline \partial \; _ {w} \psi $
 +
vanish to infinite order at $  z = w $;  
 +
and
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560107.png" />.
+
c) $  \psi (z, w) = {- \psi (w, z) } bar $.
  
 
Similar results have been obtained for certain weakly pseudo-convex domains, see [[#References|[a1]]], [[#References|[a3]]], [[#References|[a4]]].
 
Similar results have been obtained for certain weakly pseudo-convex domains, see [[#References|[a1]]], [[#References|[a3]]], [[#References|[a4]]].

Revision as of 10:58, 29 May 2020


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

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
How to Cite This Entry:
Bergman kernel function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bergman_kernel_function&oldid=46016
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article