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''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

<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>

<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>

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

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b01556026.png" />.

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:

<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>

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" />

<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 <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 <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

<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>

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

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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.

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:

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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" />:

<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>

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

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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" />.

====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 <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

<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>

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

<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>

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

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" />;

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

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560107.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015560/b015560108.png" />-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>