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A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729302.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729303.png" /> complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729304.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729305.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729307.png" />, that satisfies the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729308.png" /> is upper semi-continuous (cf. [[Semi-continuous function|Semi-continuous function]]) everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p0729309.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293010.png" /> is a [[Subharmonic function|subharmonic function]] of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293011.png" /> in each connected component of the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293012.png" /> for any fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293014.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293015.png" /> is called a plurisuperharmonic function if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293016.png" /> is plurisubharmonic. The plurisubharmonic functions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293017.png" /> constitute a proper subclass of the class of subharmonic functions, while these two classes coincide for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293018.png" />. The most important examples of plurisubharmonic functions are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293023.png" /> is a [[Holomorphic function|holomorphic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293024.png" />.
+
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$#A+1 = 189 n = 0
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$#C+1 = 189 : ~/encyclopedia/old_files/data/P072/P.0702930 Plurisubharmonic function
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For an upper semi-continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293026.png" />, to be plurisubharmonic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293027.png" />, it is necessary and sufficient that for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293030.png" />, there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293031.png" /> such that the following inequality holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293032.png" />:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/p/p072/p072930/p07293033.png" /></td> </tr></table>
+
A real-valued function  $  u = u( z) $,
 +
$  - \infty \leq  u < + \infty $,
 +
of  $  n $
 +
complex variables  $  z = ( z _ {1} \dots z _ {n} ) $
 +
in a domain  $  D $
 +
of the complex space  $  \mathbf C  ^ {n} $,
 +
$  n \geq  1 $,
 +
that satisfies the following conditions: 1)  $  u( z) $
 +
is upper semi-continuous (cf. [[Semi-continuous function|Semi-continuous function]]) everywhere in  $  D $;  
 +
and 2)  $  u( z  ^ {0} + \lambda a) $
 +
is a [[Subharmonic function|subharmonic function]] of the variable  $  \lambda \in \mathbf C $
 +
in each connected component of the open set  $  \{ {\lambda \in \mathbf C } : {z  ^ {0} + \lambda a \in D } \} $
 +
for any fixed points  $  z  ^ {0} \in D $,
 +
$  a \in \mathbf C  ^ {n} $.  
 +
A function  $  v( z) $
 +
is called a plurisuperharmonic function if  $  - v( z) $
 +
is plurisubharmonic. The plurisubharmonic functions for  $  n > 1 $
 +
constitute a proper subclass of the class of subharmonic functions, while these two classes coincide for  $  n= 1 $.  
 +
The most important examples of plurisubharmonic functions are  $  \mathop{\rm ln}  | f( z) | $,
 +
$  \mathop{\rm ln}  ^ {+}  | f( z) | $,
 +
$  | f( z) |  ^ {p} $,
 +
p \geq  0 $,
 +
where  $  f( z) $
 +
is a [[Holomorphic function|holomorphic function]] in  $  D $.
  
The following criterion is more convenient for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293034.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293035.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293036.png" /> is a plurisubharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293037.png" /> if and only if the [[Hermitian form|Hermitian form]] (the Hessian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293038.png" />, cf. [[Hessian of a function|Hessian of a function]])
+
For an upper semi-continuous function  $  u( z) $,
 +
$  u( z) < + \infty $,
 +
to be plurisubharmonic in a domain  $  D \subset  \mathbf C  ^ {n} $,
 +
it is necessary and sufficient that for every fixed  $  z \in D $,
 +
$  a \in \mathbf C  ^ {n} $,
 +
$  | a | = 1 $,
 +
there exists a number  $  \delta = \delta ( z, a) > 0 $
 +
such that the following inequality holds for  $  0 < r < \delta $:
  
<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/p/p072/p072930/p07293039.png" /></td> </tr></table>
+
$$
 +
u( z)  \leq 
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi } u( z + re ^ {i \phi } a)  d \phi .
 +
$$
  
is positive semi-definite at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293040.png" />.
+
The following criterion is more convenient for functions  $  u( z) $
 +
of class  $  C  ^ {2} ( D) $:  
 +
$  u( z) $
 +
is a plurisubharmonic function in  $  D $
 +
if and only if the [[Hermitian form|Hermitian form]] (the Hessian of  $  u $,
 +
cf. [[Hessian of a function|Hessian of a function]])
  
The following hold for plurisubharmonic functions, in addition to the general properties of subharmonic functions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293041.png" /> is plurisubharmonic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293042.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293043.png" /> is a plurisubharmonic function in a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293044.png" />; b) a linear combination of plurisubharmonic functions with positive coefficients is plurisubharmonic; c) the limit of a uniformly-convergent or monotone decreasing sequence of plurisubharmonic functions is plurisubharmonic; d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293045.png" /> is a plurisubharmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293046.png" /> if and only if it can be represented as the limit of a decreasing sequence of plurisubharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293047.png" /> of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293048.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293049.png" /> are domains such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293051.png" />; e) for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293052.png" /> the mean value
+
$$
 +
H(( z ; u) a, \overline{a}\; ) = \sum _ {j,k= 1 } ^ { n } 
 +
\frac{\partial  ^ {2} u }{
 +
\partial  z _ {j} \partial  \overline{z}\; _ {k} }
 +
a _ {j} \overline{a}\; {} _ {k}  $$
  
<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/p/p072/p072930/p07293053.png" /></td> </tr></table>
+
is positive semi-definite at each point  $  z \in D $.
  
over a sphere of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293055.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293056.png" />, is an increasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293057.png" /> that is convex with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293058.png" /> on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293059.png" />, if the sphere
+
The following hold for plurisubharmonic functions, in addition to the general properties of subharmonic functions: a)  $  u( z) $
 +
is plurisubharmonic in a domain  $  D $
 +
if and only if  $  u( z) $
 +
is a plurisubharmonic function in a neighbourhood of each point  $  z \in D $;
 +
b) a linear combination of plurisubharmonic functions with positive coefficients is plurisubharmonic; c) the limit of a uniformly-convergent or monotone decreasing sequence of plurisubharmonic functions is plurisubharmonic; d)  $  u( z) $
 +
is a plurisubharmonic function in a domain  $  D $
 +
if and only if it can be represented as the limit of a decreasing sequence of plurisubharmonic functions  $  \{ u _ {k} ( z) \} _ {k=} 1  ^  \infty  $
 +
of the classes  $  C  ^  \infty  ( D _ {k} ) $,
 +
respectively, where  $  D _ {k} $
 +
are domains such that $  D _ {k} \subset  \overline{D}\; {} _ {k} \subset  D _ {k+} 1 $
 +
and  $  \cup _ {k=} 1 ^  \infty  D _ {k} = D $;
 +
e) for any point  $  z  ^ {0} \in D $
 +
the mean value
  
<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/p/p072/p072930/p07293060.png" /></td> </tr></table>
+
$$
 +
J ( z  ^ {0} , r; u)  =
 +
\frac{1}{\sigma _ {2n} }
 +
\int\limits _ {| a | = 1 } u( z  ^ {0} +
 +
ra)  da
 +
$$
  
is located in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293061.png" />, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293062.png" />; f) a plurisubharmonic function remains plurisubharmonic under holomorphic mappings; g) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293063.png" /> is a continuous plurisubharmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293064.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293065.png" /> is a closed connected analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293066.png" /> (cf. [[Analytic set|Analytic set]]) and if the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293067.png" /> attains a maximum on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293069.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293070.png" />.
+
over a sphere of radius  $  r $,  
 +
where  $  \sigma _ {2n} = 2 \pi  ^ {n} /( n- 1)! $
 +
is the area of the unit sphere in $  \mathbf R  ^ {2n} $,  
 +
is an increasing function of $  r $
 +
that is convex with respect to  $  \mathop{\rm ln}  r $
 +
on the segment  $  0 \leq  r \leq  R $,
 +
if the sphere
  
The following proper subclasses of the class of plurisubharmonic functions are also significant for applications. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293071.png" /> is called strictly plurisubharmonic if there exists a convex increasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293073.png" />,
+
$$
 +
V( z  ^ {0} , R)  = \{ {z \in \mathbf C  ^ {n} } : {| z- z  ^ {0} | < R } \}
 +
$$
  
<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/p/p072/p072930/p07293074.png" /></td> </tr></table>
+
is located in  $  D $,
 +
in which case  $  u( z  ^ {0} ) \leq  J( z  ^ {0} , r; u) $;  
 +
f) a plurisubharmonic function remains plurisubharmonic under holomorphic mappings; g) if  $  u( z) $
 +
is a continuous plurisubharmonic function in a domain  $  D $,
 +
if  $  E $
 +
is a closed connected analytic subset of  $  D $(
 +
cf. [[Analytic set|Analytic set]]) and if the restriction  $  u \mid  _ {E} $
 +
attains a maximum on  $  E $,
 +
then  $  u( z) = \textrm{ const } $
 +
on  $  E $.
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293075.png" /> is a plurisubharmonic function. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293076.png" /> one obtains logarithmically-plurisubharmonic functions.
+
The following proper subclasses of the class of plurisubharmonic functions are also significant for applications. A function  $  u( z) $
 +
is called strictly plurisubharmonic if there exists a convex increasing function $  \phi ( t) $,  
 +
$  - \infty < t < + \infty $,
  
The class of plurisubharmonic functions and the above subclasses are important in describing various features of holomorphic functions and domains in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293077.png" />, as well as in more general analytic spaces [[#References|[1]]]–[[#References|[4]]], [[#References|[7]]]. For example, the class of Hartogs functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293078.png" /> is defined as the smallest class of real-valued functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293079.png" /> containing all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293080.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293081.png" /> is a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293082.png" />, and closed under the following operations:
+
$$
 +
\lim\limits _ {t\rightarrow+ \infty } 
 +
\frac{\phi ( t) }{t}
 +
  = + \infty ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293083.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293085.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293086.png" />;
+
such that  $  \phi  ^ {-} 1 ( u( z)) $
 +
is a plurisubharmonic function. In particular, for  $  \phi ( t) = e  ^ {t} $
 +
one obtains logarithmically-plurisubharmonic functions.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293087.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293089.png" /> for every domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293091.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293092.png" />;
+
The class of plurisubharmonic functions and the above subclasses are important in describing various features of holomorphic functions and domains in the complex space  $  \mathbf C  ^ {n} $,
 +
as well as in more general analytic spaces [[#References|[1]]]–[[#References|[4]]], [[#References|[7]]]. For example, the class of Hartogs functions  $  H( D) $
 +
is defined as the smallest class of real-valued functions in  $  D $
 +
containing all functions  $  \mathop{\rm ln}  | f( z) | $,  
 +
where  $  f( z) $
 +
is a holomorphic function in  $  D $,  
 +
and closed under the following operations:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293093.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293096.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293097.png" />;
+
$  \alpha $)
 +
$  u _ {1} , u _ {2} \in H( D) $,  
 +
$  \lambda _ {1} , \lambda _ {2} \geq  0 $
 +
imply $  \lambda _ {1} u _ {1} + \lambda _ {2} u _ {2} \in H( D) $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293098.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p07293099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930100.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930101.png" />;
+
$  \beta $)
 +
$  u _ {k} \in H( D) $,
 +
$  u _ {k} \leq  M( D _ {1} ) $
 +
for every domain  $  D _ {1} \subset  \overline{D}\; _ {1} \subset  D $,
 +
$  k = 1, 2 \dots $
 +
imply  $  \sup \{ {u _ {k} ( z) } : {k= 1, 2 ,\dots } \} \in H( D) $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930102.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930103.png" /> for every subdomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930104.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930105.png" />.
+
$  \gamma $)
 +
$  u _ {k} \in H( D) $,
 +
$  u _ {k} \geq  u _ {k+} 1 $,
 +
$  k = 1, 2 \dots $
 +
imply  $  \lim\limits _ {k \rightarrow \infty }  u _ {k} ( z) \in H( D) $;
  
Upper semi-continuous Hartogs functions are plurisubharmonic, but not every plurisubharmonic function is a Hartogs function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930106.png" /> is a [[Domain of holomorphy|domain of holomorphy]], the classes of upper semi-continuous Hartogs functions and plurisubharmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930107.png" /> coincide [[#References|[5]]], [[#References|[6]]].
+
$  \delta $)
 +
$  u \in H( D) $,
 +
$  z \in D $
 +
imply  $  \lim\limits _ {z _ {1}  \rightarrow z }  \sup  u( z _ {1} ) \in H( D) $;
 +
 
 +
$  \epsilon $)
 +
$  u \in H( D _ {1} ) $
 +
for every subdomain  $  D _ {1} \subset  \overline{D}\; _ {1} \subset  D $
 +
implies  $  u \in H( D) $.
 +
 
 +
Upper semi-continuous Hartogs functions are plurisubharmonic, but not every plurisubharmonic function is a Hartogs function. If $  D $
 +
is a [[Domain of holomorphy|domain of holomorphy]], the classes of upper semi-continuous Hartogs functions and plurisubharmonic functions in $  D $
 +
coincide [[#References|[5]]], [[#References|[6]]].
  
 
See also [[Pluriharmonic function|Pluriharmonic function]].
 
See also [[Pluriharmonic function|Pluriharmonic function]].
Line 46: Line 169:
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of many complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Lelong,  "Fonctions plurisousharmonique; mesures de Radon associées. Applications aux fonctions analytiques" , ''Colloque sur les fonctions de plusieurs variables, Brussels 1953'' , G. Thone &amp; Masson  (1953)  pp. 21–40</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H.J. Bremermann,  "Complex convexity"  ''Trans. Amer. Math. Soc.'' , '''82'''  (1956)  pp. 17–51</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H.J. Bremermann,  "On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions"  ''Math. Ann.'' , '''131'''  (1956)  pp. 76–86</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H.J. Bremermann,  "Note on plurisubharmonic and Hartogs functions"  ''Proc. Amer. Math. Soc.'' , '''7'''  (1956)  pp. 771–775</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.D. Solomentsev,  "Harmonic and subharmonic functions and their generalizations"  ''Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie''  (1964)  pp. 83–100  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of many complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Lelong,  "Fonctions plurisousharmonique; mesures de Radon associées. Applications aux fonctions analytiques" , ''Colloque sur les fonctions de plusieurs variables, Brussels 1953'' , G. Thone &amp; Masson  (1953)  pp. 21–40</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H.J. Bremermann,  "Complex convexity"  ''Trans. Amer. Math. Soc.'' , '''82'''  (1956)  pp. 17–51</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H.J. Bremermann,  "On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions"  ''Math. Ann.'' , '''131'''  (1956)  pp. 76–86</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H.J. Bremermann,  "Note on plurisubharmonic and Hartogs functions"  ''Proc. Amer. Math. Soc.'' , '''7'''  (1956)  pp. 771–775</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.D. Solomentsev,  "Harmonic and subharmonic functions and their generalizations"  ''Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie''  (1964)  pp. 83–100  (In Russian)</TD></TR></table>
  
 +
====Comments====
 +
A function  $  u \in C  ^ {2} ( D) $
 +
is strictly plurisubharmonic if and only if the complex Hessian  $  H(( z;  u) a, \overline{a}\; ) $
 +
is a positive-definite Hermitian form on  $  \mathbf C  ^ {n} $.
  
 +
The Hessian has also an interpretation for arbitrary plurisubharmonic functions  $  u $.
 +
For every  $  a \in \mathbf C  ^ {n} $,
 +
$  H(( z;  u) a, \overline{a}\; ) $
 +
can be viewed as a distribution (cf. [[Generalized function|Generalized function]]), which is positive and hence can be represented by a measure. This is in complete analogy with the interpretation of the Laplacian of subharmonic functions.
  
====Comments====
+
However, in this setting one usually introduces currents, cf. [[#References|[a2]]]. Let  $  C _ {0}  ^  \infty  ( p, q) ( D) $
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930108.png" /> is strictly plurisubharmonic if and only if the complex Hessian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930109.png" /> is a positive-definite Hermitian form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930110.png" />.
+
denote the space of compactly-supported differential forms  $  \phi = \sum _ {| I| = p,| J| = q }  \phi _ {I,J}  dz _ {I} \wedge d \overline{z}\; {} _ {J} $
 
+
on $  D $
The Hessian has also an interpretation for arbitrary plurisubharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930111.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930113.png" /> can be viewed as a distribution (cf. [[Generalized function|Generalized function]]), which is positive and hence can be represented by a measure. This is in complete analogy with the interpretation of the Laplacian of subharmonic functions.
+
of degree  $  p $
 +
in  $  \{ dz _ {1} \dots dz _ {n} \} $
 +
and degree  $  q $
 +
in  $  \{ d \overline{z}\; _ {1} \dots d \overline{z}\; _ {n} \} $(
 +
cf. [[Differential form|Differential form]]). The exterior differential operators  $  \partial  $,  
 +
$  \overline \partial \; $
 +
and $  d $
 +
are defined by:
  
However, in this setting one usually introduces currents, cf. [[#References|[a2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930114.png" /> denote the space of compactly-supported differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930115.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930116.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930117.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930118.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930119.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930120.png" /> (cf. [[Differential form|Differential form]]). The exterior differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930123.png" /> are defined by:
+
$$
 +
\partial  \phi  = \sum _ { k= } 1 ^ { n }  \
 +
\sum _ {\begin{array}{c}
 +
{| I| = p } \\
 +
{| J| = q }
 +
\end{array}
 +
}
  
<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/p/p072/p072930/p072930124.png" /></td> </tr></table>
+
\frac{\partial  \phi _ {I,J} }{\partial  z _ {k} }
 +
\
 +
dz _ {k} \wedge d \overline{z}\; {} _ {J}  \in \
 +
C _ {0}  ^  \infty  ( p+ 1, q) ,
 +
$$
  
<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/p/p072/p072930/p072930125.png" /></td> </tr></table>
+
$$
 +
\overline \partial \; \phi  = \sum _ { k= } 1 ^ { n }  \sum _ {\begin{array}{c}
 +
{| I| = p }
 +
\\
 +
{| J| = q }
 +
\end{array}
 +
}
 +
\frac{\partial  \phi _ {I,J} }{\partial  \overline{z}\; {} _ {k} }
 +
\
 +
d \overline{z}\; {} _ {k} \wedge d \overline{z}\; {} _ {J}  \in  C _ {0}  ^  \infty  ( p, q+ 1) ,
 +
$$
  
<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/p/p072/p072930/p072930126.png" /></td> </tr></table>
+
$$
 +
d \phi  = \partial  \phi + \overline \partial \; \phi .
 +
$$
  
The forms in the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930127.png" /> are called closed, the forms in the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930128.png" /> are called exact. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930129.png" />, the set of exact forms is contained in the set of closed forms. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930131.png" />-form is called positive of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930134.png" /> if for every system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930135.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930136.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930138.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930139.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930140.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930142.png" /> the Euclidean volume element.
+
The forms in the kernel of $  d $
 +
are called closed, the forms in the image of $  d $
 +
are called exact. As $  dd = 0 $,  
 +
the set of exact forms is contained in the set of closed forms. A $  ( p, p) $-
 +
form is called positive of degree p $
 +
if for every system $  a _ {1} \dots a _ {n-} p $
 +
of $  ( 1, 0) $-
 +
forms $  a _ {i} = \sum _ {j=} 1  ^ {n} a _ {ij}  dz _ {j} $,  
 +
$  a _ {ij} \in \mathbf C $,  
 +
the $  ( n, n) $-
 +
form  $  \phi \wedge ia _ {1} \wedge \overline{a}\; {} _ {1} \wedge \dots \wedge ia _ {n-} p \wedge \overline{a}\; {} _ {n-} p = g  dV $,  
 +
with $  g \geq  0 $
 +
and $  dV $
 +
the Euclidean volume element.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930144.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930146.png" />-current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930147.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930148.png" /> is a linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930149.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930150.png" /> with the property that for every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930151.png" /> there are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930152.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930153.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930155.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930156.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930157.png" /> are extended via duality; e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930158.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930159.png" />-current, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930160.png" />. Closed and exact currents are defined as for differential forms. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930161.png" />-current is called positive if for every system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930162.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930163.png" />-forms as above and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930164.png" />,
+
Let $  p  ^  \prime  = n- p $,  
 +
$  q  ^  \prime  = n- q $.  
 +
A $  ( p ^  \prime  , q  ^  \prime  ) $-
 +
current $  t $
 +
on $  D $
 +
is a linear form $  t $
 +
on $  C _ {0}  ^  \infty  ( p, q)( D) $
 +
with the property that for every compact set $  K \subset  D $
 +
there are constants $  C, k $
 +
such that $  | \langle  t, \phi \rangle | < C  \sup _ {I, J, \alpha ,z }  | D  ^  \alpha  \phi _ {I,J} ( z) | $
 +
for $  z \in K $
 +
and $  | \alpha | \leq  k $,  
 +
where $  D  ^  \alpha  = \partial  ^ {| \alpha | } / ( \partial  z _ {1} ^ {\alpha _ {1} } {} \dots \partial  \overline{z}\; {} _ {n} ^ {\alpha _ {2n} } ) $.  
 +
The operators $  d , \partial  , \overline \partial \; $
 +
are extended via duality; e.g., if $  t $
 +
is a $  ( p ^  \prime  , q  ^  \prime  ) $-
 +
current, then $  \langle  dt, \phi \rangle = (- 1) ^ {p ^  \prime  + q  ^  \prime  } \langle  t, d \phi \rangle $.  
 +
Closed and exact currents are defined as for differential forms. A $  ( p  ^  \prime  , p ^  \prime  ) $-
 +
current is called positive if for every system $  a _ {1} \dots a _ {p} $
 +
of $  ( 1, 0) $-
 +
forms as above and for every $  \phi \in C _ {0}  ^  \infty  ( D) $,
  
<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/p/p072/p072930/p072930165.png" /></td> </tr></table>
+
$$
 +
< t, \phi  ia _ {1} \wedge \overline{a}\; {} _ {1} \wedge \dots \wedge
 +
ia _ {p} \wedge \overline{a}\; {} _ {p} > \geq  0 .
 +
$$
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930166.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930167.png" /> gives rise to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930168.png" />-current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930169.png" /> via integration: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930170.png" />. A [[Complex manifold|complex manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930171.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930172.png" /> gives rise to a positive closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930173.png" />-current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930174.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930175.png" />, the current of integration along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930176.png" />:
+
A $  ( p ^  \prime  , q  ^  \prime  ) $-
 +
form $  \psi $
 +
gives rise to a $  ( p ^  \prime  , q  ^  \prime  ) $-
 +
current $  t _  \psi  $
 +
via integration: $  \langle  t _  \psi  , \phi \rangle = \int _ {D} \phi \wedge \psi $.  
 +
A [[Complex manifold|complex manifold]] $  M \subset  D $
 +
of dimension p $
 +
gives rise to a positive closed $  ( p  ^  \prime  , p ^  \prime  ) $-
 +
current $  [ M] $
 +
on $  D $,  
 +
the current of integration along $  M $:
  
<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/p/p072/p072930/p072930177.png" /></td> </tr></table>
+
$$
 +
\langle  [ M ] , \phi \rangle  = \int\limits _ { M } \phi .
 +
$$
  
The current of integration has also been defined for analytic varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930178.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930179.png" /> (cf. [[Analytic manifold|Analytic manifold]]): one defines the current of integration for the set of regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930180.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930181.png" /> and shows that it can be extended to a positive closed current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930182.png" />. A plurisubharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930183.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930184.png" />, hence identifies with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930185.png" />-current. Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930186.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930187.png" />-current, which turns out to be positive and closed. Conversely, a positive closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930188.png" />-current is locally of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930189.png" />. The current of integration on an irreducible variety of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930190.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930191.png" /> is a holomorphic function with gradient not identically vanishing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930192.png" />, equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072930/p072930193.png" />. See also [[Residue of an analytic function|Residue of an analytic function]] and [[Residue form|Residue form]].
+
The current of integration has also been defined for analytic varieties $  Y $
 +
in $  D $(
 +
cf. [[Analytic manifold|Analytic manifold]]): one defines the current of integration for the set of regular points of $  Y $
 +
on $  D \setminus  \{ \textrm{ singular  points  of  }  Y \} $
 +
and shows that it can be extended to a positive closed current on $  D $.  
 +
A plurisubharmonic function $  h $
 +
is in $  L _ { \mathop{\rm loc}  }  ^ {1} $,  
 +
hence identifies with a $  ( 0, 0) $-
 +
current. Therefore $  i \partial  \overline \partial \; h $
 +
is a $  ( 1, 1) $-
 +
current, which turns out to be positive and closed. Conversely, a positive closed $  ( 1, 1) $-
 +
current is locally of the form $  i \partial  \overline \partial \; h $.  
 +
The current of integration on an irreducible variety of the form $  Y = \{ {z } : {f( z) = 0 } \} $,  
 +
where $  f $
 +
is a holomorphic function with gradient not identically vanishing on $  Y $,  
 +
equals $  ( i / \pi ) \partial  \overline \partial \;  \mathop{\rm log}  | f | $.  
 +
See also [[Residue of an analytic function|Residue of an analytic function]] and [[Residue form|Residue form]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras and Jensen measures" , Cambridge Univ. Press  (1979)  pp. Chapts. 5; 6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Lelong,  L. Gruman,  "Entire functions of several complex variables" , Springer  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.I. Ronkin,  "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.M. Range,  "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. VI, Par. 6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.M. Chirka,  "Complex analytic sets" , Kluwer  (1989)  pp. 292ff  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras and Jensen measures" , Cambridge Univ. Press  (1979)  pp. Chapts. 5; 6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Lelong,  L. Gruman,  "Entire functions of several complex variables" , Springer  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.I. Ronkin,  "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.M. Range,  "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. VI, Par. 6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.M. Chirka,  "Complex analytic sets" , Kluwer  (1989)  pp. 292ff  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


A real-valued function $ u = u( z) $, $ - \infty \leq u < + \infty $, of $ n $ complex variables $ z = ( z _ {1} \dots z _ {n} ) $ in a domain $ D $ of the complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, that satisfies the following conditions: 1) $ u( z) $ is upper semi-continuous (cf. Semi-continuous function) everywhere in $ D $; and 2) $ u( z ^ {0} + \lambda a) $ is a subharmonic function of the variable $ \lambda \in \mathbf C $ in each connected component of the open set $ \{ {\lambda \in \mathbf C } : {z ^ {0} + \lambda a \in D } \} $ for any fixed points $ z ^ {0} \in D $, $ a \in \mathbf C ^ {n} $. A function $ v( z) $ is called a plurisuperharmonic function if $ - v( z) $ is plurisubharmonic. The plurisubharmonic functions for $ n > 1 $ constitute a proper subclass of the class of subharmonic functions, while these two classes coincide for $ n= 1 $. The most important examples of plurisubharmonic functions are $ \mathop{\rm ln} | f( z) | $, $ \mathop{\rm ln} ^ {+} | f( z) | $, $ | f( z) | ^ {p} $, $ p \geq 0 $, where $ f( z) $ is a holomorphic function in $ D $.

For an upper semi-continuous function $ u( z) $, $ u( z) < + \infty $, to be plurisubharmonic in a domain $ D \subset \mathbf C ^ {n} $, it is necessary and sufficient that for every fixed $ z \in D $, $ a \in \mathbf C ^ {n} $, $ | a | = 1 $, there exists a number $ \delta = \delta ( z, a) > 0 $ such that the following inequality holds for $ 0 < r < \delta $:

$$ u( z) \leq \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } u( z + re ^ {i \phi } a) d \phi . $$

The following criterion is more convenient for functions $ u( z) $ of class $ C ^ {2} ( D) $: $ u( z) $ is a plurisubharmonic function in $ D $ if and only if the Hermitian form (the Hessian of $ u $, cf. Hessian of a function)

$$ H(( z ; u) a, \overline{a}\; ) = \sum _ {j,k= 1 } ^ { n } \frac{\partial ^ {2} u }{ \partial z _ {j} \partial \overline{z}\; _ {k} } a _ {j} \overline{a}\; {} _ {k} $$

is positive semi-definite at each point $ z \in D $.

The following hold for plurisubharmonic functions, in addition to the general properties of subharmonic functions: a) $ u( z) $ is plurisubharmonic in a domain $ D $ if and only if $ u( z) $ is a plurisubharmonic function in a neighbourhood of each point $ z \in D $; b) a linear combination of plurisubharmonic functions with positive coefficients is plurisubharmonic; c) the limit of a uniformly-convergent or monotone decreasing sequence of plurisubharmonic functions is plurisubharmonic; d) $ u( z) $ is a plurisubharmonic function in a domain $ D $ if and only if it can be represented as the limit of a decreasing sequence of plurisubharmonic functions $ \{ u _ {k} ( z) \} _ {k=} 1 ^ \infty $ of the classes $ C ^ \infty ( D _ {k} ) $, respectively, where $ D _ {k} $ are domains such that $ D _ {k} \subset \overline{D}\; {} _ {k} \subset D _ {k+} 1 $ and $ \cup _ {k=} 1 ^ \infty D _ {k} = D $; e) for any point $ z ^ {0} \in D $ the mean value

$$ J ( z ^ {0} , r; u) = \frac{1}{\sigma _ {2n} } \int\limits _ {| a | = 1 } u( z ^ {0} + ra) da $$

over a sphere of radius $ r $, where $ \sigma _ {2n} = 2 \pi ^ {n} /( n- 1)! $ is the area of the unit sphere in $ \mathbf R ^ {2n} $, is an increasing function of $ r $ that is convex with respect to $ \mathop{\rm ln} r $ on the segment $ 0 \leq r \leq R $, if the sphere

$$ V( z ^ {0} , R) = \{ {z \in \mathbf C ^ {n} } : {| z- z ^ {0} | < R } \} $$

is located in $ D $, in which case $ u( z ^ {0} ) \leq J( z ^ {0} , r; u) $; f) a plurisubharmonic function remains plurisubharmonic under holomorphic mappings; g) if $ u( z) $ is a continuous plurisubharmonic function in a domain $ D $, if $ E $ is a closed connected analytic subset of $ D $( cf. Analytic set) and if the restriction $ u \mid _ {E} $ attains a maximum on $ E $, then $ u( z) = \textrm{ const } $ on $ E $.

The following proper subclasses of the class of plurisubharmonic functions are also significant for applications. A function $ u( z) $ is called strictly plurisubharmonic if there exists a convex increasing function $ \phi ( t) $, $ - \infty < t < + \infty $,

$$ \lim\limits _ {t\rightarrow+ \infty } \frac{\phi ( t) }{t} = + \infty , $$

such that $ \phi ^ {-} 1 ( u( z)) $ is a plurisubharmonic function. In particular, for $ \phi ( t) = e ^ {t} $ one obtains logarithmically-plurisubharmonic functions.

The class of plurisubharmonic functions and the above subclasses are important in describing various features of holomorphic functions and domains in the complex space $ \mathbf C ^ {n} $, as well as in more general analytic spaces [1][4], [7]. For example, the class of Hartogs functions $ H( D) $ is defined as the smallest class of real-valued functions in $ D $ containing all functions $ \mathop{\rm ln} | f( z) | $, where $ f( z) $ is a holomorphic function in $ D $, and closed under the following operations:

$ \alpha $) $ u _ {1} , u _ {2} \in H( D) $, $ \lambda _ {1} , \lambda _ {2} \geq 0 $ imply $ \lambda _ {1} u _ {1} + \lambda _ {2} u _ {2} \in H( D) $;

$ \beta $) $ u _ {k} \in H( D) $, $ u _ {k} \leq M( D _ {1} ) $ for every domain $ D _ {1} \subset \overline{D}\; _ {1} \subset D $, $ k = 1, 2 \dots $ imply $ \sup \{ {u _ {k} ( z) } : {k= 1, 2 ,\dots } \} \in H( D) $;

$ \gamma $) $ u _ {k} \in H( D) $, $ u _ {k} \geq u _ {k+} 1 $, $ k = 1, 2 \dots $ imply $ \lim\limits _ {k \rightarrow \infty } u _ {k} ( z) \in H( D) $;

$ \delta $) $ u \in H( D) $, $ z \in D $ imply $ \lim\limits _ {z _ {1} \rightarrow z } \sup u( z _ {1} ) \in H( D) $;

$ \epsilon $) $ u \in H( D _ {1} ) $ for every subdomain $ D _ {1} \subset \overline{D}\; _ {1} \subset D $ implies $ u \in H( D) $.

Upper semi-continuous Hartogs functions are plurisubharmonic, but not every plurisubharmonic function is a Hartogs function. If $ D $ is a domain of holomorphy, the classes of upper semi-continuous Hartogs functions and plurisubharmonic functions in $ D $ coincide [5], [6].

See also Pluriharmonic function.

References

[1] V.S. Vladimirov, "Methods of the theory of many complex variables" , M.I.T. (1966) (Translated from Russian)
[2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
[3] P. Lelong, "Fonctions plurisousharmonique; mesures de Radon associées. Applications aux fonctions analytiques" , Colloque sur les fonctions de plusieurs variables, Brussels 1953 , G. Thone & Masson (1953) pp. 21–40
[4] H.J. Bremermann, "Complex convexity" Trans. Amer. Math. Soc. , 82 (1956) pp. 17–51
[5] H.J. Bremermann, "On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions" Math. Ann. , 131 (1956) pp. 76–86
[6] H.J. Bremermann, "Note on plurisubharmonic and Hartogs functions" Proc. Amer. Math. Soc. , 7 (1956) pp. 771–775
[7] E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie (1964) pp. 83–100 (In Russian)

Comments

A function $ u \in C ^ {2} ( D) $ is strictly plurisubharmonic if and only if the complex Hessian $ H(( z; u) a, \overline{a}\; ) $ is a positive-definite Hermitian form on $ \mathbf C ^ {n} $.

The Hessian has also an interpretation for arbitrary plurisubharmonic functions $ u $. For every $ a \in \mathbf C ^ {n} $, $ H(( z; u) a, \overline{a}\; ) $ can be viewed as a distribution (cf. Generalized function), which is positive and hence can be represented by a measure. This is in complete analogy with the interpretation of the Laplacian of subharmonic functions.

However, in this setting one usually introduces currents, cf. [a2]. Let $ C _ {0} ^ \infty ( p, q) ( D) $ denote the space of compactly-supported differential forms $ \phi = \sum _ {| I| = p,| J| = q } \phi _ {I,J} dz _ {I} \wedge d \overline{z}\; {} _ {J} $ on $ D $ of degree $ p $ in $ \{ dz _ {1} \dots dz _ {n} \} $ and degree $ q $ in $ \{ d \overline{z}\; _ {1} \dots d \overline{z}\; _ {n} \} $( cf. Differential form). The exterior differential operators $ \partial $, $ \overline \partial \; $ and $ d $ are defined by:

$$ \partial \phi = \sum _ { k= } 1 ^ { n } \ \sum _ {\begin{array}{c} {| I| = p } \\ {| J| = q } \end{array} } \frac{\partial \phi _ {I,J} }{\partial z _ {k} } \ dz _ {k} \wedge d \overline{z}\; {} _ {J} \in \ C _ {0} ^ \infty ( p+ 1, q) , $$

$$ \overline \partial \; \phi = \sum _ { k= } 1 ^ { n } \sum _ {\begin{array}{c} {| I| = p } \\ {| J| = q } \end{array} } \frac{\partial \phi _ {I,J} }{\partial \overline{z}\; {} _ {k} } \ d \overline{z}\; {} _ {k} \wedge d \overline{z}\; {} _ {J} \in C _ {0} ^ \infty ( p, q+ 1) , $$

$$ d \phi = \partial \phi + \overline \partial \; \phi . $$

The forms in the kernel of $ d $ are called closed, the forms in the image of $ d $ are called exact. As $ dd = 0 $, the set of exact forms is contained in the set of closed forms. A $ ( p, p) $- form is called positive of degree $ p $ if for every system $ a _ {1} \dots a _ {n-} p $ of $ ( 1, 0) $- forms $ a _ {i} = \sum _ {j=} 1 ^ {n} a _ {ij} dz _ {j} $, $ a _ {ij} \in \mathbf C $, the $ ( n, n) $- form $ \phi \wedge ia _ {1} \wedge \overline{a}\; {} _ {1} \wedge \dots \wedge ia _ {n-} p \wedge \overline{a}\; {} _ {n-} p = g dV $, with $ g \geq 0 $ and $ dV $ the Euclidean volume element.

Let $ p ^ \prime = n- p $, $ q ^ \prime = n- q $. A $ ( p ^ \prime , q ^ \prime ) $- current $ t $ on $ D $ is a linear form $ t $ on $ C _ {0} ^ \infty ( p, q)( D) $ with the property that for every compact set $ K \subset D $ there are constants $ C, k $ such that $ | \langle t, \phi \rangle | < C \sup _ {I, J, \alpha ,z } | D ^ \alpha \phi _ {I,J} ( z) | $ for $ z \in K $ and $ | \alpha | \leq k $, where $ D ^ \alpha = \partial ^ {| \alpha | } / ( \partial z _ {1} ^ {\alpha _ {1} } {} \dots \partial \overline{z}\; {} _ {n} ^ {\alpha _ {2n} } ) $. The operators $ d , \partial , \overline \partial \; $ are extended via duality; e.g., if $ t $ is a $ ( p ^ \prime , q ^ \prime ) $- current, then $ \langle dt, \phi \rangle = (- 1) ^ {p ^ \prime + q ^ \prime } \langle t, d \phi \rangle $. Closed and exact currents are defined as for differential forms. A $ ( p ^ \prime , p ^ \prime ) $- current is called positive if for every system $ a _ {1} \dots a _ {p} $ of $ ( 1, 0) $- forms as above and for every $ \phi \in C _ {0} ^ \infty ( D) $,

$$ < t, \phi ia _ {1} \wedge \overline{a}\; {} _ {1} \wedge \dots \wedge ia _ {p} \wedge \overline{a}\; {} _ {p} > \geq 0 . $$

A $ ( p ^ \prime , q ^ \prime ) $- form $ \psi $ gives rise to a $ ( p ^ \prime , q ^ \prime ) $- current $ t _ \psi $ via integration: $ \langle t _ \psi , \phi \rangle = \int _ {D} \phi \wedge \psi $. A complex manifold $ M \subset D $ of dimension $ p $ gives rise to a positive closed $ ( p ^ \prime , p ^ \prime ) $- current $ [ M] $ on $ D $, the current of integration along $ M $:

$$ \langle [ M ] , \phi \rangle = \int\limits _ { M } \phi . $$

The current of integration has also been defined for analytic varieties $ Y $ in $ D $( cf. Analytic manifold): one defines the current of integration for the set of regular points of $ Y $ on $ D \setminus \{ \textrm{ singular points of } Y \} $ and shows that it can be extended to a positive closed current on $ D $. A plurisubharmonic function $ h $ is in $ L _ { \mathop{\rm loc} } ^ {1} $, hence identifies with a $ ( 0, 0) $- current. Therefore $ i \partial \overline \partial \; h $ is a $ ( 1, 1) $- current, which turns out to be positive and closed. Conversely, a positive closed $ ( 1, 1) $- current is locally of the form $ i \partial \overline \partial \; h $. The current of integration on an irreducible variety of the form $ Y = \{ {z } : {f( z) = 0 } \} $, where $ f $ is a holomorphic function with gradient not identically vanishing on $ Y $, equals $ ( i / \pi ) \partial \overline \partial \; \mathop{\rm log} | f | $. See also Residue of an analytic function and Residue form.

References

[a1] T.W. Gamelin, "Uniform algebras and Jensen measures" , Cambridge Univ. Press (1979) pp. Chapts. 5; 6
[a2] P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1980)
[a3] L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) (Translated from Russian)
[a4] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6
[a5] E.M. Chirka, "Complex analytic sets" , Kluwer (1989) pp. 292ff (Translated from Russian)
How to Cite This Entry:
Plurisubharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plurisubharmonic_function&oldid=48192
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article