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The natural brand of [[Potential theory|potential theory]] in the setting of function theory of several complex variables (cf. also [[Analytic function|Analytic function]]). The basic objects are plurisubharmonic functions (cf. also [[Plurisubharmonic function|Plurisubharmonic function]]). These are studied much from the same perspective as subharmonic functions (cf. also [[Subharmonic function|Subharmonic function]]) are studied in potential theory on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300701.png" />. General references are [[#References|[a1]]], [[#References|[a10]]], [[#References|[a16]]], [[#References|[a23]]].
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300702.png" /> on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300703.png" /> is called plurisubharmonic if it is subharmonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300704.png" />, viewed as a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300705.png" />, and if the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300706.png" /> to every complex line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300707.png" /> is subharmonic (cf. also [[Plurisubharmonic function|Plurisubharmonic function]]; [[Subharmonic function|Subharmonic function]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300708.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p1300709.png" /> on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007011.png" /> is plurisubharmonic if and only if
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<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/p130/p130070/p13007012.png" /></td> </tr></table>
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The natural brand of [[Potential theory|potential theory]] in the setting of function theory of several complex variables (cf. also [[Analytic function|Analytic function]]). The basic objects are plurisubharmonic functions (cf. also [[Plurisubharmonic function|Plurisubharmonic function]]). These are studied much from the same perspective as subharmonic functions (cf. also [[Subharmonic function|Subharmonic function]]) are studied in potential theory on ${\bf R} ^ { n }$. General references are [[#References|[a1]]], [[#References|[a10]]], [[#References|[a16]]], [[#References|[a23]]].
  
is a non-negative [[Hermitian matrix|Hermitian matrix]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007013.png" />. One denotes the set of plurisubharmonic functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007014.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007015.png" />. Plurisubharmonic functions can be defined on domains in complex manifolds via local coordinates (cf. also [[Analytic manifold|Analytic manifold]]).
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A function $u$ on a domain $D \subset \mathbf{C} ^ { x }$ is called plurisubharmonic if it is subharmonic on $D$, viewed as a domain in $\mathbf{R} ^ { 2 n }$, and if the restriction of $u$ to every complex line in $D$ is subharmonic (cf. also [[Plurisubharmonic function|Plurisubharmonic function]]; [[Subharmonic function|Subharmonic function]]). If $u$ is $C ^ { 2 }$ on a domain $D \subset \mathbf{C} ^ { x }$, then $u$ is plurisubharmonic if and only if
  
Plurisubharmonic functions are precisely the subharmonic functions invariant under a holomorphic change of coordinates. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007016.png" /> is holomorphic on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007018.png" /> (cf. also [[Analytic function|Analytic function]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007019.png" /> is plurisubharmonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007020.png" />. Moreover, every plurisubharmonic function can locally be written as
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\begin{equation*} \left( \frac { \partial ^ { 2 } u } { \partial z _ { i } \partial \overline{z _ { j } }} \right) \end{equation*}
  
<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/p130/p130070/p13007021.png" /></td> </tr></table>
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is a non-negative [[Hermitian matrix|Hermitian matrix]] on $D$. One denotes the set of plurisubharmonic functions on a domain $D \subset \mathbf{C} ^ { x }$ by $\operatorname{PSH} ( D )$. Plurisubharmonic functions can be defined on domains in complex manifolds via local coordinates (cf. also [[Analytic manifold|Analytic manifold]]).
  
for suitable holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007022.png" />, see [[#References|[a7]]]. Plurisubharmonic functions were formally introduced by P. Lelong, [[#References|[a19]]], and K. Oka, [[#References|[a22]]], although related ideas stem from the end of the nineteenth century.
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Plurisubharmonic functions are precisely the subharmonic functions invariant under a holomorphic change of coordinates. If $f$ is holomorphic on a domain $D$ in $\mathbf{C} ^ { n }$ (cf. also [[Analytic function|Analytic function]]), then $\operatorname { log } | f |$ is plurisubharmonic on $D$. Moreover, every plurisubharmonic function can locally be written as
  
The analogue of the [[Laplace operator|Laplace operator]] on domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007023.png" /> is the Monge–Ampère operator:
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\begin{equation*} \operatorname { limsup } _ { j \rightarrow \infty } \frac { 1 } { j } \operatorname { log } | f _ { j } |, \end{equation*}
  
<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/p130/p130070/p13007024.png" /></td> </tr></table>
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for suitable holomorphic functions $f_j$, see [[#References|[a7]]]. Plurisubharmonic functions were formally introduced by P. Lelong, [[#References|[a19]]], and K. Oka, [[#References|[a22]]], although related ideas stem from the end of the nineteenth century.
  
This operator is originally only defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007025.png" /> plurisubharmonic functions (cf. also [[Monge–Ampère equation|Monge–Ampère equation]]). Due to the non-linearity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007026.png" /> it is impossible to extend it to a well-defined operator on all plurisubharmonic functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007027.png" /> in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007029.png" /> is a decreasing sequence of plurisubharmonic functions with limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007030.png" />, see [[#References|[a9]]]. Nevertheless, the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007031.png" /> can be enlarged to include all bounded plurisubharmonic functions, [[#References|[a3]]]. The most recent result (as of 2000) in this direction is in [[#References|[a11]]].
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The analogue of the [[Laplace operator|Laplace operator]] on domains in $\mathbf{C}$ is the Monge–Ampère operator:
  
On strongly pseudo-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007032.png" /> (cf. also [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]), the following [[Dirichlet problem|Dirichlet problem]] for the Monge–Ampère operator was solved by E. Bedford and B.A. Taylor [[#References|[a3]]]: Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007033.png" /> continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007035.png" /> continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007036.png" />, there exists a continuous plurisubharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007038.png" />, continuous up to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007039.png" />, such that
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\begin{equation*} M f = \operatorname { det } \left( \frac { \partial ^ { 2 } f } { \partial z _ { i } \partial \overline{z}_ { j } } \right) . \end{equation*}
  
<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/p130/p130070/p13007040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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This operator is originally only defined for $C ^ { 2 }$ plurisubharmonic functions (cf. also [[Monge–Ampère equation|Monge–Ampère equation]]). Due to the non-linearity of $M$ it is impossible to extend it to a well-defined operator on all plurisubharmonic functions on a domain $D$ in such a way that $\operatorname { lim } _ { n \rightarrow \infty } M ( u _ { n } ) = M ( u )$ if $\{ u _ { n } \}$ is a decreasing sequence of plurisubharmonic functions with limit $u$, see [[#References|[a9]]]. Nevertheless, the domain of $M$ can be enlarged to include all bounded plurisubharmonic functions, [[#References|[a3]]]. The most recent result (as of 2000) in this direction is in [[#References|[a11]]].
  
This result has been extended by weakening the conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007041.png" />, and replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007042.png" /> by certain positive measures; see e.g. [[#References|[a5]]], [[#References|[a18]]]. In [[#References|[a11]]], large classes of plurisubharmonic functions on which the Monge–Ampère operator is well defined are determined and necessary and sufficient conditions on a positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007043.png" /> are given, so that the problem (a1) has a solution within such a class.
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On strongly pseudo-convex domains $D$ (cf. also [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]), the following [[Dirichlet problem|Dirichlet problem]] for the Monge–Ampère operator was solved by E. Bedford and B.A. Taylor [[#References|[a3]]]: Given $f$ continuous on $\partial D$ and $\phi$ continuous on $D$, there exists a continuous plurisubharmonic function $u$ on $D$, continuous up to the boundary of $D$, such that
  
The regularity of this Dirichlet problem is quite bad. The following example is due to T. Gamelin and N. Sibony: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007044.png" /> be the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007045.png" />,
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\begin{equation} \tag{a1} \left\{ \begin{array} { c } { M ( u ) = \phi } &amp; {\text { on } D , } \\ { u |_{ \partial D = f.} } \end{array} \right. \end{equation}
  
<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/p130/p130070/p13007046.png" /></td> </tr></table>
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This result has been extended by weakening the conditions on $D$, and replacing $\phi$ by certain positive measures; see e.g. [[#References|[a5]]], [[#References|[a18]]]. In [[#References|[a11]]], large classes of plurisubharmonic functions on which the Monge–Ampère operator is well defined are determined and necessary and sufficient conditions on a positive measure $\phi$ are given, so that the problem (a1) has a solution within such a class.
  
<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/p130/p130070/p13007047.png" /></td> </tr></table>
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The regularity of this Dirichlet problem is quite bad. The following example is due to T. Gamelin and N. Sibony: Let $D$ be the unit ball in $\mathbf{C} ^ { 2 }$,
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\begin{equation*} f ( z _ { 1 } , z _ { 2 } ) = \left( | z _ { 1 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 } = \left( | z _ { 2 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 }, \end{equation*}
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\begin{equation*} ( z _ { 1 } , z _ { 2 } ) \in \partial D. \end{equation*}
  
 
Then the function
 
Then 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/p/p130/p130070/p13007048.png" /></td> </tr></table>
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\begin{equation*} u ( z _ { 1 } , z _ { 2 } ) = \left\{ \begin{array} { c l } { 0 } &amp; { \text { if } | z _ { 1 } | ^ { 2 } , | z _ { 2 } | ^ { 2 } &lt; \frac { 1 } { 2 } } ,\\ { \operatorname { max } \left\{ \left( | z _ { 1 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 } \right. }, &amp; { \left. \left( | z _ { 2 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 } \right\} } \\ { \text { elsewhere on } D, } \end{array} \right. \end{equation*}
  
satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007051.png" />.
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satisfies $M u = 0$ on $D$, $u| _ { \partial D } = f$.
  
However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007053.png" /> are both smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007055.png" />, then was shown in [[#References|[a8]]] that there exists a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007056.png" /> satisfying (a1).
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However, if $f$ and $\phi$ are both smooth and $\phi &gt; 0$ on $D$, then was shown in [[#References|[a8]]] that there exists a smooth $u$ satisfying (a1).
  
There have been defined several capacity functions (cf. also [[Capacity|Capacity]]; [[Capacity potential|Capacity potential]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007057.png" /> that all share the property that sets of capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007058.png" /> are precisely the pluripolar sets, i.e. sets that are locally contained in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007059.png" /> locus of plurisubharmonic functions. See [[#References|[a4]]], [[#References|[a10]]], [[#References|[a23]]], [[#References|[a24]]]. Firstly, the classical construction of [[Logarithmic capacity|logarithmic capacity]] carries over: Let
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There have been defined several capacity functions (cf. also [[Capacity|Capacity]]; [[Capacity potential|Capacity potential]]) on $\mathbf{C} ^ { n }$ that all share the property that sets of capacity $0$ are precisely the pluripolar sets, i.e. sets that are locally contained in the $- \infty$ locus of plurisubharmonic functions. See [[#References|[a4]]], [[#References|[a10]]], [[#References|[a23]]], [[#References|[a24]]]. Firstly, the classical construction of [[Logarithmic capacity|logarithmic capacity]] carries over: Let
  
<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/p130/p130070/p13007060.png" /></td> </tr></table>
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\begin{equation*} \mathcal{L} = \{ u \in \operatorname { PSH } ( \mathbf{C} ^ { n } ) : u - \operatorname { log } ( 1 + | z | ) = O ( 1 ) ( z \rightarrow \infty ) \}. \end{equation*}
  
For a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007062.png" />, define the Green function with pole at infinity by
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For a bounded set $E$ in $\mathbf{C} ^ { n }$, define the Green function with pole at infinity by
  
<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/p130/p130070/p13007063.png" /></td> </tr></table>
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\begin{equation*} L _ { E } ( z ) = \operatorname { sup } \{ v ( z ) : v \in \mathcal{L} , v \leq 0 \text { on } E \}. \end{equation*}
  
Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007064.png" />, the upper semi-continuous regularization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007065.png" />. Then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007066.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007067.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007068.png" /> one defines the Robin function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007069.png" /> by
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Set $L _ { E } ^ { * } ( z ) = \operatorname { limsup } _ { w \rightarrow z } L _ { E } ( w )$, the upper semi-continuous regularization of $L_{E}$. Then either $L _ { E } ^ { * } \equiv \infty$ or $L \in \operatorname { PSH } ( \mathbf{C} ^ { n } )$. For $u \in \mathcal{L}$ one defines the Robin function on $\mathbf{C} ^ { n }$ by
  
<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/p130/p130070/p13007070.png" /></td> </tr></table>
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\begin{equation*} \rho _ { u } ( z ) = \limsup  _ { t \in \text{C} } ( u ( t z ) - \operatorname { log } | t z | ). \end{equation*}
  
Next the logarithmic capacity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007071.png" /> is defined as
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Next the logarithmic capacity of $E$ is defined as
  
<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/p130/p130070/p13007072.png" /></td> </tr></table>
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\begin{equation*} \operatorname { Cap } ( E ) = \operatorname { exp } \left( - \operatorname { sup } _ { z \in \text{C} ^ { n } } \rho _ { L _ { E } } ( z ) \right). \end{equation*}
  
It is, however, a non-trivial result that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007073.png" /> is a Choquet capacity (cf. [[Capacity|Capacity]]), see [[#References|[a17]]]. Another important (relative) capacity is the Monge–Ampère capacity introduced by Bedford and Taylor, [[#References|[a4]]]. It is defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007074.png" /> be a strictly pseudo-convex domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007075.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007076.png" /> be a compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007077.png" />. The Monge–Ampère capacity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007078.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007079.png" /> is
+
It is, however, a non-trivial result that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007073.png"/> is a Choquet capacity (cf. [[Capacity|Capacity]]), see [[#References|[a17]]]. Another important (relative) capacity is the Monge–Ampère capacity introduced by Bedford and Taylor, [[#References|[a4]]]. It is defined as follows: Let $\Omega$ be a strictly pseudo-convex domain in $\mathbf{C} ^ { n }$ and let $K$ be a compact subset of $\Omega$. The Monge–Ampère capacity of $K$ relative to $\Omega$ is
  
<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/p130/p130070/p13007080.png" /></td> </tr></table>
+
\begin{equation*} C ( K , \Omega ) = \end{equation*}
  
<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/p130/p130070/p13007081.png" /></td> </tr></table>
+
\begin{equation*} = \operatorname { sup } \left\{ \int _ { K } M ( u ) d V : u \in \operatorname { PSH } ( \Omega ) , 0 &lt; u &lt; 1 \right\}. \end{equation*}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007082.png" /> is an arbitrary subset, one defines
+
If $E \subset \Omega$ is an arbitrary subset, one defines
  
<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/p130/p130070/p13007083.png" /></td> </tr></table>
+
\begin{equation*} C ( E , \Omega ) = \operatorname { sup } \{ C ( K ) : K \subset \Omega \}. \end{equation*}
  
It is shown in [[#References|[a4]]] that plurisubharmonic functions are quasi-continuous, i.e. continuous outside an open set of arbitrarily small capacity. Another application is a new proof of the following Josefson theorem [[#References|[a14]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007084.png" /> is pluripolar, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007085.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007086.png" />.
+
It is shown in [[#References|[a4]]] that plurisubharmonic functions are quasi-continuous, i.e. continuous outside an open set of arbitrarily small capacity. Another application is a new proof of the following Josefson theorem [[#References|[a14]]]: If $E \subset \mathbf{C} ^ { n }$ is pluripolar, then there exists a $u \in \operatorname { PSH } ( \mathbf{C} ^ { n } )$ with $u | _ { E } = - \infty$.
  
 
Although there is no analogue of the Riesz decomposition theorem (cf. also [[Riesz theorem(2)|Riesz theorem]]; [[Riesz decomposition theorem|Riesz decomposition theorem]]), there are notions of Green functions.
 
Although there is no analogue of the Riesz decomposition theorem (cf. also [[Riesz theorem(2)|Riesz theorem]]; [[Riesz decomposition theorem|Riesz decomposition theorem]]), there are notions of Green functions.
  
1) The (Klimek or pluricomplex) Green function on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007087.png" /> with pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007088.png" /> is the function
+
1) The (Klimek or pluricomplex) Green function on a domain $\Omega \subset {\bf C} ^ { n }$ with pole at $w \in \Omega$ is 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/p/p130/p130070/p13007089.png" /></td> </tr></table>
+
\begin{equation*} G ( z , w ) = \end{equation*}
  
<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/p130/p130070/p13007090.png" /></td> </tr></table>
+
\begin{equation*} = \operatorname { sup } \left\{ h ( z ) : \begin{array}{ c c } { h \in \operatorname{PSH}(\Omega), \, h&lt;0,} \\{h ( \zeta ) - \operatorname { log } \| \zeta - w \| = O ( 1 ) ( \zeta \rightarrow w )} \end{array} \right\}. \end{equation*}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007091.png" /> is hyperconvex, i.e. pseudo-convex and admitting a bounded plurisubharmonic exhaustion function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007092.png" /> is negative and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007093.png" /> fixed, tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007094.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007095.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007097.png" /> is the [[Dirac distribution|Dirac distribution]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007098.png" />; see [[#References|[a12]]], [[#References|[a15]]] for more details.
+
If $\Omega$ is hyperconvex, i.e. pseudo-convex and admitting a bounded plurisubharmonic exhaustion function, then $G ( z , w )$ is negative and, for $w$ fixed, tends to $0$ if $z \rightarrow \partial \Omega$. Moreover, $M ( G ( z , w ) ) = ( 2 \pi ) ^ { n } \delta _ { w }$, where $\delta _ { W }$ is the [[Dirac distribution|Dirac distribution]] at $w$; see [[#References|[a12]]], [[#References|[a15]]] for more details.
  
2) The symmetric Green function on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p13007099.png" /> is the function
+
2) The symmetric Green function on a domain $\Omega \subset {\bf C} ^ { n }$ is 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/p/p130/p130070/p130070100.png" /></td> </tr></table>
+
\begin{equation*} W ( z , w ) = \operatorname { sup } h ( z , w ) \end{equation*}
  
 
where the supremum is taken over
 
where the supremum is taken over
  
<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/p130/p130070/p130070101.png" /></td> </tr></table>
+
\begin{equation*} h \in \operatorname { SPSH } ( \Omega \times \Omega ) , h &lt; 0, \end{equation*}
  
<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/p130/p130070/p130070102.png" /></td> </tr></table>
+
\begin{equation*} h ( z , w ) - \operatorname { log } \| z - w \| \leq \end{equation*}
  
<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/p130/p130070/p130070103.png" /></td> </tr></table>
+
\begin{equation*} \leq - \operatorname { log } ( \operatorname { max } \{ \operatorname { dist } ( z , \partial \Omega ) , \operatorname { dist } ( w , \partial \Omega ) \} ). \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070104.png" /> stands for the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070105.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070106.png" /> that are plurisubharmonic in each of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070108.png" /> separately, when the other is kept fixed. On strictly pseudo-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070109.png" />, the symmetric Green function is negative and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070110.png" /> fixed, tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070111.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070112.png" />.
+
Here, $\operatorname { SPSH } ( \Omega \times \Omega )$ stands for the functions $f ( z , w )$ on $\Omega \times \Omega$ that are plurisubharmonic in each of the variables $z$, $w$ separately, when the other is kept fixed. On strictly pseudo-convex domains $\Omega$, the symmetric Green function is negative and, for $w$ fixed, tends to $0$ as $z \rightarrow \partial \Omega$.
  
In general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070113.png" />, and there need not be equality, see [[#References|[a2]]]. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070114.png" /> need not be symmetric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070115.png" /> need not be a [[Fundamental solution|fundamental solution]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070116.png" />. However, on bounded convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070117.png" />. This is based on work of L. Lempert [[#References|[a20]]], [[#References|[a21]]] showing that on bounded convex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070118.png" /> the Kobayashi distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070119.png" /> (cf. [[Hyperbolic metric|Hyperbolic metric]]), the Lempert functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070120.png" /> and the Carathéodory distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070121.png" /> (cf. also [[Green function|Green function]]) coincide. The relation between these objects and the Green functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070122.png" /> is (see e.g. [[#References|[a10]]])
+
In general $W \leq G$, and there need not be equality, see [[#References|[a2]]]. In particular, $G$ need not be symmetric and $W$ need not be a [[Fundamental solution|fundamental solution]] of $M$. However, on bounded convex domains $G = W$. This is based on work of L. Lempert [[#References|[a20]]], [[#References|[a21]]] showing that on bounded convex domains in $\mathbf{C} ^ { n }$ the Kobayashi distance $K ( z , w )$ (cf. [[Hyperbolic metric|Hyperbolic metric]]), the Lempert functional $\delta ( z , w )$ and the Carathéodory distance $C ( z , w )$ (cf. also [[Green function|Green function]]) coincide. The relation between these objects and the Green functions on a domain $\Omega$ is (see e.g. [[#References|[a10]]])
  
<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/p130/p130070/p130070123.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { log } \operatorname { tanh } C ( z , w ) \leq W ( z , w ) \leq \end{equation*}
  
<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/p130/p130070/p130070124.png" /></td> </tr></table>
+
\begin{equation*} \leq G ( z , w ) \leq \operatorname { log } \operatorname { tanh } \delta ( z , w ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070125.png" /> is the Lempert functional
+
where $\delta$ is the Lempert functional
  
<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/p130/p130070/p130070126.png" /></td> </tr></table>
+
\begin{equation*} \delta ( z , w ) = \operatorname { inf } _ { f \in \mathcal{F} } \{ \operatorname { log } | \xi | : f ( \xi ) = z , f ( 0 ) = w \}, \end{equation*}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070127.png" /> the family of holomorphic mappings from the unit disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070128.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070129.png" />.
+
with $\mathcal{F}$ the family of holomorphic mappings from the unit disc in $\mathbf{C}$ to $\Omega$.
  
 
The Green function is instrumental in the following result of Z. Błocki and P. Pflug, [[#References|[a6]]], which is one of the first applications outside pluripotential theory: Every bounded hyperconvex domain is complete in the Bergman metric (cf. [[Bergman spaces|Bergman spaces]]).
 
The Green function is instrumental in the following result of Z. Błocki and P. Pflug, [[#References|[a6]]], which is one of the first applications outside pluripotential theory: Every bounded hyperconvex domain is complete in the Bergman metric (cf. [[Bergman spaces|Bergman spaces]]).
Line 108: Line 116:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Bedford,  "Survey of pluri-potential theory" , ''Several Complex Variables (Stockholm, 1987/8)'' , ''Math. Notes'' , '''38''' , Princeton Univ. Press  (1993)  pp. 48–97</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Bedford,  J.P. Demailly,  "Two counterexamples concerning the pluri-complex Green function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070130.png" />"  ''Indiana Univ. Math. J.'' , '''37'''  (1988)  pp. 865–867</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Bedford,  B.A. Taylor,  "The Dirichlet problem for a complex Monge–Ampère equation"  ''Invent. Math.'' , '''37'''  (1976)  pp. 1–44</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Bedford,  B.A. Taylor,  "A new capacity for plurisubharmonic functions"  ''Acta Math.'' , '''149'''  (1982)  pp. 1–40</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Z. Błocki,  "The complex Monge–Ampère equation in hyperconvex domain"  ''Ann. Scuola Norm. Sup. Pisa'' , '''23'''  (1996)  pp. 721–747</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Z. Błocki,  P. Pflug,  "Hyperconvexity and Bergman completeness"  ''Nagoya Math. J.'' , '''151'''  (1998)  pp. 221–225</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Bremermann,  "On the conjecture of equivalence of plurisubharmonic functions and Hartogs functions"  ''Math. Ann.'' , '''131'''  (1956)  pp. 76–86</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Caffarelli,  J.J. Kohn,  L. Nirenberg,  J. Spruck,  "The Dirichlet problem for nonlinear second order elliptic equations. II. Complex Monge–Ampère, and uniform elliptic, equations"  ''Commun. Pure Appl. Math.'' , '''38'''  (1985)  pp. 209–252</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  U. Cegrell,  "Discontinuité de l'opérateur de Monge Ampère complexe"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''296'''  (1983)  pp. 869–871</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  U. Cegrell,  "Capacities in complex analysis" , Vieweg  (1988)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  U. Cegrell,  "Pluricomplex energy"  ''Acta Math.'' , '''180'''  (1998)  pp. 187–217</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J.P. Demailly,  "Mesures de Monge–Ampère et mesures pluriharmoniques"  ''Math. Z.'' , '''194'''  (1987)  pp. 519–564</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  G. Herbort,  "The Bergman metric on hyperconvex domains"  ''Math. Z.'' , '''232'''  (1999)  pp. 183–196</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  B. Josefson,  "On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070131.png" />"  ''Ark. Mat.'' , '''16'''  (1978)  pp. 109–115</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  M. Klimek,  "Extremal plurisubharmonic functions and invariant pseudodistances"  ''Bull. Soc. Math. France'' , '''113'''  (1985)  pp. 231–240</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  M. Klimek,  "Pluripotential theory" , Clarendon Press/Oxford Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  S. Kołodziej,  "The logarithmic capacity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070132.png" />"  ''Ann. Polon. Math.'' , '''48'''  (1988)  pp. 253–267</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  S. Kołodziej,  "The complex Monge–Ampère equation"  ''Acta Math.'' , '''180'''  (1998)  pp. 69–117</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  P. Lelong,  "Les fonctions plurisousharmonique"  ''Ann. Sci. École Norm. Sup.'' , '''62'''  (1945)  pp. 301–338</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  L. Lempert,  "La métrique de Kobayashi et la représentation des domaines sur la boule"  ''Bull. Soc. Math. France'' , '''109'''  (1981)  pp. 427–474</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  L. Lempert,  "Holomorphic retracts and intrinsic metrics in convex domains"  ''Anal. Math.'' , '''8''' :  4  (1982)  pp. 257–261</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  K. Oka,  "Sur les fonctions analytiques de plusieurs variables VI. Domaines pseudoconvexes"  ''Tôhoku Math. J.'' , '''49'''  (1942)  pp. 15–52</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  A. Sadullaev,  "Plurisubharmonic measures and capacities on complex manifolds"  ''Russian Math. Surveys'' , '''36'''  (1981)  pp. 61–119  ''Uspekhi Mat. Nauk.'' , '''36'''  (1981)  pp. 53–105</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  J. Siciak,  "Extremal functions and capacities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130070/p130070133.png" />"  ''Sophia Kokyuroku Math.'' , '''14'''  (1982)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  E. Bedford,  "Survey of pluri-potential theory" , ''Several Complex Variables (Stockholm, 1987/8)'' , ''Math. Notes'' , '''38''' , Princeton Univ. Press  (1993)  pp. 48–97</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  E. Bedford,  J.P. Demailly,  "Two counterexamples concerning the pluri-complex Green function in $\mathbf{C} ^ { n }$"  ''Indiana Univ. Math. J.'' , '''37'''  (1988)  pp. 865–867</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E. Bedford,  B.A. Taylor,  "The Dirichlet problem for a complex Monge–Ampère equation"  ''Invent. Math.'' , '''37'''  (1976)  pp. 1–44</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  E. Bedford,  B.A. Taylor,  "A new capacity for plurisubharmonic functions"  ''Acta Math.'' , '''149'''  (1982)  pp. 1–40</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  Z. Błocki,  "The complex Monge–Ampère equation in hyperconvex domain"  ''Ann. Scuola Norm. Sup. Pisa'' , '''23'''  (1996)  pp. 721–747</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  Z. Błocki,  P. Pflug,  "Hyperconvexity and Bergman completeness"  ''Nagoya Math. J.'' , '''151'''  (1998)  pp. 221–225</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  H. Bremermann,  "On the conjecture of equivalence of plurisubharmonic functions and Hartogs functions"  ''Math. Ann.'' , '''131'''  (1956)  pp. 76–86</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  L. Caffarelli,  J.J. Kohn,  L. Nirenberg,  J. Spruck,  "The Dirichlet problem for nonlinear second order elliptic equations. II. Complex Monge–Ampère, and uniform elliptic, equations"  ''Commun. Pure Appl. Math.'' , '''38'''  (1985)  pp. 209–252</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  U. Cegrell,  "Discontinuité de l'opérateur de Monge Ampère complexe"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''296'''  (1983)  pp. 869–871</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  U. Cegrell,  "Capacities in complex analysis" , Vieweg  (1988)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  U. Cegrell,  "Pluricomplex energy"  ''Acta Math.'' , '''180'''  (1998)  pp. 187–217</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J.P. Demailly,  "Mesures de Monge–Ampère et mesures pluriharmoniques"  ''Math. Z.'' , '''194'''  (1987)  pp. 519–564</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  G. Herbort,  "The Bergman metric on hyperconvex domains"  ''Math. Z.'' , '''232'''  (1999)  pp. 183–196</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  B. Josefson,  "On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on $\mathbf{C} ^ { n }$" ''Ark. Mat.'' , '''16'''  (1978)  pp. 109–115</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  M. Klimek,  "Extremal plurisubharmonic functions and invariant pseudodistances"  ''Bull. Soc. Math. France'' , '''113'''  (1985)  pp. 231–240</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  M. Klimek,  "Pluripotential theory" , Clarendon Press/Oxford Univ. Press  (1991)</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  S. Kołodziej,  "The logarithmic capacity in $\mathbf{C} ^ { n }$" ''Ann. Polon. Math.'' , '''48'''  (1988)  pp. 253–267</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  S. Kołodziej,  "The complex Monge–Ampère equation"  ''Acta Math.'' , '''180'''  (1998)  pp. 69–117</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  P. Lelong,  "Les fonctions plurisousharmonique"  ''Ann. Sci. École Norm. Sup.'' , '''62'''  (1945)  pp. 301–338</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  L. Lempert,  "La métrique de Kobayashi et la représentation des domaines sur la boule"  ''Bull. Soc. Math. France'' , '''109'''  (1981)  pp. 427–474</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  L. Lempert,  "Holomorphic retracts and intrinsic metrics in convex domains"  ''Anal. Math.'' , '''8''' :  4  (1982)  pp. 257–261</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  K. Oka,  "Sur les fonctions analytiques de plusieurs variables VI. Domaines pseudoconvexes"  ''Tôhoku Math. J.'' , '''49'''  (1942)  pp. 15–52</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  A. Sadullaev,  "Plurisubharmonic measures and capacities on complex manifolds"  ''Russian Math. Surveys'' , '''36'''  (1981)  pp. 61–119  ''Uspekhi Mat. Nauk.'' , '''36'''  (1981)  pp. 53–105</td></tr><tr><td valign="top">[a24]</td> <td valign="top">  J. Siciak,  "Extremal functions and capacities in $\mathbf{C} ^ { n }$"  ''Sophia Kokyuroku Math.'' , '''14'''  (1982)</td></tr></table>

Revision as of 17:00, 1 July 2020

The natural brand of potential theory in the setting of function theory of several complex variables (cf. also Analytic function). The basic objects are plurisubharmonic functions (cf. also Plurisubharmonic function). These are studied much from the same perspective as subharmonic functions (cf. also Subharmonic function) are studied in potential theory on ${\bf R} ^ { n }$. General references are [a1], [a10], [a16], [a23].

A function $u$ on a domain $D \subset \mathbf{C} ^ { x }$ is called plurisubharmonic if it is subharmonic on $D$, viewed as a domain in $\mathbf{R} ^ { 2 n }$, and if the restriction of $u$ to every complex line in $D$ is subharmonic (cf. also Plurisubharmonic function; Subharmonic function). If $u$ is $C ^ { 2 }$ on a domain $D \subset \mathbf{C} ^ { x }$, then $u$ is plurisubharmonic if and only if

\begin{equation*} \left( \frac { \partial ^ { 2 } u } { \partial z _ { i } \partial \overline{z _ { j } }} \right) \end{equation*}

is a non-negative Hermitian matrix on $D$. One denotes the set of plurisubharmonic functions on a domain $D \subset \mathbf{C} ^ { x }$ by $\operatorname{PSH} ( D )$. Plurisubharmonic functions can be defined on domains in complex manifolds via local coordinates (cf. also Analytic manifold).

Plurisubharmonic functions are precisely the subharmonic functions invariant under a holomorphic change of coordinates. If $f$ is holomorphic on a domain $D$ in $\mathbf{C} ^ { n }$ (cf. also Analytic function), then $\operatorname { log } | f |$ is plurisubharmonic on $D$. Moreover, every plurisubharmonic function can locally be written as

\begin{equation*} \operatorname { limsup } _ { j \rightarrow \infty } \frac { 1 } { j } \operatorname { log } | f _ { j } |, \end{equation*}

for suitable holomorphic functions $f_j$, see [a7]. Plurisubharmonic functions were formally introduced by P. Lelong, [a19], and K. Oka, [a22], although related ideas stem from the end of the nineteenth century.

The analogue of the Laplace operator on domains in $\mathbf{C}$ is the Monge–Ampère operator:

\begin{equation*} M f = \operatorname { det } \left( \frac { \partial ^ { 2 } f } { \partial z _ { i } \partial \overline{z}_ { j } } \right) . \end{equation*}

This operator is originally only defined for $C ^ { 2 }$ plurisubharmonic functions (cf. also Monge–Ampère equation). Due to the non-linearity of $M$ it is impossible to extend it to a well-defined operator on all plurisubharmonic functions on a domain $D$ in such a way that $\operatorname { lim } _ { n \rightarrow \infty } M ( u _ { n } ) = M ( u )$ if $\{ u _ { n } \}$ is a decreasing sequence of plurisubharmonic functions with limit $u$, see [a9]. Nevertheless, the domain of $M$ can be enlarged to include all bounded plurisubharmonic functions, [a3]. The most recent result (as of 2000) in this direction is in [a11].

On strongly pseudo-convex domains $D$ (cf. also Pseudo-convex and pseudo-concave), the following Dirichlet problem for the Monge–Ampère operator was solved by E. Bedford and B.A. Taylor [a3]: Given $f$ continuous on $\partial D$ and $\phi$ continuous on $D$, there exists a continuous plurisubharmonic function $u$ on $D$, continuous up to the boundary of $D$, such that

\begin{equation} \tag{a1} \left\{ \begin{array} { c } { M ( u ) = \phi } & {\text { on } D , } \\ { u |_{ \partial D = f.} } \end{array} \right. \end{equation}

This result has been extended by weakening the conditions on $D$, and replacing $\phi$ by certain positive measures; see e.g. [a5], [a18]. In [a11], large classes of plurisubharmonic functions on which the Monge–Ampère operator is well defined are determined and necessary and sufficient conditions on a positive measure $\phi$ are given, so that the problem (a1) has a solution within such a class.

The regularity of this Dirichlet problem is quite bad. The following example is due to T. Gamelin and N. Sibony: Let $D$ be the unit ball in $\mathbf{C} ^ { 2 }$,

\begin{equation*} f ( z _ { 1 } , z _ { 2 } ) = \left( | z _ { 1 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 } = \left( | z _ { 2 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 }, \end{equation*}

\begin{equation*} ( z _ { 1 } , z _ { 2 } ) \in \partial D. \end{equation*}

Then the function

\begin{equation*} u ( z _ { 1 } , z _ { 2 } ) = \left\{ \begin{array} { c l } { 0 } & { \text { if } | z _ { 1 } | ^ { 2 } , | z _ { 2 } | ^ { 2 } < \frac { 1 } { 2 } } ,\\ { \operatorname { max } \left\{ \left( | z _ { 1 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 } \right. }, & { \left. \left( | z _ { 2 } | ^ { 2 } - \frac { 1 } { 2 } \right) ^ { 2 } \right\} } \\ { \text { elsewhere on } D, } \end{array} \right. \end{equation*}

satisfies $M u = 0$ on $D$, $u| _ { \partial D } = f$.

However, if $f$ and $\phi$ are both smooth and $\phi > 0$ on $D$, then was shown in [a8] that there exists a smooth $u$ satisfying (a1).

There have been defined several capacity functions (cf. also Capacity; Capacity potential) on $\mathbf{C} ^ { n }$ that all share the property that sets of capacity $0$ are precisely the pluripolar sets, i.e. sets that are locally contained in the $- \infty$ locus of plurisubharmonic functions. See [a4], [a10], [a23], [a24]. Firstly, the classical construction of logarithmic capacity carries over: Let

\begin{equation*} \mathcal{L} = \{ u \in \operatorname { PSH } ( \mathbf{C} ^ { n } ) : u - \operatorname { log } ( 1 + | z | ) = O ( 1 ) ( z \rightarrow \infty ) \}. \end{equation*}

For a bounded set $E$ in $\mathbf{C} ^ { n }$, define the Green function with pole at infinity by

\begin{equation*} L _ { E } ( z ) = \operatorname { sup } \{ v ( z ) : v \in \mathcal{L} , v \leq 0 \text { on } E \}. \end{equation*}

Set $L _ { E } ^ { * } ( z ) = \operatorname { limsup } _ { w \rightarrow z } L _ { E } ( w )$, the upper semi-continuous regularization of $L_{E}$. Then either $L _ { E } ^ { * } \equiv \infty$ or $L \in \operatorname { PSH } ( \mathbf{C} ^ { n } )$. For $u \in \mathcal{L}$ one defines the Robin function on $\mathbf{C} ^ { n }$ by

\begin{equation*} \rho _ { u } ( z ) = \limsup _ { t \in \text{C} } ( u ( t z ) - \operatorname { log } | t z | ). \end{equation*}

Next the logarithmic capacity of $E$ is defined as

\begin{equation*} \operatorname { Cap } ( E ) = \operatorname { exp } \left( - \operatorname { sup } _ { z \in \text{C} ^ { n } } \rho _ { L _ { E } } ( z ) \right). \end{equation*}

It is, however, a non-trivial result that is a Choquet capacity (cf. Capacity), see [a17]. Another important (relative) capacity is the Monge–Ampère capacity introduced by Bedford and Taylor, [a4]. It is defined as follows: Let $\Omega$ be a strictly pseudo-convex domain in $\mathbf{C} ^ { n }$ and let $K$ be a compact subset of $\Omega$. The Monge–Ampère capacity of $K$ relative to $\Omega$ is

\begin{equation*} C ( K , \Omega ) = \end{equation*}

\begin{equation*} = \operatorname { sup } \left\{ \int _ { K } M ( u ) d V : u \in \operatorname { PSH } ( \Omega ) , 0 < u < 1 \right\}. \end{equation*}

If $E \subset \Omega$ is an arbitrary subset, one defines

\begin{equation*} C ( E , \Omega ) = \operatorname { sup } \{ C ( K ) : K \subset \Omega \}. \end{equation*}

It is shown in [a4] that plurisubharmonic functions are quasi-continuous, i.e. continuous outside an open set of arbitrarily small capacity. Another application is a new proof of the following Josefson theorem [a14]: If $E \subset \mathbf{C} ^ { n }$ is pluripolar, then there exists a $u \in \operatorname { PSH } ( \mathbf{C} ^ { n } )$ with $u | _ { E } = - \infty$.

Although there is no analogue of the Riesz decomposition theorem (cf. also Riesz theorem; Riesz decomposition theorem), there are notions of Green functions.

1) The (Klimek or pluricomplex) Green function on a domain $\Omega \subset {\bf C} ^ { n }$ with pole at $w \in \Omega$ is the function

\begin{equation*} G ( z , w ) = \end{equation*}

\begin{equation*} = \operatorname { sup } \left\{ h ( z ) : \begin{array}{ c c } { h \in \operatorname{PSH}(\Omega), \, h<0,} \\{h ( \zeta ) - \operatorname { log } \| \zeta - w \| = O ( 1 ) ( \zeta \rightarrow w )} \end{array} \right\}. \end{equation*}

If $\Omega$ is hyperconvex, i.e. pseudo-convex and admitting a bounded plurisubharmonic exhaustion function, then $G ( z , w )$ is negative and, for $w$ fixed, tends to $0$ if $z \rightarrow \partial \Omega$. Moreover, $M ( G ( z , w ) ) = ( 2 \pi ) ^ { n } \delta _ { w }$, where $\delta _ { W }$ is the Dirac distribution at $w$; see [a12], [a15] for more details.

2) The symmetric Green function on a domain $\Omega \subset {\bf C} ^ { n }$ is the function

\begin{equation*} W ( z , w ) = \operatorname { sup } h ( z , w ) \end{equation*}

where the supremum is taken over

\begin{equation*} h \in \operatorname { SPSH } ( \Omega \times \Omega ) , h < 0, \end{equation*}

\begin{equation*} h ( z , w ) - \operatorname { log } \| z - w \| \leq \end{equation*}

\begin{equation*} \leq - \operatorname { log } ( \operatorname { max } \{ \operatorname { dist } ( z , \partial \Omega ) , \operatorname { dist } ( w , \partial \Omega ) \} ). \end{equation*}

Here, $\operatorname { SPSH } ( \Omega \times \Omega )$ stands for the functions $f ( z , w )$ on $\Omega \times \Omega$ that are plurisubharmonic in each of the variables $z$, $w$ separately, when the other is kept fixed. On strictly pseudo-convex domains $\Omega$, the symmetric Green function is negative and, for $w$ fixed, tends to $0$ as $z \rightarrow \partial \Omega$.

In general $W \leq G$, and there need not be equality, see [a2]. In particular, $G$ need not be symmetric and $W$ need not be a fundamental solution of $M$. However, on bounded convex domains $G = W$. This is based on work of L. Lempert [a20], [a21] showing that on bounded convex domains in $\mathbf{C} ^ { n }$ the Kobayashi distance $K ( z , w )$ (cf. Hyperbolic metric), the Lempert functional $\delta ( z , w )$ and the Carathéodory distance $C ( z , w )$ (cf. also Green function) coincide. The relation between these objects and the Green functions on a domain $\Omega$ is (see e.g. [a10])

\begin{equation*} \operatorname { log } \operatorname { tanh } C ( z , w ) \leq W ( z , w ) \leq \end{equation*}

\begin{equation*} \leq G ( z , w ) \leq \operatorname { log } \operatorname { tanh } \delta ( z , w ), \end{equation*}

where $\delta$ is the Lempert functional

\begin{equation*} \delta ( z , w ) = \operatorname { inf } _ { f \in \mathcal{F} } \{ \operatorname { log } | \xi | : f ( \xi ) = z , f ( 0 ) = w \}, \end{equation*}

with $\mathcal{F}$ the family of holomorphic mappings from the unit disc in $\mathbf{C}$ to $\Omega$.

The Green function is instrumental in the following result of Z. Błocki and P. Pflug, [a6], which is one of the first applications outside pluripotential theory: Every bounded hyperconvex domain is complete in the Bergman metric (cf. Bergman spaces).

A more elementary proof is given in [a13].

References

[a1] E. Bedford, "Survey of pluri-potential theory" , Several Complex Variables (Stockholm, 1987/8) , Math. Notes , 38 , Princeton Univ. Press (1993) pp. 48–97
[a2] E. Bedford, J.P. Demailly, "Two counterexamples concerning the pluri-complex Green function in $\mathbf{C} ^ { n }$" Indiana Univ. Math. J. , 37 (1988) pp. 865–867
[a3] E. Bedford, B.A. Taylor, "The Dirichlet problem for a complex Monge–Ampère equation" Invent. Math. , 37 (1976) pp. 1–44
[a4] E. Bedford, B.A. Taylor, "A new capacity for plurisubharmonic functions" Acta Math. , 149 (1982) pp. 1–40
[a5] Z. Błocki, "The complex Monge–Ampère equation in hyperconvex domain" Ann. Scuola Norm. Sup. Pisa , 23 (1996) pp. 721–747
[a6] Z. Błocki, P. Pflug, "Hyperconvexity and Bergman completeness" Nagoya Math. J. , 151 (1998) pp. 221–225
[a7] H. Bremermann, "On the conjecture of equivalence of plurisubharmonic functions and Hartogs functions" Math. Ann. , 131 (1956) pp. 76–86
[a8] L. Caffarelli, J.J. Kohn, L. Nirenberg, J. Spruck, "The Dirichlet problem for nonlinear second order elliptic equations. II. Complex Monge–Ampère, and uniform elliptic, equations" Commun. Pure Appl. Math. , 38 (1985) pp. 209–252
[a9] U. Cegrell, "Discontinuité de l'opérateur de Monge Ampère complexe" C.R. Acad. Sci. Paris Sér. I Math. , 296 (1983) pp. 869–871
[a10] U. Cegrell, "Capacities in complex analysis" , Vieweg (1988)
[a11] U. Cegrell, "Pluricomplex energy" Acta Math. , 180 (1998) pp. 187–217
[a12] J.P. Demailly, "Mesures de Monge–Ampère et mesures pluriharmoniques" Math. Z. , 194 (1987) pp. 519–564
[a13] G. Herbort, "The Bergman metric on hyperconvex domains" Math. Z. , 232 (1999) pp. 183–196
[a14] B. Josefson, "On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on $\mathbf{C} ^ { n }$" Ark. Mat. , 16 (1978) pp. 109–115
[a15] M. Klimek, "Extremal plurisubharmonic functions and invariant pseudodistances" Bull. Soc. Math. France , 113 (1985) pp. 231–240
[a16] M. Klimek, "Pluripotential theory" , Clarendon Press/Oxford Univ. Press (1991)
[a17] S. Kołodziej, "The logarithmic capacity in $\mathbf{C} ^ { n }$" Ann. Polon. Math. , 48 (1988) pp. 253–267
[a18] S. Kołodziej, "The complex Monge–Ampère equation" Acta Math. , 180 (1998) pp. 69–117
[a19] P. Lelong, "Les fonctions plurisousharmonique" Ann. Sci. École Norm. Sup. , 62 (1945) pp. 301–338
[a20] L. Lempert, "La métrique de Kobayashi et la représentation des domaines sur la boule" Bull. Soc. Math. France , 109 (1981) pp. 427–474
[a21] L. Lempert, "Holomorphic retracts and intrinsic metrics in convex domains" Anal. Math. , 8 : 4 (1982) pp. 257–261
[a22] K. Oka, "Sur les fonctions analytiques de plusieurs variables VI. Domaines pseudoconvexes" Tôhoku Math. J. , 49 (1942) pp. 15–52
[a23] A. Sadullaev, "Plurisubharmonic measures and capacities on complex manifolds" Russian Math. Surveys , 36 (1981) pp. 61–119 Uspekhi Mat. Nauk. , 36 (1981) pp. 53–105
[a24] J. Siciak, "Extremal functions and capacities in $\mathbf{C} ^ { n }$" Sophia Kokyuroku Math. , 14 (1982)
How to Cite This Entry:
Pluripotential theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pluripotential_theory&oldid=50352
This article was adapted from an original article by Jan Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article