View source for Pluripotential theory

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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]]].

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

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

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]]).

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

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

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.

The analogue of the [[Laplace operator|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|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]]].

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

\begin{equation} \tag{a1} \left\{ \begin{array} { c } { M ( u ) = \phi } &amp; {\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. [[#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.

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

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 $\mathbf{C} ^ { n }$ 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

\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 <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

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

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

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

It is shown in [[#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.

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&lt;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|Dirac distribution]] at $w$; see [[#References|[a12]]], [[#References|[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 &lt; 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 [[#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]]])

\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, [[#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]]).

A more elementary proof is given in [[#References|[a13]]].

====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 $\mathbf{C} ^ { n }$"  ''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>