# Plurisubharmonic function

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 , . 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 , .