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 [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)