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) |
Plurisubharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plurisubharmonic_function&oldid=48192