Plurisubharmonic function

A real-valued function , , of complex variables in a domain of the complex space , , that satisfies the following conditions: 1) is upper semi-continuous (cf. Semi-continuous function) everywhere in ; and 2) is a subharmonic function of the variable in each connected component of the open set for any fixed points , . A function is called a plurisuperharmonic function if is plurisubharmonic. The plurisubharmonic functions for constitute a proper subclass of the class of subharmonic functions, while these two classes coincide for . The most important examples of plurisubharmonic functions are , , , , where is a holomorphic function in .

For an upper semi-continuous function , , to be plurisubharmonic in a domain , it is necessary and sufficient that for every fixed , , , there exists a number such that the following inequality holds for :

The following criterion is more convenient for functions of class : is a plurisubharmonic function in if and only if the Hermitian form (the Hessian of , cf. Hessian of a function)

is positive semi-definite at each point .

The following hold for plurisubharmonic functions, in addition to the general properties of subharmonic functions: a) is plurisubharmonic in a domain if and only if is a plurisubharmonic function in a neighbourhood of each point ; 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) is a plurisubharmonic function in a domain if and only if it can be represented as the limit of a decreasing sequence of plurisubharmonic functions of the classes , respectively, where are domains such that and ; e) for any point the mean value

over a sphere of radius , where is the area of the unit sphere in , is an increasing function of that is convex with respect to on the segment , if the sphere

is located in , in which case ; f) a plurisubharmonic function remains plurisubharmonic under holomorphic mappings; g) if is a continuous plurisubharmonic function in a domain , if is a closed connected analytic subset of (cf. Analytic set) and if the restriction attains a maximum on , then on .

The following proper subclasses of the class of plurisubharmonic functions are also significant for applications. A function is called strictly plurisubharmonic if there exists a convex increasing function , ,

such that is a plurisubharmonic function. In particular, for 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 , as well as in more general analytic spaces [1]–[4], [7]. For example, the class of Hartogs functions is defined as the smallest class of real-valued functions in containing all functions , where is a holomorphic function in , and closed under the following operations:

) , imply ;

) , for every domain , imply ;

) , , imply ;

) , imply ;

) for every subdomain implies .

Upper semi-continuous Hartogs functions are plurisubharmonic, but not every plurisubharmonic function is a Hartogs function. If is a domain of holomorphy, the classes of upper semi-continuous Hartogs functions and plurisubharmonic functions in 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 is strictly plurisubharmonic if and only if the complex Hessian is a positive-definite Hermitian form on .

The Hessian has also an interpretation for arbitrary plurisubharmonic functions . For every , 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 denote the space of compactly-supported differential forms on of degree in and degree in (cf. Differential form). The exterior differential operators , and are defined by:

The forms in the kernel of are called closed, the forms in the image of are called exact. As , the set of exact forms is contained in the set of closed forms. A -form is called positive of degree if for every system of -forms , , the -form , with and the Euclidean volume element.

Let , . A -current on is a linear form on with the property that for every compact set there are constants such that for and , where . The operators are extended via duality; e.g., if is a -current, then . Closed and exact currents are defined as for differential forms. A -current is called positive if for every system of -forms as above and for every ,

A -form gives rise to a -current via integration: . A complex manifold of dimension gives rise to a positive closed -current on , the current of integration along :

The current of integration has also been defined for analytic varieties in (cf. Analytic manifold): one defines the current of integration for the set of regular points of on and shows that it can be extended to a positive closed current on . A plurisubharmonic function is in , hence identifies with a -current. Therefore is a -current, which turns out to be positive and closed. Conversely, a positive closed -current is locally of the form . The current of integration on an irreducible variety of the form , where is a holomorphic function with gradient not identically vanishing on , equals . 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)