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)