Subharmonic function

A function $ u = u( x): D \rightarrow [ - \infty , \infty ) $ of the points $ x = ( x _ {1} \dots x _ {n} ) $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, defined in a domain $ D \subset \mathbf R ^ {n} $ and possessing the following properties: 1) $ u( x) $ is upper semi-continuous in $ D $; 2) for any point $ x _ {0} \in D $ there exist values $ r > 0 $, arbitrarily small, such that

$$ u( x _ {0} ) \leq I( u; x _ {0} , r) = \ \frac{1}{s _ {n} r ^ {n-1} } \int\limits _ {S( x _ {0} ,r) } u( x) d \sigma ( x), $$

where $ I( u; x _ {0} , r) $ is the mean value of the function $ u( x) $ over the area of the sphere $ S( x _ {0} , r) $ with centre $ x _ {0} $ of radius $ r $ and $ s _ {n} = 2 \pi ^ {n/2} \Gamma ( n/2) $ is the area of the unit sphere in $ \mathbf R ^ {n} $; and 3) $ u( x) \not\equiv - \infty $ (this condition is sometimes dropped). In this definition of a subharmonic function, the mean value $ I( u; x _ {0} , r) $ over the area of the sphere can be replaced by the mean value

$$ J( u; x _ {0} , r) = \frac{1}{v _ {n} r ^ {n} } \int\limits _ {B( x _ {0} ,r) } u( x) dv( x) $$

over the volume of the ball $ B( x _ {0} , r) $, where $ v _ {n} = s _ {n} /n $ is the volume of the unit ball in $ \mathbf R ^ {n} $.

An equivalent definition of a subharmonic function, which explains the name "subharmonic function" is obtained by replacing condition 2) by 2'): If $ \Delta $ is a relatively-compact subdomain of $ D $ and $ v( x) $ is a harmonic function in $ \Delta $ that is continuous on the closure $ \overline \Delta \; $ and is such that

$$ \tag{1 } u( x) \leq v( x) $$

on the boundary $ \partial \Delta $, then the inequality (1) holds everywhere in $ \Delta $ ($ v( x) $ is called a harmonic majorant of the subharmonic function $ u( x) $ in $ \Delta $). If the function $ u( x) $ belongs to the class $ C ^ {2} ( D) $, then for it to be subharmonic in $ D $ it is necessary and sufficient that the result of applying the Laplace operator, $ \Delta u $, be non-negative in $ D $.

The idea of a subharmonic function was expounded in essence by H. Poincaré in the balayage method. Subharmonic functions are also found in the work of F. Hartogs [1] on the theory of analytic functions of several complex variables; the systematic study of subharmonic functions began with the work of F. Riesz . The close connection between subharmonic functions and analytic functions $ f( z) $ of one or several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n \geq 1 $, and the consequent possible use of subharmonic functions for the study of analytic functions is related to the fact that the modulus $ | f( z) | $ and the logarithm $ \mathop{\rm ln} | f( z) | $ of the modulus of an analytic function are subharmonic functions. On the other hand, condition 2') shows that subharmonic functions can be considered as the analogue of convex functions of one real variable (cf. Convex function (of a real variable)).

Simple properties of subharmonic functions.

1) If $ u _ {1} \dots u _ {m} $ are subharmonic functions in $ D $ and $ \lambda _ {1} \dots \lambda _ {m} $ are non-negative numbers, then the linear combination $ \sum _ {k=1} ^ {m} \lambda _ {k} u _ {k} $ is a subharmonic function in $ D $.

2) The upper envelope $ \sup \{ {u _ {k} ( x) } : {1 \leq k \leq m } \} $ of a finite family of subharmonic functions $ \{ u _ {k} \} _ {k=1} ^ {m} $ is a subharmonic function. If the upper envelope of an infinite family of subharmonic functions is upper semi-continuous, then it is also a subharmonic function.

3) Uniformly-converging and monotone-decreasing sequences of subharmonic functions converge to subharmonic functions.

4) If $ u( x) $ is a subharmonic function in $ D $ and $ \phi ( u) $ is a convex non-decreasing function on the domain of values $ E $ of the function $ u $ in $ D $, or if $ u( x) $ is a harmonic function in $ D $ and $ \phi ( u) $ is a convex function on $ E $, then $ \phi ( u( x)) $ is a subharmonic function in $ D $. In particular, if $ u( x) $ is a subharmonic function in $ D $, then $ e ^ {\lambda u( x) } $, $ \lambda > 0 $, and $ [ u ^ {+} ( x)] ^ {k} $, $ k \geq 1 $, where $ u ^ {+} ( x) = \max \{ u( x), 0 \} $, are subharmonic functions in $ D $; if $ u( x) $ is a harmonic function in $ D $, then $ | u( x) | ^ {k} $, $ k \geq 1 $, is a subharmonic function in $ D $.

5) The maximum principle: If $ u( x) $ is a subharmonic function in $ D $ and for any boundary point $ \xi \in \partial D $ and any $ \epsilon > 0 $ there is a neighbourhood $ V = V( \xi ) $ such that $ u( x) < \epsilon $ in $ D \cap V $, then either $ u( x) < 0 $ or $ u( x) \equiv 0 $ in $ D $. This property also holds for unbounded domains $ D $, where for $ \xi = \infty \in \partial D $ the neighbourhood $ V $ is taken to mean the exterior of a sphere, $ V = V( \infty ) = \{ {x \in \mathbf R ^ {n} } : {| x | > R } \} $.

6) If $ u( x) $ is a subharmonic function in a domain $ D $ of the complex plane $ \mathbf C $ and $ z = z( w) $ is a holomorphic mapping of a domain $ D ^ \prime \subset \mathbf C ^ {n} $ into $ D $, then $ u( z( w)) $ is a subharmonic function in $ D ^ \prime $.

7) A function $ u( x) $ is harmonic in the domain $ D \subset \mathbf R ^ {n} $ if and only if $ u( x) $ and $ - u( x) $ are subharmonic functions in $ D $ (cf. Harmonic function).

8) If $ u( x) $ is a subharmonic function on the whole plane $ \mathbf R ^ {2} $ that is bounded above, then $ u( x) = \textrm{ const } $ (in $ \mathbf R ^ {n} $ when $ n \geq 3 $ this property does not hold).

The Perron method for solving the Dirichlet problem for harmonic functions is based on the properties 2), 5) and 7).

The convexity properties of the mean values of subharmonic functions are of great importance: If $ u( x) $ is a subharmonic function in an annulus $ r _ {1} \leq r = | x - x _ {0} | \leq r _ {2} $, $ 0 < r _ {1} < r _ {2} $, then the mean values $ I( u; x _ {0} , r) $, $ J( u; x _ {0} , r) $, as well as the maximum

$$ M( u; x _ {0} , r) = \max \{ {u( x ) } : {| x- x _ {0} | = r } \} , $$

are convex functions in $ \mathop{\rm ln} r $ for $ n= 2 $, or in $ r ^ {2-n} $ for $ n \geq 3 $, on the interval $ r _ {1} \leq r \leq r _ {2} $; if $ u( x) $ is a subharmonic function in the disc (ball) $ 0 \leq r = | x- x _ {0} | \leq r _ {2} $, then, moreover, $ I( u; x _ {0} , r) $ and $ J( u; x _ {0} , r) $ are continuous non-decreasing functions in $ r $ on the interval $ 0 \leq r \leq r _ {2} $; moreover

$$ I( u; x _ {0} , 0) = J( u; x _ {0} , 0) = u( x _ {0} ); $$

in the latter instance,

$$ u( x _ {0} ) \leq J( u; x _ {0} , r) \leq I( u; x _ {0} , r) $$

for $ 0 \leq r \leq r _ {2} $. The mean values $ I( u; x _ {0} , r) $ and $ J( u; x _ {0} , r) $, considered as functions of the point $ x _ {0} $ for fixed $ u $ and $ r $, are subharmonic functions in the corresponding subdomain $ D ^ \prime \subset D $, and $ J( u; x _ {0} , r) $ is continuous. By forming iterations of a sufficiently-high order,

$$ u _ {m} ^ {(k)} ( x) = J \left ( u _ {m} ^ {( k- 1)} ; x, \frac{1}{m} \right ) ,\ \ u _ {m} ^ {( 0)} ( x) = u( x), $$

it is possible to obtain a monotone-decreasing sequence of subharmonic functions $ \{ u _ {m} ^ {(k)} ( x) \} _ {m= m _ {0} } ^ \infty $ of any degree of smoothness that converges, as $ m \rightarrow \infty $, to an arbitrarily given subharmonic function $ u( x) $.

The Newton potential and logarithmic potential of non-negative masses, when written with a minus sign, are subharmonic functions everywhere in the space $ \mathbf R ^ {n} $. On the other hand, one of the basic theorems in the theory of subharmonic functions is the Riesz local representation theorem: An arbitrary subharmonic function can be represented as the sum of a harmonic function and a potential with a minus sign (see ). More precisely, if $ u( x) $ is a subharmonic function in a domain $ D \subset \mathbf R ^ {n} $, then there exists a unique non-negative Borel measure $ \mu $ on $ D $ (a measure associated with $ u( x) $, or a Riesz measure) such that for any compact set $ E \subset D $ the representation

$$ \tag{2 } u( x) = \int\limits _ { E } K( x- \xi ) d \mu ( \xi ) + h( x) $$

is valid, where $ K( x- \xi ) = \mathop{\rm ln} | x- \xi | $ when $ n= 2 $, $ K( x- \xi ) = - | x- \xi | ^ {2-n} $ when $ n \geq 3 $, and where $ h( x) $ is a harmonic function in the interior of $ E $. The Riesz theorem establishes a close link between the theory of subharmonic functions and potential theory.

If $ E $ is a regular closed domain $ \overline{G}\; $ bounded, for example, by a Lyapunov surface, and having a Green function $ g( x, \xi ) $, then as well as (2) a representation using the Green potential is valid:

$$ \tag{3 } u( x) = - \int\limits _ {\overline{G}\; } g( x, \xi ) d \mu ( \xi ) + w( x) , $$

where $ w( x) $ is the least harmonic majorant of the subharmonic function $ u( x) $ in the domain $ G $.

A representation in the form (3), generally speaking, does not hold in the whole domain of definition $ D $ of $ u( x) $, and in the theory of subharmonic functions great importance is attached to the question of distinguishing the class of subharmonic functions $ u( x) $ that allow a representation (3) in the whole domain $ D $, i.e. the question of distinguishing the class $ A $ of subharmonic functions $ u( x) $ that have a harmonic majorant in the whole domain $ D $. For example, if $ D = B( 0, R) $ is a ball (disc) and there exists a constant $ C = C( u) $ such that

$$ \tag{4 } \int\limits _ {S( 0,R) } u ^ {+} ( r \xi ) d \sigma ( \xi ) < C( u) < + \infty ,\ \ 0 < r < 1 , $$

then $ u( x) $ allows a representation (3) in $ D $, and the least harmonic majorant $ w( x) $ is, in turn, represented by a Poisson–Stieltjes integral:

$$ \tag{5 } w( x) = \int\limits \frac{R ^ {n-2} ( R ^ {2} - | x | ^ {2} ) }{| x- \xi | ^ {n} } d \nu ( \xi ), $$

where $ \nu $ is a Borel measure of arbitrary sign concentrated on the boundary sphere (circle) $ \partial D = S( 0, R) $ (a boundary measure) and normalized by the condition $ \int d \nu ( \xi ) = 1 $.

With regard to (5), it is important in practical applications to know under which conditions the boundary measure $ \nu $ has only a negative singular component, i.e. under which conditions the component $ \nu ^ {+} $ in the canonical decomposition $ \nu = \nu ^ {+} - \nu ^ {-} $ is absolutely continuous. This question is answered by introducing the class of strictly-subharmonic functions (see [11]–[13], [15], as well as [10], where generalizations are also examined). Let $ \psi ( y) $ be an increasing concave function in $ y $ for which $ \lim\limits _ {y \rightarrow + \infty } y/ \psi ( y) = + \infty $. A function $ u( x) $, $ x \in D $, is said to be strictly subharmonic relative to $ \psi ( y) $ if $ \psi ( u( x)) $ is a subharmonic function. For example, logarithmically-subharmonic functions (cf. Logarithmically-subharmonic function) $ u( x) \geq 0 $, for which $ \mathop{\rm ln} u( x) $ is a subharmonic function, belong to the class of strictly-subharmonic functions. If condition (4) is fulfilled for a strictly-subharmonic function $ u( x) $ in the ball $ D $, then $ u( x) $ can be represented in the form (3) in $ D $, and the boundary measure $ \nu $ is characterized by the decomposition

$$ d \nu ( \xi ) = u( \xi ) d \sigma ( \xi ) - d \nu ^ {-} ( \xi ) ,\ \ \xi \in \partial D, $$

where $ u( \xi ) $ are the radial boundary values of the function $ u( x) $ (which exist almost everywhere with respect to the Lebesgue measure on the sphere $ \partial D = S( 0, R) $), and $ \nu ^ {-} \geq 0 $ is the singular component of the measure $ \nu $.

Subharmonic functions of class $ A $ in the ball $ D $ have radial boundary values almost everywhere on $ \partial D = S( 0, R) $. However, examples have been constructed of bounded, continuous subharmonic functions in $ D $ that do not have non-tangential boundary values anywhere on $ \partial D $, a phenomenon that does not occur for harmonic functions. For non-tangential boundary values to exist, apart from (4) further conditions have to be imposed on the associated measure $ \mu $ in $ D $ (see, for example, [14]).

One of the essential questions in the theory of subharmonic functions and its applications is the characterization of the boundary properties of functions of different subclasses of the class $ A $. The general method of introducing these subclasses consists of the fact that for strictly-subharmonic functions $ u( x) $ relative to a concave function $ \psi ( y) $, any non-decreasing function $ \phi ( y) $ is examined that is convex relative to $ \psi ( y) $ and that is such that $ \lim\limits _ {y \rightarrow + \infty } \phi ( y)/ \psi ( y) = + \infty $, and the class $ A _ \phi $ is introduced. For a sphere it is defined by the condition

$$ \int\limits _ {S( 0,R) } \phi ^ {+} ( u( r \xi )) d \sigma ( \xi ) < C( u, \phi ) < + \infty ,\ \ 0 < r < 1. $$

For the boundary properties of subharmonic functions, see [3]–[5], [10]–[16].

For functions that can be represented as the difference between two subharmonic functions, the concept of characteristic in the sense of R. Nevanlinna has been introduced, and the theory of functions of bounded characteristic (cf. Function of bounded characteristic) has been generalized (see [3], ).

A distinctive generalization of the theory of entire functions (cf. Entire function) is the theory of subharmonic functions in the whole space $ \mathbf R ^ {n} $. Here, generalizations of the Weierstrass and Hadamard classical representation theorems of entire functions have been obtained, along with the theory of the growth and value distribution, the theory of asymptotic values and asymptotic paths, etc. (see ).

In the theory of analytic functions of several complex variables, the study of the subclasses of plurisubharmonic functions and pluriharmonic functions (cf. Plurisubharmonic function; Pluriharmonic function) is of considerable importance (see [17]). For axiomatic generalizations of subharmonic functions, see [9].

References

[1]	F. Hartogs, "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen welche nach Potenzen einer Veränderlichen fortschreiten" Math. Ann. , 62 (1906) pp. 1–88
[2a]	F. Riesz, "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel I" Acta Math. , 48 (1926) pp. 329–343
[2b]	F. Riesz, "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel II" Acta Math. , 54 (1930) pp. 321–360
[3]	I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian)
[4]	I.I. Privalov, "Boundary properties of single-valued analytic functions" , Moscow (1941) (In Russian)
[5]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[6]	T. Radó, "Subharmonic functions" , Springer (1937)
[7a]	W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)
[7b]	W.K. Hayman, "Subharmonic functions" , 2 , Acad. Press (1989)
[8]	M. Brelot, "Etude des fonctions sousharmoniques au voisinage d'un point" , Hermann (1934)
[9]	M. Brélot, "Lectures on potential theory" , Tata Inst. (1967)
[10]	M. Heins, "Hardy classes on Riemann surfaces" , Springer (1969)
[11]	E.D. Solomentsev, "On some classes of subharmonic functions" Izv. Akad. Nauk SSSR , 5–6 (1938) pp. 571–582 (In Russian)
[12]	I.I. Privalov, P.I. Kuznetsov, "Sur les problèmes limites et les classes différentes de fonctions harmoniques et sousharmoniques définies dans une domaine arbitraire" Mat. Sb. , 6 : 3 (1939) pp. 345–376 (In Russian) (French abstract)
[13]	E.D. Solomentsev, "Classes of functions subharmonic on a half-space" Vestnik Moskov. Gos. Univ. Ser. Mat.-Mekh. Astron. : 5 (1959) pp. 73–91 (In Russian)
[14]	E.D. Solomentsev, "On boundary values of subharmonic functions" Czech. Math. J. , 8 : 4 (1958) pp. 520–536 (In Russian) (French abstract)
[15]	L. Gårding, L. Hormander, "Strongly subharmonic functions" Math. Scand. , 15 (1964) pp. 93–96
[16]	E.D. Solomentsev, "Harmonic and subharmonic functions and their generalization" Itogi Nauk Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962 (1964) pp. 83–100 (In Russian)
[17]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

Axiomatic potential theory can be founded upon the properties 1) and 3), completed by some additional properties of the set $ S $ of negative subharmonic functions: a) the upper semi-continuous regularization of the supremum of a subset of $ S $ is subharmonic; b) (the Riesz decomposition property) for any $ u , v, w \in S $, $ u \geq v+ w $, there exist $ v ^ \prime , w ^ \prime \in S $ such that $ u= v ^ \prime + w ^ \prime $, $ v ^ \prime \geq v $, $ w ^ \prime \geq w $; and c) any $ u \in S $ is the limit of a decreasing sequence in a sufficiently nice subcone of continuous subharmonic functions. See [a1].

References

[a1]	J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2]	L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) (Translated from Russian)
[a3]	O.D. Kellogg, "Foundations of potential theory" , Dover, reprint (1954) pp. 315ff (Re-issue: Springer, 1967)