Subharmonic function
A function of the points
of a Euclidean space
,
, defined in a domain
and possessing the following properties: 1)
is upper semi-continuous in
; 2) for any point
there exist values
, arbitrarily small, such that
![]() |
where is the mean value of the function
over the area of the sphere
with centre
of radius
and
is the area of the unit sphere in
; and 3)
(this condition is sometimes dropped). In this definition of a subharmonic function, the mean value
over the area of the sphere can be replaced by the mean value
![]() |
over the volume of the ball , where
is the volume of the unit ball in
.
An equivalent definition of a subharmonic function, which explains the name "subharmonic function" is obtained by replacing condition 2) by 2'): If is a relatively-compact subdomain of
and
is a harmonic function in
that is continuous on the closure
and is such that
![]() | (1) |
on the boundary , then the inequality (1) holds everywhere in
(
is called a harmonic majorant of the subharmonic function
in
). If the function
belongs to the class
, then for it to be subharmonic in
it is necessary and sufficient that the result of applying the Laplace operator,
, be non-negative in
.
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 of one or several complex variables
,
, and the consequent possible use of subharmonic functions for the study of analytic functions is related to the fact that the modulus
and the logarithm
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 are subharmonic functions in
and
are non-negative numbers, then the linear combination
is a subharmonic function in
.
2) The upper envelope of a finite family of subharmonic functions
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 is a subharmonic function in
and
is a convex non-decreasing function on the domain of values
of the function
in
, or if
is a harmonic function in
and
is a convex function on
, then
is a subharmonic function in
. In particular, if
is a subharmonic function in
, then
,
, and
,
, where
, are subharmonic functions in
; if
is a harmonic function in
, then
,
, is a subharmonic function in
.
5) The maximum principle: If is a subharmonic function in
and for any boundary point
and any
there is a neighbourhood
such that
in
, then either
or
in
. This property also holds for unbounded domains
, where for
the neighbourhood
is taken to mean the exterior of a sphere,
.
6) If is a subharmonic function in a domain
of the complex plane
and
is a holomorphic mapping of a domain
into
, then
is a subharmonic function in
.
7) A function is harmonic in the domain
if and only if
and
are subharmonic functions in
(cf. Harmonic function).
8) If is a subharmonic function on the whole plane
that is bounded above, then
(in
when
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 is a subharmonic function in an annulus
,
, then the mean values
,
, as well as the maximum
![]() |
are convex functions in for
, or in
for
, on the interval
; if
is a subharmonic function in the disc (ball)
, then, moreover,
and
are continuous non-decreasing functions in
on the interval
; moreover
![]() |
in the latter instance,
![]() |
for . The mean values
and
, considered as functions of the point
for fixed
and
, are subharmonic functions in the corresponding subdomain
, and
is continuous. By forming iterations of a sufficiently-high order,
![]() |
it is possible to obtain a monotone-decreasing sequence of subharmonic functions of any degree of smoothness that converges, as
, to an arbitrarily given subharmonic function
.
The Newton potential and logarithmic potential of non-negative masses, when written with a minus sign, are subharmonic functions everywhere in the space . 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
is a subharmonic function in a domain
, then there exists a unique non-negative Borel measure
on
(a measure associated with
, or a Riesz measure) such that for any compact set
the representation
![]() | (2) |
is valid, where when
,
when
, and where
is a harmonic function in the interior of
. The Riesz theorem establishes a close link between the theory of subharmonic functions and potential theory.
If is a regular closed domain
bounded, for example, by a Lyapunov surface, and having a Green function
, then as well as (2) a representation using the Green potential is valid:
![]() | (3) |
where is the least harmonic majorant of the subharmonic function
in the domain
.
A representation in the form (3), generally speaking, does not hold in the whole domain of definition of
, and in the theory of subharmonic functions great importance is attached to the question of distinguishing the class of subharmonic functions
that allow a representation (3) in the whole domain
, i.e. the question of distinguishing the class
of subharmonic functions
that have a harmonic majorant in the whole domain
. For example, if
is a ball (disc) and there exists a constant
such that
![]() | (4) |
then allows a representation (3) in
, and the least harmonic majorant
is, in turn, represented by a Poisson–Stieltjes integral:
![]() | (5) |
where is a Borel measure of arbitrary sign concentrated on the boundary sphere (circle)
(a boundary measure) and normalized by the condition
.
With regard to (5), it is important in practical applications to know under which conditions the boundary measure has only a negative singular component, i.e. under which conditions the component
in the canonical decomposition
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
be an increasing concave function in
for which
. A function
,
, is said to be strictly subharmonic relative to
if
is a subharmonic function. For example, logarithmically-subharmonic functions (cf. Logarithmically-subharmonic function)
, for which
is a subharmonic function, belong to the class of strictly-subharmonic functions. If condition (4) is fulfilled for a strictly-subharmonic function
in the ball
, then
can be represented in the form (3) in
, and the boundary measure
is characterized by the decomposition
![]() |
where are the radial boundary values of the function
(which exist almost everywhere with respect to the Lebesgue measure on the sphere
), and
is the singular component of the measure
.
Subharmonic functions of class in the ball
have radial boundary values almost everywhere on
. However, examples have been constructed of bounded, continuous subharmonic functions in
that do not have non-tangential boundary values anywhere on
, 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
in
(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 . The general method of introducing these subclasses consists of the fact that for strictly-subharmonic functions
relative to a concave function
, any non-decreasing function
is examined that is convex relative to
and that is such that
, and the class
is introduced. For a sphere it is defined by the condition
![]() |
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 . 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 of negative subharmonic functions: a) the upper semi-continuous regularization of the supremum of a subset of
is subharmonic; b) (the Riesz decomposition property) for any
,
, there exist
such that
,
,
; and c) any
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) |
Subharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subharmonic_function&oldid=13674