Riesz theorem(2)
Riesz's theorem on the representation of a subharmonic function: If is a subharmonic function in a domain
of a Euclidean space
,
, then there exists a unique positive Borel measure
on
such that for any relatively compact set
the Riesz representation of
as the sum of a potential and a harmonic function
is valid:
![]() | (1) |
where
![]() |
and
is the distance between the points
(see ). The measure
is called the associated measure for the function
or the Riesz measure.
If is the closure of a domain
and if, moreover, there exists a generalized Green function
, then formula (1) can be written in the form
![]() | (2) |
where is the least harmonic majorant of
in
.
Formulas (1) and (2) can be extended under certain additional conditions to the entire domain (see Subharmonic function, and also , ).
Riesz's theorem on the mean value of a subharmonic function: If is a subharmonic function in a spherical shell
, then its mean value
over the area of the sphere
with centre at
and radius
,
, that is,
![]() |
where is the area of
, is a convex function with respect to
for
and with respect to
for
. If
is a subharmonic function in the entire ball
, then
is, furthermore, a non-decreasing continuous function with respect to
under the condition that
(see ).
Riesz's theorem on analytic functions of Hardy classes ,
: If
is a regular analytic function in the unit disc
of Hardy class
,
(see Boundary properties of analytic functions; Hardy classes), then the following relations hold:
![]() |
![]() |
where is an arbitrary set of positive measure on the circle
, and
are the boundary values of
on
. Moreover,
if and only if its integral is continuous in the closed disc
and is absolutely continuous on
(see [2]).
Theorems 1)–3) were proved by F. Riesz (see , [2]).
References
[1a] | F. Riesz, "Sur les fonctions sous harmoniques et leur rapport à la theorie du potentiel I" Acta Math. , 48 (1926) pp. 329–343 |
[1b] | F. Riesz, "Sur les fonctions sous harmoniques et leur rapport à la theorie du potentiel II" Acta Math. , 54 (1930) pp. 321–360 |
[2] | F. Riesz, "Ueber die Randwerte einer analytischer Funktion" Math. Z. , 18 (1923) pp. 87–95 |
[3] | I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian) |
[4] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[5] | W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976) |
Comments
In abstract potential theory, a potential on an open set is a superharmonic function
on
such that any harmonic minorant of
is negative on
. The Riesz representation theorem now takes the form: Any superharmonic function on
can be written uniquely as the sum of a potential and a harmonic function on
, see [a2].
In an ordered Banach space , the Riesz interpolation property means that, for any
, there exists a
such that
. An equivalent form is the decomposition property: for
there exist
and
such that
and
,
. These properties are used in the theory of Choquet simplexes (cf. Choquet simplex) and in the fine theory of hyperharmonic functions, see [a1] and [a2].
References
[a1] | L. Asimow, A.J. Ellis, "Convexity theory and its applications in functional analysis" , Acad. Press (1980) |
[a2] | C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) |
Riesz theorem(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_theorem(2)&oldid=12058