Riesz theorem(2)

Riesz's theorem on the representation of a subharmonic function: If $ u $ is a subharmonic function in a domain $ D $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, then there exists a unique positive Borel measure $ \mu $ on $ D $ such that for any relatively compact set $ K \subset D $ the Riesz representation of $ u $ as the sum of a potential and a harmonic function $ h $ is valid:

$$ \tag{1 } u( x) = - \int\limits _ { K } E _ {n} (| x- y |) d \mu ( y) + h( x), $$

where

$$ E _ {2} (| x- y |) = \mathop{\rm ln} \frac{1}{| x- y | } ,\ \ E _ {n} (| x- y |) = \frac{1}{| x- y | ^ {n-} 2 } , $$

$ n \geq 3 $ and $ | x- y | $ is the distance between the points $ x, y \in \mathbf R ^ {n} $( see ). The measure $ \mu $ is called the associated measure for the function $ u $ or the Riesz measure.

If $ K = \overline{H}\; $ is the closure of a domain $ H $ and if, moreover, there exists a generalized Green function $ g( x, y; H) $, then formula (1) can be written in the form

$$ \tag{2 } u( x) = - \int\limits _ {\overline{H}\; } g( x, y; H) d \mu ( y) + h ^ \star ( x) , $$

where $ h ^ \star $ is the least harmonic majorant of $ u $ in $ H $.

Formulas (1) and (2) can be extended under certain additional conditions to the entire domain $ D $( see Subharmonic function, and also , ).

Riesz's theorem on the mean value of a subharmonic function: If $ u $ is a subharmonic function in a spherical shell $ \{ {x \in \mathbf R ^ {n} } : {0 \leq r \leq | x- x _ {0} | \leq R } \} $, then its mean value $ J( p) $ over the area of the sphere $ S _ {n} ( x _ {0} , \rho ) $ with centre at $ x _ {0} $ and radius $ \rho $, $ r \leq \rho \leq R $, that is,

$$ J( \rho ) = J( \rho ; x _ {0} , u) = \ \frac{1}{\sigma _ {n} ( \rho ) } \int\limits _ {S _ {n} ( x _ {0} , \rho ) } u( y) d \sigma _ {n} ( y) , $$

where $ \sigma _ {n} ( \rho ) $ is the area of $ S _ {n} ( x _ {0} , \rho ) $, is a convex function with respect to $ 1/ \rho ^ {n-} 2 $ for $ n \geq 3 $ and with respect to $ \mathop{\rm ln} \rho $ for $ n= 2 $. If $ u $ is a subharmonic function in the entire ball $ \{ {x \in \mathbf R ^ {n} } : {| x- x _ {0} | \leq R } \} $, then $ J( \rho ) $ is, furthermore, a non-decreasing continuous function with respect to $ \rho $ under the condition that $ J( 0) = u( x _ {0} ) $( see ).

Riesz's theorem on analytic functions of Hardy classes $ H ^ \delta $, $ \delta > 0 $: If $ f( z) $ is a regular analytic function in the unit disc $ D= \{ {z = re ^ {i \theta } \in \mathbf C } : {| z | < 1 } \} $ of Hardy class $ H ^ \delta $, $ \delta > 0 $( see Boundary properties of analytic functions; Hardy classes), then the following relations hold:

$$ \lim\limits _ {r \rightarrow 1 } \int\limits _ { E } | f( re ^ {i \theta } ) | ^ \delta d \theta = \ \int\limits _ { E } | f( e ^ {i \theta } ) | ^ \delta d \theta , $$

$$ \lim\limits _ {r \rightarrow 1 } \int\limits _ { 0 } ^ { {2 } \pi } | f( re ^ {i \theta } ) - f( e ^ {i \theta } ) | ^ \delta d \theta = 0, $$

where $ E $ is an arbitrary set of positive measure on the circle $ \Gamma = \{ {z = e ^ {i \theta } } : {| z | = 1 } \} $, and $ f( e ^ {i \theta } ) $ are the boundary values of $ f( z) $ on $ \Gamma $. Moreover, $ f( z) \in H ^ {1} $ if and only if its integral is continuous in the closed disc $ D \cup \Gamma $ and is absolutely continuous on $ \Gamma $( 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 $ U $ is a superharmonic function $ u \geq 0 $ on $ U $ such that any harmonic minorant of $ u $ is negative on $ U $. The Riesz representation theorem now takes the form: Any superharmonic function on $ U $ can be written uniquely as the sum of a potential and a harmonic function on $ U $, see [a2].

In an ordered Banach space $ E $, the Riesz interpolation property means that, for any $ a, b \leq d , e $, there exists a $ c \in E $ such that $ a, b \leq c \leq d, e $. An equivalent form is the decomposition property: for $ 0 \leq a \leq b+ c $ there exist $ d $ and $ e $ such that $ a = d+ e $ and $ d \leq b $, $ e \leq c $. 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)