# 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)

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].