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Riesz's theorem on the representation of a subharmonic function: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823201.png" /> is a [[Subharmonic function|subharmonic function]] in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823202.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823204.png" />, then there exists a unique positive [[Borel measure|Borel measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823205.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823206.png" /> such that for any relatively compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823207.png" /> the Riesz representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823208.png" /> as the sum of a [[Potential|potential]] and a [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r0823209.png" /> is valid:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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 +
Riesz's theorem on the representation of a subharmonic function: If  $  u $
 +
is a [[Subharmonic function|subharmonic function]] in a domain  $  D $
 +
of a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,
 +
then there exists a unique positive [[Borel measure|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|potential]] and a [[Harmonic function|harmonic function]]  $  h $
 +
is valid:
 +
 
 +
$$ \tag{1 }
 +
u( x)  = - \int\limits _ { K } E _ {n} (| x- y |)  d \mu ( y) + h( x),
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232011.png" /></td> </tr></table>
+
$$
 +
E _ {2} (| x- y |)  =   \mathop{\rm ln} 
 +
\frac{1}{| x- y | }
 +
,\ \
 +
E _ {n} (| x- y |)  =
 +
\frac{1}{| x- y |  ^ {n-} 2 }
 +
,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232013.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232014.png" /> (see ). The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232015.png" /> is called the associated measure for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232016.png" /> or the Riesz measure.
+
$  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232017.png" /> is the closure of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232018.png" /> and if, moreover, there exists a generalized [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232019.png" />, then formula (1) can be written in the form
+
If $  K = \overline{H}\; $
 +
is the closure of a domain $  H $
 +
and if, moreover, there exists a generalized [[Green function|Green function]] $  g( x, y;  H) $,  
 +
then formula (1) can be written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
u( x)  = - \int\limits _ {\overline{H}\; } g( x, y; H)  d \mu ( y) + h  ^  \star  ( x) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232021.png" /> is the least [[Harmonic majorant|harmonic majorant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232023.png" />.
+
where $  h  ^  \star  $
 +
is the least [[Harmonic majorant|harmonic majorant]] of $  u $
 +
in $  H $.
  
Formulas (1) and (2) can be extended under certain additional conditions to the entire domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232024.png" /> (see [[Subharmonic function|Subharmonic function]], and also , ).
+
Formulas (1) and (2) can be extended under certain additional conditions to the entire domain $  D $(
 +
see [[Subharmonic function|Subharmonic function]], and also , ).
  
Riesz's theorem on the mean value of a subharmonic function: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232025.png" /> is a subharmonic function in a spherical shell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232026.png" />, then its mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232027.png" /> over the area of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232028.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232029.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232031.png" />, that is,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232032.png" /></td> </tr></table>
+
$$
 +
J( \rho )  = J( \rho ; x _ {0} , u)  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232033.png" /> is the area of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232034.png" />, is a convex function with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232036.png" /> and with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232039.png" /> is a subharmonic function in the entire ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232041.png" /> is, furthermore, a non-decreasing continuous function with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232042.png" /> under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232043.png" /> (see ).
+
\frac{1}{\sigma _ {n} ( \rho ) }
 +
\int\limits _ {S _ {n} ( x _ {0} , \rho ) } u( y)  d \sigma _ {n} ( y) ,
 +
$$
  
Riesz's theorem on analytic functions of Hardy classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232045.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232046.png" /> is a regular [[Analytic function|analytic function]] in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232047.png" /> of Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232049.png" /> (see [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Hardy classes|Hardy classes]]), then the following relations hold:
+
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 ).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232050.png" /></td> </tr></table>
+
Riesz's theorem on analytic functions of Hardy classes  $  H  ^  \delta  $,
 +
$  \delta > 0 $:
 +
If  $  f( z) $
 +
is a regular [[Analytic function|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|Boundary properties of analytic functions]]; [[Hardy classes|Hardy classes]]), then the following relations hold:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232051.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {r \rightarrow 1 }  \int\limits _ { E } | f( re ^ {i \theta } )
 +
|  ^  \delta  d \theta  = \
 +
\int\limits _ { E } | f( e ^ {i \theta } ) |  ^  \delta  d \theta ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232052.png" /> is an arbitrary set of positive measure on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232053.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232054.png" /> are the boundary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232055.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232056.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232057.png" /> if and only if its integral is continuous in the closed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232058.png" /> and is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232059.png" /> (see [[#References|[2]]]).
+
$$
 +
\lim\limits _ {r \rightarrow 1 }  \int\limits _ { 0 } ^ { {2 }  \pi } | f( re ^ {i
 +
\theta } ) - f( e ^ {i \theta } ) |  ^  \delta  d \theta  = 0,
 +
$$
 +
 
 +
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 [[#References|[2]]]).
  
 
Theorems 1)–3) were proved by F. Riesz (see , [[#References|[2]]]).
 
Theorems 1)–3) were proved by F. Riesz (see , [[#References|[2]]]).
Line 35: Line 122:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sous harmoniques et leur rapport à la theorie du potentiel I"  ''Acta Math.'' , '''48'''  (1926)  pp. 329–343</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sous harmoniques et leur rapport à la theorie du potentiel II"  ''Acta Math.'' , '''54'''  (1930)  pp. 321–360</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  "Ueber die Randwerte einer analytischer Funktion"  ''Math. Z.'' , '''18'''  (1923)  pp. 87–95</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Privalov,  "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sous harmoniques et leur rapport à la theorie du potentiel I"  ''Acta Math.'' , '''48'''  (1926)  pp. 329–343</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sous harmoniques et leur rapport à la theorie du potentiel II"  ''Acta Math.'' , '''54'''  (1930)  pp. 321–360</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  "Ueber die Randwerte einer analytischer Funktion"  ''Math. Z.'' , '''18'''  (1923)  pp. 87–95</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Privalov,  "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In abstract potential theory, a [[Potential|potential]] on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232060.png" /> is a [[Superharmonic function|superharmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232061.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232062.png" /> such that any harmonic minorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232063.png" /> is negative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232064.png" />. The Riesz representation theorem now takes the form: Any superharmonic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232065.png" /> can be written uniquely as the sum of a potential and a harmonic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232066.png" />, see [[#References|[a2]]].
+
In abstract potential theory, a [[Potential|potential]] on an open set $  U $
 +
is a [[Superharmonic function|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 [[#References|[a2]]].
  
In an ordered Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232067.png" />, the Riesz interpolation property means that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232068.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232070.png" />. An equivalent form is the decomposition property: for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232071.png" /> there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232073.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082320/r08232076.png" />. These properties are used in the theory of Choquet simplexes (cf. [[Choquet simplex|Choquet simplex]]) and in the fine theory of hyperharmonic functions, see [[#References|[a1]]] and [[#References|[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|Choquet simplex]]) and in the fine theory of hyperharmonic functions, see [[#References|[a1]]] and [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Asimow,  A.J. Ellis,  "Convexity theory and its applications in functional analysis" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Asimow,  A.J. Ellis,  "Convexity theory and its applications in functional analysis" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


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)
How to Cite This Entry:
Riesz theorem(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_theorem(2)&oldid=48570
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article