Difference between revisions of "Riesz theorem(2)"
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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 | ||
− | + | $$ | |
+ | E _ {2} (| x- y |) = \mathop{\rm ln} | ||
+ | \frac{1}{| x- y | } | ||
+ | ,\ \ | ||
+ | E _ {n} (| x- y |) = | ||
+ | \frac{1}{| x- y | ^ {n-} 2 } | ||
+ | , | ||
+ | $$ | ||
− | + | $ 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 | + | 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 | ||
− | + | $$ \tag{2 } | |
+ | u( x) = - \int\limits _ {\overline{H}\; } g( x, y; H) d \mu ( y) + h ^ \star ( x) , | ||
+ | $$ | ||
− | where | + | 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 | + | 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 | + | 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|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: | ||
− | + | $$ | |
+ | \lim\limits _ {r \rightarrow 1 } \int\limits _ { E } | f( re ^ {i \theta } ) | ||
+ | | ^ \delta d \theta = \ | ||
+ | \int\limits _ { E } | f( e ^ {i \theta } ) | ^ \delta d \theta , | ||
+ | $$ | ||
− | + | $$ | |
+ | \lim\limits _ {r \rightarrow 1 } \int\limits _ { 0 } ^ { {2 } \pi } | f( re ^ {i | ||
+ | \theta } ) - f( e ^ {i \theta } ) | ^ \delta d \theta = 0, | ||
+ | $$ | ||
+ | |||
+ | 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 [[#References|[2]]]). | ||
Theorems 1)–3) were proved by F. Riesz (see , [[#References|[2]]]). | Theorems 1)–3) were proved by F. Riesz (see , [[#References|[2]]]). | ||
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====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 | + | 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 | + | 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) |
Riesz theorem(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_theorem(2)&oldid=12058