View source for Riesz theorem(2)

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:

<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>

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>

<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.

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

<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>

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" />.

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 , ).

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,

<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>

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 ).

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:

<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>

<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>

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]]]).

Theorems 1)–3) were proved by F. Riesz (see , [[#References|[2]]]).

====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>



====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 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]]].

====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>