# Riesz theorem(2)

Riesz's theorem on the representation of a subharmonic function: If is a subharmonic function in a domain of a Euclidean space , , then there exists a unique positive Borel measure on such that for any relatively compact set the Riesz representation of as the sum of a potential and a harmonic function is valid:

(1) |

where

and is the distance between the points (see ). The measure is called the associated measure for the function or the Riesz measure.

If is the closure of a domain and if, moreover, there exists a generalized Green function , then formula (1) can be written in the form

(2) |

where is the least harmonic majorant of in .

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

Riesz's theorem on the mean value of a subharmonic function: If is a subharmonic function in a spherical shell , then its mean value over the area of the sphere with centre at and radius , , that is,

where is the area of , is a convex function with respect to for and with respect to for . If is a subharmonic function in the entire ball , then is, furthermore, a non-decreasing continuous function with respect to under the condition that (see ).

Riesz's theorem on analytic functions of Hardy classes , : If is a regular analytic function in the unit disc of Hardy class , (see Boundary properties of analytic functions; Hardy classes), then the following relations hold:

where is an arbitrary set of positive measure on the circle , and are the boundary values of on . Moreover, if and only if its integral is continuous in the closed disc and is absolutely continuous on (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 is a superharmonic function on such that any harmonic minorant of is negative on . The Riesz representation theorem now takes the form: Any superharmonic function on can be written uniquely as the sum of a potential and a harmonic function on , see [a2].

In an ordered Banach space , the Riesz interpolation property means that, for any , there exists a such that . An equivalent form is the decomposition property: for there exist and such that and , . 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). E.D. Solomentsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Riesz_theorem(2)&oldid=12058