# Riesz theorem(2)

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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 ).

Theorems 1)–3) were proved by F. Riesz (see , ).

How to Cite This Entry:
Riesz theorem(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_theorem(2)&oldid=12058
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article