Privalov operators

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Privalov parameters

Operators that allow one to express the condition of harmonicity of a function without using partial derivatives (cf. Harmonic function). Let be a locally integrable function in a bounded domain of a Euclidean space , ; let denote the volume of the ball of radius with centre , lying in ; and let

The upper and lower Privalov operators and are defined, respectively, by the formulas

If the upper and lower Privalov operators coincide, then the Privalov operator is defined by

If the function has continuous partial derivatives up to and including the second order at , then the Privalov operator exists at and is equal to the value of the Laplace operator: . Privalov's theorem says: If a function , continuous in a domain , satisfies everywhere in the conditions

then is harmonic in . This implies that a function , continuous in , is harmonic if and only if at every point one has , from some sufficiently small onwards, or, in other words, if and only if

The average value over the volume of a sphere can be replaced by that over the surface area.


[1] I.I. Privalov, Mat. Sb. , 32 (1925) pp. 464–471
[2] I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian)
[3] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1969)


More generally, if is lower semi-continuous, then is hyperharmonic if and only if on (the theorem of Blaschke–Privalov).

Similar results hold if the average value over the surface area is used for the operators and is replaced by .

How to Cite This Entry:
Privalov operators. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098