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Golubev-Privalov theorem

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If is a complex summable function on a closed rectifiable Jordan curve in the complex -plane, then a necessary and sufficient condition for the existence of a function , regular in the interior of the domain bounded by and whose angular boundary values coincide with almost-everywhere on , is

(1)

These conditions are known as the Golubev–Privalov conditions. That they are sufficient has been shown by V.V. Golubev [1]; that they are necessary has been shown by I.I. Privalov [2]. In other words, conditions (1) are necessary and sufficient for the integral of Cauchy–Lebesgue type (cf. Cauchy integral) constructed for the function and the curve :

to be a Cauchy–Lebesgue integral.

In a more general formulation, let be a complex Borel measure on . Then the integral of Cauchy–Stieltjes type (cf. Cauchy integral),

is a Cauchy–Stieltjes integral if and only if the generalized Golubev–Privalov conditions

(2)

are satisfied.

In other words, conditions (2) are necessary and sufficient for the existence of a regular function in such that its angular boundary values coincide almost-everywhere (with respect to Lebesgue measure) on with

where is the angle between the positive direction of the abscissa axis and the tangent to at the point and is the derivative of with respect to Lebesgue measure (arc length) on .

The Golubev–Privalov theorem is of importance in the theory of boundary properties of analytic functions.

References

[1] V.V. Golubev, "Univalent analytic functions with perfect sets of singular points" , Moscow (1916) (In Russian) (See also: V.V. Golubev, Single-valued analytic functions. Automorphic functions, Moscow, 1961 (in Russian))
[2] I.I. Privalov, "The Cauchy integral" , Saratov (1918) (In Russian)
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)


Comments

References

[a1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Golubev–Privalov theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Golubev%E2%80%93Privalov_theorem&oldid=22522