Golubev-Privalov theorem

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

$$ \tag{1 } \int\limits _ { L } z ^ {n} f ( z) dz = 0,\ \ n = 0, 1 , \dots $$

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) $ F( z) $ constructed for the function $ f( z) $ and the curve $ L $:

$$ F ( z) = \ \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } ,\ \ z \in D, $$

to be a Cauchy–Lebesgue integral.

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

$$ F ( z) = \ \frac{1}{2 \pi i } \int\limits \frac{d \mu ( \zeta ) }{\zeta - z } ,\ \ z \in D, $$

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

$$ \tag{2 } \int\limits z ^ {n} d \mu ( z) = 0,\ \ n = 0, 1 \dots $$

are satisfied.

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

$$ e ^ {- i \phi ( z) } \mu ^ \prime ( z), $$

where $ \phi ( z) $ is the angle between the positive direction of the abscissa axis and the tangent to $ L $ at the point $ z \in L $ and $ \mu ^ \prime ( z) $ is the derivative of $ \mu $ with respect to Lebesgue measure (arc length) on $ L $.

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)

References

[a1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)