# 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)