View source for Golubev-Privalov theorem

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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

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

These conditions are known as the Golubev–Privalov conditions. That they are sufficient has been shown by V.V. Golubev [[#References|[1]]]; that they are necessary has been shown by I.I. Privalov [[#References|[2]]]. In other words, conditions (1) are necessary and sufficient for the integral of Cauchy–Lebesgue type (cf. [[Cauchy integral|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|Borel measure]] on  $  L $. 
Then the integral of Cauchy–Stieltjes type (cf. [[Cauchy integral|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|boundary properties of analytic functions]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,   "The Cauchy integral" , Saratov  (1918)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>