Difference between revisions of "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) |
Golubev-Privalov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Golubev-Privalov_theorem&oldid=18613