Privalov theorem

Privalov's theorem on conjugate functions

Let

$$ f ( t) = \ { \frac{a _ {0} }{2} } + \sum _ {k = 1 } ^ \infty ( a _ {k} \cos kt + b _ {k} \sin kt) $$

be a continuous periodic function of period $ 2 \pi $ and let

$$ \widetilde{f} ( t) = \ { \frac{a _ {0} }{2} } + \sum _ {k = 1 } ^ \infty ( b _ {k} \cos kt - a _ {k} \sin kt) $$

be the function trigonometrically conjugate to $ f $( cf. also Conjugate function). Then if $ f $ satisfies a Lipschitz condition of order $ \alpha $, $ f \in \mathop{\rm Lip} \alpha $, $ 0 < \alpha \leq 1 $, then $ \widetilde{f} \in \mathop{\rm Lip} \alpha $ for $ 0 < \alpha < 1 $ and $ \widetilde{f} $ has modulus of continuity $ M( \delta , \widetilde{f} ) = \sup _ {| x _ {1} - x _ {2} | \leq \delta } | f( x _ {1} ) - f ( x _ {2} ) | $ at most $ M \delta \mathop{\rm ln} ( 1/ \delta ) $ for $ \alpha = 1 $. This theorem, proved by I.I. Privalov , has important applications in the theory of trigonometric series. It can be transferred to Lipschitz conditions in certain other metrics (cf. e.g. ).

Privalov's uniqueness theorem for analytic functions

Let $ f ( z) $ be a single-valued analytic function in a domain $ D $ of the complex $ z $- plane bounded by a rectifiable Jordan curve $ \Gamma $. If on some set $ E \subset \Gamma $ of positive Lebesgue measure on $ \Gamma $, $ f ( z) $ has non-tangential boundary values (cf. Angular boundary value) zero, then $ f ( z) \equiv 0 $ in $ D $. This theorem was proved by Privalov ; the Luzin–Privalov theorem (cf. Luzin–Privalov theorems) is a generalization of it. See also Uniqueness properties of analytic functions.

Privalov's theorem on the singular Cauchy integral

Privalov's theorem on the singular Cauchy integral, or Privalov's main lemma, is one of the basic results in the theory of integrals of Cauchy–Stieltjes type (cf. Cauchy integral). Let $ \Gamma $: $ \zeta = \zeta ( s) $, $ 0 \leq s \leq l $, be a rectifiable (closed) Jordan curve in the complex $ z $- plane; let $ l $ be the length of $ \Gamma $; let $ s $ be the arc length on $ \Gamma $ reckoned from some fixed point; let $ \phi = \phi ( s) $ be the angle between the positive direction of the $ x $- axis and the tangent to $ \Gamma $; and let $ \psi ( s) $ be a complex-valued function of bounded variation on $ \Gamma $. Let a point $ \zeta _ {0} \in \Gamma $ be defined by a value $ s _ {0} $ of the arc length, $ \zeta _ {0} = \zeta ( s _ {0} ) $, $ 0 \leq s _ {0} \leq l $, and let $ \Gamma _ \delta $ be the part of $ \Gamma $ that remains when the shorter arc with end-points $ \zeta ( s _ {0} - \delta ) $ and $ \zeta ( s _ {0} + \delta ) $ is removed from $ \Gamma $. The limit

$$ \tag{1 } \lim\limits _ {\delta \rightarrow 0 } \ { \frac{1}{2 \pi i } } \int\limits _ {\Gamma _ \delta } \frac{e ^ {i \phi ( s) } d \psi ( s) }{\zeta - \zeta _ {0} } = \ { \frac{1}{2 \pi i } } \int\limits _ \Gamma \frac{e ^ {i \phi ( s) } d \psi ( s) }{\zeta - \zeta _ {0} } , $$

if it exists and is finite, is called a Cauchy–Stieltjes singular integral. Let $ D ^ {+} $( respectively, $ D ^ {-} $) be the finite (infinite) domain bounded by $ \Gamma $. A statement of Privalov's theorem is: If for almost-all points of $ \Gamma $, with respect to the Lebesgue measure on $ \Gamma $, the singular integral (1) exists, then almost-everywhere on $ \Gamma $ the non-tangential boundary values $ F ^ { \pm } ( \zeta _ {0} ) $ of the integral of Cauchy–Stieltjes type,

$$ \tag{2 } F ( z) = \ { \frac{1}{2 \pi i } } \int\limits _ \Gamma \frac{e ^ {i \phi ( s) } d \psi ( s) }{\zeta - z } ,\ \ z \in D ^ \pm , $$

exist, taken respectively from $ D ^ {+} $ or $ D ^ {-} $, and almost-everywhere the Sokhotskii formulas

$$ \tag{3 } F ^ { \pm } ( \zeta _ {0} ) = \ { \frac{1}{2 \pi i } } \int\limits _ \Gamma \frac{e ^ {i \phi ( s) } d \psi ( s) }{\zeta - \zeta _ {0} } \pm { \frac{1}{2} } \psi ^ \prime ( s _ {0} ) $$

hold. Conversely, if almost-everywhere on $ \Gamma $ the non-tangential boundary value $ F ^ { + } ( \zeta _ {0} ) $( or $ F ^ { - } ( \zeta _ {0} ) $) of the integral (2) exists, then almost-everywhere on $ \Gamma $ the singular integral (1) and the boundary value from the other side, $ F ^ { - } ( \zeta _ {0} ) $( respectively, $ F ^ { + } ( \zeta _ {0} ) $) exist and relation (3) holds. This theorem was established by Privalov for integrals of Cauchy–Lebesgue type (i.e. in the case of an absolutely-continuous function $ \psi ( s) $, cf. ), and later in the general case . It plays a basic role in the theory of singular integral equations and discontinuous boundary problems of analytic function theory (cf. ).

Privalov's theorem on boundary values of integrals of Cauchy–Lebesgue type

If a Jordan curve $ \Gamma $ is piecewise smooth and without cusps and if a complex-valued function $ f ( \zeta ) $, $ \zeta \in \Gamma $, satisfies a Lipschitz condition

$$ | f ( \zeta _ {1} ) - f ( \zeta _ {2} ) | < \ C | \zeta _ {1} - \zeta _ {2} | ^ \alpha ,\ \ 0 < \alpha \leq 1, $$

then the integral of Cauchy–Lebesgue type

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

is a continuous function in the closed domains $ D ^ \pm $. Moreover, the boundary values $ F ^ { \pm } ( \zeta ) $ satisfy

$$ | F ^ { \pm } ( \zeta _ {1} ) - F ^ { \pm } ( \zeta _ {2} ) | < \ C _ {1} | \zeta _ {1} - \zeta _ {2} | ^ \alpha $$

for $ 0 < \alpha < 1 $, and

$$ | F ^ { \pm } ( \zeta _ {1} ) - F ^ { \pm } ( \zeta _ {2} ) | < \ C _ {2} ( \delta ) | \zeta _ {1} - \zeta _ {2} | \mathop{\rm ln} \ { \frac{1}{| \zeta _ {1} - \zeta _ {2} | } } $$

for $ \alpha = 1 $, $ | \zeta _ {1} - \zeta _ {2} | \leq \delta < 1 $( cf. [2]).

References

[1]	I.I. Privalov, "Sur les fonctions conjuguées" Bull. Soc. Math. France , 44 (1916) pp. 100–103
[2]	I.I. Privalov, "The Cauchy integral" , Saratov (1918) (In Russian)
[3]	I.I. Privalov, "Boundary properties of single-valued analytic functions" , Moscow (1941) (In Russian)
[4]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[5]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[6]	B.V. Khvedelidze, "The method of Cauchy-type integrals in the discontinuous boundary-value problems of the theory of holomorphic functions of a complex variable" J. Soviet Math. , 7 : 3 (1977) pp. 309–414 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 7 (1975) pp. 5–162
[a1]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9