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Privalov's theorem on conjugate functions: Let
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== Privalov's theorem on conjugate functions ==
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Let
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748601.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748601.png" /></td> </tr></table>
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be the function trigonometrically conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748604.png" /> (cf. also [[Conjugate function|Conjugate function]]). Then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748605.png" /> satisfies a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748608.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748609.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486011.png" /> has modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486012.png" /> at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486014.png" />. 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. ).
 
be the function trigonometrically conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748604.png" /> (cf. also [[Conjugate function|Conjugate function]]). Then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748605.png" /> satisfies a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748608.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748609.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486011.png" /> has modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486012.png" /> at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486015.png" /> be a single-valued analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486016.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486017.png" />-plane bounded by a rectifiable Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486018.png" />. If on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486019.png" /> of positive Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486021.png" /> has non-tangential boundary values (cf. [[Angular boundary value|Angular boundary value]]) zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486023.png" />. This theorem was proved by Privalov ; the Luzin–Privalov theorem (cf. [[Luzin–Privalov theorems|Luzin–Privalov theorems]]) is a generalization of it. See also [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]].
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== Privalov's uniqueness theorem for analytic functions ==
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486015.png" /> be a single-valued analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486016.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486017.png" />-plane bounded by a rectifiable Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486018.png" />. If on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486019.png" /> of positive Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486021.png" /> has non-tangential boundary values (cf. [[Angular boundary value|Angular boundary value]]) zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486023.png" />. This theorem was proved by Privalov ; the Luzin–Privalov theorem (cf. [[Luzin–Privalov theorems|Luzin–Privalov theorems]]) is a generalization of it. See also [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]].
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== 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|Cauchy integral]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486024.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486026.png" />, be a rectifiable (closed) Jordan curve in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486027.png" />-plane; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486028.png" /> be the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486029.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486030.png" /> be the arc length on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486031.png" /> reckoned from some fixed point; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486032.png" /> be the angle between the positive direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486033.png" />-axis and the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486034.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486035.png" /> be a complex-valued function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486036.png" />. Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486037.png" /> be defined by a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486038.png" /> of the arc length, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486040.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486041.png" /> be the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486042.png" /> that remains when the shorter arc with end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486044.png" /> is removed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486045.png" />. The limit
 
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|Cauchy integral]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486024.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486026.png" />, be a rectifiable (closed) Jordan curve in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486027.png" />-plane; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486028.png" /> be the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486029.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486030.png" /> be the arc length on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486031.png" /> reckoned from some fixed point; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486032.png" /> be the angle between the positive direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486033.png" />-axis and the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486034.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486035.png" /> be a complex-valued function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486036.png" />. Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486037.png" /> be defined by a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486038.png" /> of the arc length, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486040.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486041.png" /> be the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486042.png" /> that remains when the shorter arc with end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486044.png" /> is removed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486045.png" />. The limit
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hold. Conversely, if almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486058.png" /> the non-tangential boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486059.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486060.png" />) of the integral (2) exists, then almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486061.png" /> the singular integral (1) and the boundary value from the other side, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486062.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486063.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486064.png" />, 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. ).
 
hold. Conversely, if almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486058.png" /> the non-tangential boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486059.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486060.png" />) of the integral (2) exists, then almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486061.png" /> the singular integral (1) and the boundary value from the other side, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486062.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486063.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486064.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486065.png" /> is piecewise smooth and without cusps and if a complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486067.png" />, satisfies a Lipschitz condition
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== Privalov's theorem on boundary values of integrals of Cauchy–Lebesgue type ==
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 +
If a Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486065.png" /> is piecewise smooth and without cusps and if a complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486067.png" />, satisfies a Lipschitz condition
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486068.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486068.png" /></td> </tr></table>
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for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486076.png" /> (cf. [[#References|[2]]]).
 
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486076.png" /> (cf. [[#References|[2]]]).
  
====References====
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==References==
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Privalov,  "Sur les fonctions conjuguées"  ''Bull. Soc. Math. France'' , '''44'''  (1916)  pp. 100–103</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. Privalov,  "Boundary properties of single-valued analytic functions" , Moscow  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Privalov,  "Sur les fonctions conjuguées"  ''Bull. Soc. Math. France'' , '''44'''  (1916)  pp. 100–103</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. Privalov,  "Boundary properties of single-valued analytic functions" , Moscow  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>
 
 
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>
 

Revision as of 19:32, 12 April 2019

Privalov's theorem on conjugate functions

Let

be a continuous periodic function of period and let

be the function trigonometrically conjugate to (cf. also Conjugate function). Then if satisfies a Lipschitz condition of order , , , then for and has modulus of continuity at most for . 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 be a single-valued analytic function in a domain of the complex -plane bounded by a rectifiable Jordan curve . If on some set of positive Lebesgue measure on , has non-tangential boundary values (cf. Angular boundary value) zero, then in . 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 : , , be a rectifiable (closed) Jordan curve in the complex -plane; let be the length of ; let be the arc length on reckoned from some fixed point; let be the angle between the positive direction of the -axis and the tangent to ; and let be a complex-valued function of bounded variation on . Let a point be defined by a value of the arc length, , , and let be the part of that remains when the shorter arc with end-points and is removed from . The limit

(1)

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

(2)

exist, taken respectively from or , and almost-everywhere the Sokhotskii formulas

(3)

hold. Conversely, if almost-everywhere on the non-tangential boundary value (or ) of the integral (2) exists, then almost-everywhere on the singular integral (1) and the boundary value from the other side, (respectively, ) 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 , 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 is piecewise smooth and without cusps and if a complex-valued function , , satisfies a Lipschitz condition

then the integral of Cauchy–Lebesgue type

is a continuous function in the closed domains . Moreover, the boundary values satisfy

for , and

for , (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
How to Cite This Entry:
Privalov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Privalov_theorem&oldid=43777
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article