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== Privalov's theorem on conjugate functions ==
 
== Privalov's theorem on conjugate functions ==
  
 
Let
 
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>
+
$$
 +
f ( t)  = \
 +
{
 +
\frac{a _ {0} }{2}
 +
} +
 +
\sum _ {k = 1 } ^  \infty 
 +
( a _ {k}  \cos  kt + b _ {k}  \sin  kt)
 +
$$
  
be a continuous periodic function of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p0748602.png" /> and let
+
be a continuous periodic function of period $  2 \pi $
 +
and 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/p0748603.png" /></td> </tr></table>
+
$$
 +
\widetilde{f}  ( t)  = \
 +
{
 +
\frac{a _ {0} }{2}
 +
} +
 +
\sum _ {k = 1 } ^  \infty 
 +
( b _ {k}  \cos  kt - a _ {k}  \sin  kt)
 +
$$
  
be the function trigonometrically conjugate to <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 $  f $(
 +
cf. also [[Conjugate function|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 ==
 
== 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]].
+
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|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|Luzin–Privalov theorems]]) is a generalization of it. See also [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]].
  
 
== Privalov's theorem on the singular Cauchy integral ==
 
== 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 $  \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  }
  
<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/p07486046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{e ^ {i \phi ( s) }  d \psi ( s) }{\zeta - \zeta _ {0} }
 +
  = \
 +
{
 +
\frac{1}{2 \pi i }
 +
}
 +
\int\limits _  \Gamma
  
if it exists and is finite, is called a Cauchy–Stieltjes singular integral. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486047.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486048.png" />) be the finite (infinite) domain bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486049.png" />. A statement of Privalov's theorem is: If for almost-all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486050.png" />, with respect to the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486051.png" />, the singular integral (1) exists, then almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486052.png" /> the non-tangential boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486053.png" /> of the integral of Cauchy–Stieltjes type,
+
\frac{e ^ {i \phi ( s) }  d \psi ( s) }{\zeta - \zeta _ {0} }
 +
,
 +
$$
  
<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/p07486054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
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,
  
exist, taken respectively from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486055.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486056.png" />, and almost-everywhere the [[Sokhotskii formulas|Sokhotskii formulas]]
+
$$ \tag{2 }
 +
F ( z)  = \
 +
{
 +
\frac{1}{2 \pi i }
 +
}
 +
\int\limits _  \Gamma
  
<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/p07486057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{e ^ {i \phi ( s) }  d \psi ( s) }{\zeta - z }
 +
,\ \
 +
z \in D  ^  \pm  ,
 +
$$
  
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. ).
+
exist, taken respectively from  $  D  ^ {+} $
 +
or  $  D  ^ {-} $,
 +
and almost-everywhere the [[Sokhotskii formulas|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 ==
 
== 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
+
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
  
<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>
+
$$
 +
| f ( \zeta _ {1} ) - f ( \zeta _ {2} ) |  < \
 +
C  | \zeta _ {1} - \zeta _ {2} |  ^  \alpha  ,\ \
 +
0 < \alpha \leq  1,
 +
$$
  
 
then the integral of Cauchy–Lebesgue type
 
then the integral of Cauchy–Lebesgue type
  
<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/p07486069.png" /></td> </tr></table>
+
$$
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486070.png" />. Moreover, the boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486071.png" /> satisfy
+
is a continuous function in the closed domains $  D  ^  \pm  $.  
 +
Moreover, the boundary values $  F ^ { \pm  } ( \zeta ) $
 +
satisfy
  
<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/p07486072.png" /></td> </tr></table>
+
$$
 +
| F ^ { \pm  } ( \zeta _ {1} ) - F ^ { \pm  } ( \zeta _ {2} ) |  < \
 +
C _ {1}  | \zeta _ {1} - \zeta _ {2} |  ^  \alpha
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074860/p07486073.png" />, and
+
for $  0 < \alpha < 1 $,  
 +
and
  
<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/p07486074.png" /></td> </tr></table>
+
$$
 +
| F ^ { \pm  } ( \zeta _ {1} ) - F ^ { \pm  } ( \zeta _ {2} ) |  < \
 +
C _ {2} ( \delta )  | \zeta _ {1} - \zeta _ {2} |  \mathop{\rm ln} \
 +
{
 +
\frac{1}{| \zeta _ {1} - \zeta _ {2} | }
 +
}
 +
$$
  
for <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 $  \alpha = 1 $,  
 +
$  | \zeta _ {1} - \zeta _ {2} | \leq  \delta < 1 $(
 +
cf. [[#References|[2]]]).
  
 
==References==
 
==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><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>
 
<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>

Latest revision as of 08:07, 6 June 2020


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
How to Cite This Entry:
Privalov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Privalov_theorem&oldid=48296
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article