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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858301.png" /> of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858302.png" /> holomorphic in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858303.png" /> with rectifiable Jordan boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858304.png" />, such that for every function in it there is a sequence of closed rectifiable Jordan curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858306.png" /> with the following properties:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858307.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858308.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s0858309.png" /> in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583010.png" /> is the bounded domain with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583011.png" />, then
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{{TEX|done}}
  
<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/s/s085/s085830/s08583012.png" /></td> </tr></table>
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The set  $  E _ {p} ( G) $
 +
of all functions  $  f( z) $
 +
holomorphic in a simply-connected domain  $  G \subset  \mathbf C $
 +
with rectifiable Jordan boundary  $  \Gamma $,
 +
such that for every function in it there is a sequence of closed rectifiable Jordan curves  $  \Gamma _ {n} ( f  ) \subset  G $,
 +
$  n = 1, 2 \dots $
 +
with the following properties:
 +
 
 +
1)  $  \Gamma _ {n} ( f  ) $
 +
tends to  $  \Gamma $
 +
as  $  n \rightarrow \infty $
 +
in the sense that if  $  G _ {n} ( f  ) $
 +
is the bounded domain with boundary  $  \Gamma _ {n} ( f  ) $,
 +
then
 +
 
 +
$$
 +
G _ {1} ( f  )  \subset  \dots \subset  G _ {n} ( f  )  \subset  G \ \
 +
\textrm{ and } \  \cup _ {n= 1 } ^  \infty  G _ {n} ( f  )  = G;
 +
$$
  
 
2)
 
2)
  
<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/s/s085/s085830/s08583013.png" /></td> </tr></table>
+
$$
 +
\sup _ { n }  \left \{ \int\limits _ {\Gamma _ {n} ( f  ) }
 +
| f( z) |  ^ {p}  | dz | \right \}  <   \infty \  ( p> 0 \
 +
\textrm{ fixed } ).
 +
$$
  
This definition was proposed by M.V. Keldysh and M.A. Lavrent'ev [[#References|[2]]], and is equivalent to V.I. Smirnov's definition [[#References|[1]]] in which curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583014.png" /> are used instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583015.png" />. These curves are the images of the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583016.png" /> under some univalent [[Conformal mapping|conformal mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583017.png" /> from the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583018.png" /> onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583019.png" />, and the supremum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583020.png" />.
+
This definition was proposed by M.V. Keldysh and M.A. Lavrent'ev [[#References|[2]]], and is equivalent to V.I. Smirnov's definition [[#References|[1]]] in which curves $  \gamma ( \rho ) $
 +
are used instead of $  \Gamma _ {n} ( f  ) $.  
 +
These curves are the images of the circles $  | w | = \rho < 1 $
 +
under some univalent [[Conformal mapping|conformal mapping]] $  z= \phi ( w) $
 +
from the disc $  | w | < 1 $
 +
onto the domain $  G $,  
 +
and the supremum is taken over all $  \rho \in ( 0, 1) $.
  
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583021.png" /> are the best known and most thoroughly studied generalization of the [[Hardy classes|Hardy classes]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583022.png" />, and are connected with them by the following relation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583023.png" /> if and only if
+
The classes $  E _ {p} ( G) $
 +
are the best known and most thoroughly studied generalization of the [[Hardy classes|Hardy classes]] $  H _ {p} $,  
 +
and are connected with them by the following relation: $  f \in E _ {p} ( G) $
 +
if and only if
  
<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/s/s085/s085830/s08583024.png" /></td> </tr></table>
+
$$
 +
f( \phi ( w))( \phi  ^  \prime  ( w))  ^ {1/p}  \in  H _ {p} .
 +
$$
  
The properties of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583025.png" /> are closest to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583026.png" /> in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583027.png" /> is a [[Smirnov domain|Smirnov domain]]. They have been generalized to domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583028.png" /> with boundaries of finite Hausdorff length. See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]].
+
The properties of the classes $  E _ {p} ( G) $
 +
are closest to those of $  H _ {p} $
 +
in the case when $  G $
 +
is a [[Smirnov domain|Smirnov domain]]. They have been generalized to domains $  G $
 +
with boundaries of finite Hausdorff length. See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "Sur les formules de Cauchy et de Green et quelques problèmes qui s'y rattachent"  ''Izv. Akad. Nauk SSSR. Otdel. Mat. i Estestv. Nauk'' , '''3'''  (1932)  pp. 337–372</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Keldysh,  M.A. Lavrent'ev,  "Sur la répresentation conforme des domaines limités par des courbes rectifiables"  ''Ann. Sci. Ecole Norm. Sup.'' , '''54'''  (1937)  pp. 1–38</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><TR><TD valign="top">[4]</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><TR><TD valign="top">[5]</TD> <TD valign="top">  P.L. Duren,  "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583029.png" /> spaces" , Acad. Press  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "Sur les formules de Cauchy et de Green et quelques problèmes qui s'y rattachent"  ''Izv. Akad. Nauk SSSR. Otdel. Mat. i Estestv. Nauk'' , '''3'''  (1932)  pp. 337–372</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Keldysh,  M.A. Lavrent'ev,  "Sur la répresentation conforme des domaines limités par des courbes rectifiables"  ''Ann. Sci. Ecole Norm. Sup.'' , '''54'''  (1937)  pp. 1–38</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><TR><TD valign="top">[4]</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><TR><TD valign="top">[5]</TD> <TD valign="top">  P.L. Duren,  "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085830/s08583029.png" /> spaces" , Acad. Press  (1970)</TD></TR></table>

Latest revision as of 01:28, 26 April 2022


The set $ E _ {p} ( G) $ of all functions $ f( z) $ holomorphic in a simply-connected domain $ G \subset \mathbf C $ with rectifiable Jordan boundary $ \Gamma $, such that for every function in it there is a sequence of closed rectifiable Jordan curves $ \Gamma _ {n} ( f ) \subset G $, $ n = 1, 2 \dots $ with the following properties:

1) $ \Gamma _ {n} ( f ) $ tends to $ \Gamma $ as $ n \rightarrow \infty $ in the sense that if $ G _ {n} ( f ) $ is the bounded domain with boundary $ \Gamma _ {n} ( f ) $, then

$$ G _ {1} ( f ) \subset \dots \subset G _ {n} ( f ) \subset G \ \ \textrm{ and } \ \cup _ {n= 1 } ^ \infty G _ {n} ( f ) = G; $$

2)

$$ \sup _ { n } \left \{ \int\limits _ {\Gamma _ {n} ( f ) } | f( z) | ^ {p} | dz | \right \} < \infty \ ( p> 0 \ \textrm{ fixed } ). $$

This definition was proposed by M.V. Keldysh and M.A. Lavrent'ev [2], and is equivalent to V.I. Smirnov's definition [1] in which curves $ \gamma ( \rho ) $ are used instead of $ \Gamma _ {n} ( f ) $. These curves are the images of the circles $ | w | = \rho < 1 $ under some univalent conformal mapping $ z= \phi ( w) $ from the disc $ | w | < 1 $ onto the domain $ G $, and the supremum is taken over all $ \rho \in ( 0, 1) $.

The classes $ E _ {p} ( G) $ are the best known and most thoroughly studied generalization of the Hardy classes $ H _ {p} $, and are connected with them by the following relation: $ f \in E _ {p} ( G) $ if and only if

$$ f( \phi ( w))( \phi ^ \prime ( w)) ^ {1/p} \in H _ {p} . $$

The properties of the classes $ E _ {p} ( G) $ are closest to those of $ H _ {p} $ in the case when $ G $ is a Smirnov domain. They have been generalized to domains $ G $ with boundaries of finite Hausdorff length. See also Boundary properties of analytic functions.

References

[1] V.I. Smirnov, "Sur les formules de Cauchy et de Green et quelques problèmes qui s'y rattachent" Izv. Akad. Nauk SSSR. Otdel. Mat. i Estestv. Nauk , 3 (1932) pp. 337–372
[2] M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domaines limités par des courbes rectifiables" Ann. Sci. Ecole Norm. Sup. , 54 (1937) pp. 1–38
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[5] P.L. Duren, "Theory of spaces" , Acad. Press (1970)
How to Cite This Entry:
Smirnov class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smirnov_class&oldid=14423
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article