Namespaces
Variants
Actions

Difference between revisions of "Orlicz space"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (Undo revision 48069 by Ulf Rehmann (talk))
Tag: Undo
Line 1: Line 1:
<!--
+
A [[Banach space|Banach space]] of measurable functions introduced by W. Orlicz [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702202.png" /> be a pair of complementary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702204.png" />-functions (see [[Orlicz class|Orlicz class]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702205.png" /> be a bounded closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702206.png" />. The Orlicz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702207.png" /> is the set of Lebesgue-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702208.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702209.png" /> for which
o0702201.png
 
$#A+1 = 67 n = 1
 
$#C+1 = 67 : ~/encyclopedia/old_files/data/O070/O.0700220 Orlicz space
 
Automatically converted into TeX, above some diagnostics.
 
Please remove this comment and the {{TEX|auto}} line below,
 
if TeX found to be correct.
 
-->
 
  
{{TEX|auto}}
+
<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/o/o070/o070220/o07022010.png" /></td> </tr></table>
{{TEX|done}}
 
  
A [[Banach space|Banach space]] of measurable functions introduced by W. Orlicz [[#References|[1]]]. Let  $  M( u) $
+
An Orlicz space is a complete normed space with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022011.png" />, which is called the Orlicz norm. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022014.png" /> coincides with the [[Riesz space|Riesz space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022015.png" /> and, up to a scalar factor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022016.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022017.png" />.
and $  N( u) $
 
be a pair of complementary  $  N $-
 
functions (see [[Orlicz class|Orlicz class]]) and let  $  G $
 
be a bounded closed set in  $  \mathbf R  ^ {n} $.  
 
The Orlicz space  $  L _ {M}  ^  \star  $
 
is the set of Lebesgue-measurable functions  $  x $
 
on  $  G $
 
for which
 
  
$$
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022019.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022021.png" />-functions, then the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022022.png" /> holds if and only if, for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022023.png" /> and all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022024.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022025.png" /> is fulfilled. For every Orlicz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022026.png" /> the inclusions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022027.png" /> hold. Every summable function belongs to some Orlicz space.
\| x \| _ {M}  = \sup \left \{ {\int\limits _ { G } x( t) y( t)  dt } : {
 
\int\limits _ { G } N( y( t))  dt \leq  1 } \right \}
 
< \infty .
 
$$
 
  
An Orlicz space is a complete normed space with respect to the norm  $  \| x \| _ {M} $,
+
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022028.png" /> is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022029.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022031.png" />-condition (cf. [[Orlicz class|Orlicz class]]). In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022032.png" /> is not dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022033.png" />, and the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022035.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022036.png" /> and is always separable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022037.png" />, then
which is called the Orlicz norm. When  $  M( u) = u  ^ {p} $,  
 
$  1 < p < \infty $,  
 
$  L _ {M}  ^  \star  $
 
coincides with the [[Riesz space|Riesz space]]  $  L _ {p} $
 
and, up to a scalar factor,  $  \| x \| _ {L _ {p}  } $
 
coincides with  $  \| x \| _ {M} $.
 
  
If  $  M _ {1} ( u) $
+
<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/o/o070/o070220/o07022038.png" /></td> </tr></table>
and  $  M _ {2} ( u) $
 
are  $  N $-
 
functions, then the inclusion  $  L _ {M _ {1}  }  ^  \star  \subset  L _ {M _ {2}  }  ^  \star  $
 
holds if and only if, for a certain  $  C $
 
and all sufficiently large  $  u $,
 
the inequality  $  M _ {2} ( u) \leq  M _ {1} ( Cu) $
 
is fulfilled. For every Orlicz space  $  L _ {M}  ^  \star  $
 
the inclusions  $  L _  \infty  \subset  L _ {M}  ^  \star  \subset  L _ {1} $
 
hold. Every summable function belongs to some Orlicz space.
 
 
 
The space  $  L _ {M}  ^  \star  $
 
is separable if and only if  $  M( u) $
 
satisfies the  $  \Delta _ {2} $-
 
condition (cf. [[Orlicz class|Orlicz class]]). In general,  $  L _  \infty  $
 
is not dense in  $  L _ {M}  ^  \star  $,
 
and the closure of  $  L _  \infty  $
 
in  $  L _ {M}  ^  \star  $
 
is denoted by  $  E _ {M} $
 
and is always separable. If  $  x \in L _ {M}  ^  \star  $,
 
then
 
 
 
$$
 
\lim\limits _ {\tau \rightarrow \infty }  \sup _ {mes( E) = \tau }  \| x {\chi _ {E} } \| _ {M}  =  \rho ( x, E _ {M} ),
 
$$
 
  
 
where
 
where
  
$$
+
<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/o/o070/o070220/o07022039.png" /></td> </tr></table>
\chi _ {E} ( t)  = \left \{
 
  
If $  M( u) $
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022041.png" /> are complementary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022042.png" />-functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022044.png" />, then the following analogue of the [[Hölder inequality|Hölder inequality]] holds:
and $  N( u) $
 
are complementary $  N $-
 
functions and $  x \in L _ {M}  ^  \star  $,  
 
$  y \in L _ {M}  ^  \star  $,  
 
then the following analogue of the [[Hölder inequality|Hölder inequality]] holds:
 
  
$$
+
<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/o/o070/o070220/o07022045.png" /></td> </tr></table>
\int\limits _ { G } x( t) y( t)  dt  \leq  \| x \| _ {(} M) \| y \| _ {(} N) ,
 
$$
 
  
where $  \| x \| _ {(} M) $
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022046.png" /> is the [[Luxemburg norm|Luxemburg norm]]. Every continuous linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022048.png" /> can be represented in the form
is the [[Luxemburg norm|Luxemburg norm]]. Every continuous linear functional $  f $
 
on $  E _ {M} $
 
can be represented in the form
 
  
$$
+
<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/o/o070/o070220/o07022049.png" /></td> </tr></table>
f( x)  = \int\limits _ { G } x( t) y( t)  dt,
 
$$
 
  
where $  y \in L _ {N} $
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022051.png" />.
and $  \| f \| = \| y \| _ {(} N) $.
 
  
The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces $  L _ {p} $
+
The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022052.png" /> can also be applied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022053.png" />. The following conditions are equivalent:
can also be applied to $  E _ {M} $.  
 
The following conditions are equivalent:
 
  
1) the space $  L _ {M}  ^  \star  $
+
1) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022054.png" /> is reflexive;
is reflexive;
 
  
2) $  M( u) $
+
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022056.png" /> satisfy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022057.png" />-condition;
and $  N( u) $
 
satisfy the $  \Delta _ {2} $-
 
condition;
 
  
3) an unconditional [[Basis|basis]] exists in $  L _ {M}  ^  \star  $;
+
3) an unconditional [[Basis|basis]] exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022058.png" />;
  
4) the [[Haar system|Haar system]] forms an unconditional basis in $  L _ {M}  ^  \star  $;
+
4) the [[Haar system|Haar system]] forms an unconditional basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022059.png" />;
  
5) the trigonometric system is a basis in $  L _ {M}  ^  \star  $
+
5) the trigonometric system is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022060.png" /> and the Haar system is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022061.png" />.
and the Haar system is a basis in $  E _ {M} $.
 
  
A sequence space $  l _ {M}  ^  \star  $
+
A sequence space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022062.png" /> is defined in the same way, but the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022063.png" /> depend on the asymptotics of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022064.png" /> at 0. Many of the geometric properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022066.png" /> are studied in [[#References|[5]]]; for example, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022067.png" />, the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022069.png" /> is isomorphically imbeddable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022070.png" /> can be found.
is defined in the same way, but the properties of $  l _ {M}  ^  \star  $
 
depend on the asymptotics of the function $  M( u) $
 
at 0. Many of the geometric properties of $  L _ {M}  ^  \star  $
 
and $  l _ {M}  ^  \star  $
 
are studied in [[#References|[5]]]; for example, for any function $  M( u) $,  
 
the set of all $  p $
 
such that $  l _ {p} $
 
is isomorphically imbeddable in $  L _ {M}  ^  \star  $
 
can be found.
 
  
 
Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.
 
Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.
Line 123: Line 43:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Orlicz,  "Ueber eine gewisse Klasse von Räumen vom Typus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022071.png" />"  ''Bull. Intern. Acad. Pol. Ser. A'' , '''8/9'''  (1932)  pp. 207–220</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,  Ya.B. Rutitskii,  "Convex functions and Orlicz spaces" , Noordhoff  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Gaposhkin,  "Existence of absolute bases in Orlicz spaces"  ''Funct. Anal. Appl.'' , '''1''' :  4  (1967)  pp. 278–284  ''Funkts. Anal. i Prilozhen.'' , '''1''' :  4  (1967)  pp. 26–32</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces" , '''1–2''' , Springer  (1977–1979)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Orlicz,  "Ueber eine gewisse Klasse von Räumen vom Typus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022071.png" />"  ''Bull. Intern. Acad. Pol. Ser. A'' , '''8/9'''  (1932)  pp. 207–220</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,  Ya.B. Rutitskii,  "Convex functions and Orlicz spaces" , Noordhoff  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Gaposhkin,  "Existence of absolute bases in Orlicz spaces"  ''Funct. Anal. Appl.'' , '''1''' :  4  (1967)  pp. 278–284  ''Funkts. Anal. i Prilozhen.'' , '''1''' :  4  (1967)  pp. 26–32</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces" , '''1–2''' , Springer  (1977–1979)</TD></TR></table>
 +
 +
  
 
====Comments====
 
====Comments====
 +
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR></table>

Revision as of 14:52, 7 June 2020

A Banach space of measurable functions introduced by W. Orlicz [1]. Let and be a pair of complementary -functions (see Orlicz class) and let be a bounded closed set in . The Orlicz space is the set of Lebesgue-measurable functions on for which

An Orlicz space is a complete normed space with respect to the norm , which is called the Orlicz norm. When , , coincides with the Riesz space and, up to a scalar factor, coincides with .

If and are -functions, then the inclusion holds if and only if, for a certain and all sufficiently large , the inequality is fulfilled. For every Orlicz space the inclusions hold. Every summable function belongs to some Orlicz space.

The space is separable if and only if satisfies the -condition (cf. Orlicz class). In general, is not dense in , and the closure of in is denoted by and is always separable. If , then

where

If and are complementary -functions and , , then the following analogue of the Hölder inequality holds:

where is the Luxemburg norm. Every continuous linear functional on can be represented in the form

where and .

The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces can also be applied to . The following conditions are equivalent:

1) the space is reflexive;

2) and satisfy the -condition;

3) an unconditional basis exists in ;

4) the Haar system forms an unconditional basis in ;

5) the trigonometric system is a basis in and the Haar system is a basis in .

A sequence space is defined in the same way, but the properties of depend on the asymptotics of the function at 0. Many of the geometric properties of and are studied in [5]; for example, for any function , the set of all such that is isomorphically imbeddable in can be found.

Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.

References

[1] W. Orlicz, "Ueber eine gewisse Klasse von Räumen vom Typus " Bull. Intern. Acad. Pol. Ser. A , 8/9 (1932) pp. 207–220
[2] M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian)
[3] V.F. Gaposhkin, "Existence of absolute bases in Orlicz spaces" Funct. Anal. Appl. , 1 : 4 (1967) pp. 278–284 Funkts. Anal. i Prilozhen. , 1 : 4 (1967) pp. 26–32
[4] S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[5] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979)


Comments

References

[a1] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
How to Cite This Entry:
Orlicz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orlicz_space&oldid=48069
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article