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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702101.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702102.png" /> which satisfy the condition
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{{TEX|done}}
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The set $L_M$ of functions $x(t)$ which satisfy the 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/o/o070/o070210/o0702103.png" /></td> </tr></table>
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$$\int\limits_GM(x(t))dt<\infty,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702104.png" /> is a bounded closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702106.png" /> is the [[Lebesgue measure|Lebesgue measure]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702107.png" /> is an even [[Convex function (of a real variable)|convex function (of a real variable)]] which is increasing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o0702108.png" /> positive, and
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where $G$ is a bounded closed set in $\mathbf R^n$, $dt$ is the [[Lebesgue measure|Lebesgue measure]], $M(u)$ is an even [[Convex function (of a real variable)|convex function (of a real variable)]] which is increasing for $u$ positive, 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/o/o070/o070210/o0702109.png" /></td> </tr></table>
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$$\lim_{u\to0}u^{-1}M(u)=\lim_{u\to\infty}u[M(u)]^{-1}=0.$$
  
These functions are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021011.png" />-functions. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021012.png" /> can be represented as
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These functions are called $N$-functions. The function $M(u)$ can be represented as
  
<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/o070210/o07021013.png" /></td> </tr></table>
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$$M(u)=\int\limits_0^{|u|}p(v)dv,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021014.png" /> does not decrease on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021015.png" />,
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where $p(v)=M'(v)$ does not decrease on $[0,\infty)$,
  
<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/o070210/o07021016.png" /></td> </tr></table>
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$$p(0)=\lim_{v\to0}p(v)=0,$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021017.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021018.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021019.png" /> and
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and $p(v)>0$ when $v>0$. The function $M(u)$ 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/o/o070/o070210/o07021020.png" /></td> </tr></table>
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$$N(u)=\int\limits_0^{|u|}p^{-1}(v)dv,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021021.png" /> is the inverse function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021022.png" />, are called complementary functions. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021026.png" />. For a pair of complementary functions, the Young inequality
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where $p^{-1}(v)$ is the inverse function of $p(v)$, are called complementary functions. For example, if $M(u)=u^p/p$, $1<p<\infty$, then $N(u)=n^q/q$, where $p^{-1}+q^{-1}=1$. For a pair of complementary functions, the Young inequality
  
<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/o070210/o07021027.png" /></td> </tr></table>
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$$ab\leq M(a)+N(b)$$
  
 
holds.
 
holds.
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021028.png" /> is said to satisfy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021030.png" />-condition if there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021031.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021034.png" />. An Orlicz class is linear if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021035.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021036.png" />-condition. The convexity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021037.png" /> follows from the [[Jensen inequality|Jensen inequality]].
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The function $M(u)$ is said to satisfy the $\Delta_2$-condition if there exist a $C$ and an $u_0$ such that $M(2u)\leq CM(u)$ for all $u\geq u_0$. An Orlicz class is linear if and only if $M(u)$ satisfies the $\Delta_2$-condition. The convexity of $L_M$ follows from the [[Jensen inequality|Jensen inequality]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021039.png" /> be two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021040.png" />-functions. In order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021041.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021042.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021043.png" /> and sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070210/o07021044.png" />.
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Let $M_1(u)$ and $M_2(u)$ be two $N$-functions. In order that $L_{M_1}\subset L_{M_2}$ it is necessary and sufficient that $M_2(u)\leq CM_1(u)$ for a certain $C$ and sufficiently large $u$.
  
 
Orlicz classes were examined in [[#References|[1]]] by W. Orlicz and Z. Birnbaum.
 
Orlicz classes were examined in [[#References|[1]]] by W. Orlicz and Z. Birnbaum.

Latest revision as of 14:21, 11 August 2014

The set $L_M$ of functions $x(t)$ which satisfy the condition

$$\int\limits_GM(x(t))dt<\infty,$$

where $G$ is a bounded closed set in $\mathbf R^n$, $dt$ is the Lebesgue measure, $M(u)$ is an even convex function (of a real variable) which is increasing for $u$ positive, and

$$\lim_{u\to0}u^{-1}M(u)=\lim_{u\to\infty}u[M(u)]^{-1}=0.$$

These functions are called $N$-functions. The function $M(u)$ can be represented as

$$M(u)=\int\limits_0^{|u|}p(v)dv,$$

where $p(v)=M'(v)$ does not decrease on $[0,\infty)$,

$$p(0)=\lim_{v\to0}p(v)=0,$$

and $p(v)>0$ when $v>0$. The function $M(u)$ and

$$N(u)=\int\limits_0^{|u|}p^{-1}(v)dv,$$

where $p^{-1}(v)$ is the inverse function of $p(v)$, are called complementary functions. For example, if $M(u)=u^p/p$, $1<p<\infty$, then $N(u)=n^q/q$, where $p^{-1}+q^{-1}=1$. For a pair of complementary functions, the Young inequality

$$ab\leq M(a)+N(b)$$

holds.

The function $M(u)$ is said to satisfy the $\Delta_2$-condition if there exist a $C$ and an $u_0$ such that $M(2u)\leq CM(u)$ for all $u\geq u_0$. An Orlicz class is linear if and only if $M(u)$ satisfies the $\Delta_2$-condition. The convexity of $L_M$ follows from the Jensen inequality.

Let $M_1(u)$ and $M_2(u)$ be two $N$-functions. In order that $L_{M_1}\subset L_{M_2}$ it is necessary and sufficient that $M_2(u)\leq CM_1(u)$ for a certain $C$ and sufficiently large $u$.

Orlicz classes were examined in [1] by W. Orlicz and Z. Birnbaum.

References

[1] Z. Birnbaum, W. Orlicz, "Ueber die Verallgemeinerungen des Begriffes der zueinander konjugierten Potenzen" Studia Math. , 3 (1931) pp. 1–67
[2] M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian)


Comments

References

[a1] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
How to Cite This Entry:
Orlicz class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orlicz_class&oldid=18820
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article