Orlicz class

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


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


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


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


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


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

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


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.


[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)



[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:
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article