Orlicz class

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)

References

[a1]	W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)