# Orlicz space

A Banach space of measurable functions introduced by W. Orlicz [1]. Let $M( u)$ and $N( u)$ be a pair of complementary $N$- functions (see 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

$$\| x \| _ {M} = \sup \left \{ {\int\limits _ { G } x( t) y( t) dt } : { \int\limits _ { G } N( y( t)) dt \leq 1 } \right \} < \infty .$$

An Orlicz space is a complete normed space with respect to the norm $\| x \| _ {M}$, which is called the Orlicz norm. When $M( u) = u ^ {p}$, $1 < p < \infty$, $L _ {M} ^ \star$ coincides with the Riesz space $L _ {p}$ and, up to a scalar factor, $\| x \| _ {L _ {p} }$ coincides with $\| x \| _ {M}$.

If $M _ {1} ( u)$ 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). 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

$$\chi _ {E} ( t) = \left \{ \begin{array}{ll} 1 , & t \in E, \\ 0, & t \notin E. \\ \end{array} \right .$$

If $M( u)$ 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 holds:

$$\int\limits _ { G } x( t) y( t) dt \leq \| x \| _ {(} M) \| y \| _ {(} N) ,$$

where $\| x \| _ {(} M)$ is the Luxemburg norm. Every continuous linear functional $f$ on $E _ {M}$ can be represented in the form

$$f( x) = \int\limits _ { G } x( t) y( t) dt,$$

where $y \in L _ {N}$ and $\| f \| = \| y \| _ {(} N)$.

The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces $L _ {p}$ can also be applied to $E _ {M}$. The following conditions are equivalent:

1) the space $L _ {M} ^ \star$ is reflexive;

2) $M( u)$ and $N( u)$ satisfy the $\Delta _ {2}$- condition;

3) an unconditional basis exists in $L _ {M} ^ \star$;

4) the Haar system forms an unconditional basis in $L _ {M} ^ \star$;

5) the trigonometric system is a basis in $L _ {M} ^ \star$ and the Haar system is a basis in $E _ {M}$.

A sequence space $l _ {M} ^ \star$ 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 [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.

#### 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)