# Birnbaum–Orlicz space

In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are Banach spaces. The spaces are named for Władysław Orlicz and Zygmunt William Birnbaum, who first defined them in 1931.

Besides the Lp spaces, a variety of function spaces arising naturally in analysis are Birnbaum–Orlicz spaces. One such space L log+ L, which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions f such that the integral

${\displaystyle \int _{\mathbb {R} ^{n}}|f(x)|\log ^{+}|f(x)|\,dx<\infty .}$

Here log+ is the positive part of the logarithm. Also included in the class of Birnbaum–Orlicz spaces are many of the most important Sobolev spaces.

## Formal definition

Suppose that μ is a σ-finite measure on a set X, and Φ : [0, ∞) → [0, ∞) is a Young function, i.e., a convex function such that

${\displaystyle {\frac {\Phi (x)}{x}}\to \infty ,\quad {\text{as }}x\to \infty ,}$
${\displaystyle {\frac {\Phi (x)}{x}}\to 0,\quad {\text{as }}x\to 0.}$

Let ${\displaystyle L_{\Phi }^{\dagger }}$ be the set of measurable functions f : XR such that the integral

${\displaystyle \int _{X}\Phi (|f|)\,d\mu }$

is finite, where, as usual, functions that agree almost everywhere are identified.

This may not be a vector space (it may fail to be closed under scalar multiplication). The vector space of functions spanned by ${\displaystyle L_{\Phi }^{\dagger }}$ is the Birnbaum–Orlicz space, denoted ${\displaystyle L_{\Phi }}$.

To define a norm on ${\displaystyle L_{\Phi }}$, let Ψ be the Young complement of Φ; that is,

${\displaystyle \Psi (x)=\int _{0}^{x}(\Phi ')^{-1}(t)\,dt.}$

Note that Young's inequality holds:

${\displaystyle ab\leq \Phi (a)+\Psi (b).}$

The norm is then given by

${\displaystyle \|f\|_{\Phi }=\sup \left\{\|fg\|_{1}\mid \int \Psi \circ |g|\,d\mu \leq 1\right\}.}$

Furthermore, the space ${\displaystyle L_{\Phi }}$ is precisely the space of measurable functions for which this norm is finite.

An equivalent norm (Rao & Ren 1991, §3.3) is defined on LΦ by

${\displaystyle \|f\|'_{\Phi }=\inf \left\{k\in (0,\infty )\mid \int _{X}\Phi (|f|/k)\,d\mu \leq 1\right\},}$

and likewise LΦ(μ) is the space of all measurable functions for which this norm is finite.

### Example

Here is an example where ${\displaystyle L_{\Phi }^{\dagger }}$ is not a vector space and is strictly smaller than ${\displaystyle L_{\Phi }}$. Suppose that X is the open unit interval (0,1), Φ(x) = exp(x) – 1 – x, and f(x) = log(x). Then af is in the space ${\displaystyle L_{\Phi }}$ but is only in the set ${\displaystyle L_{\Phi }^{\dagger }}$ if |a| < 1.

## Properties

• Orlicz spaces generalize Lp spaces (for ${\displaystyle 1) in the sense that if ${\displaystyle \varphi (t)=|t|^{p}}$, then ${\displaystyle \|u\|_{L^{\varphi }(X)}=\|u\|_{L^{p}(X)}}$, so ${\displaystyle L^{\varphi }(X)=L^{p}(X)}$.
• The Orlicz space ${\displaystyle L^{\varphi }(X)}$ is a Banach space — a complete normed vector space.

## Relations to Sobolev spaces

Certain Sobolev spaces are embedded in Orlicz spaces: for ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ open and bounded with Lipschitz boundary ${\displaystyle \partial X}$,

${\displaystyle W_{0}^{1,p}(X)\subseteq L^{\varphi }(X)}$

for

${\displaystyle \varphi (t):=\exp \left(|t|^{p/(p-1)}\right)-1.}$

This is the analytical content of the Trudinger inequality: For ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ open and bounded with Lipschitz boundary ${\displaystyle \partial X}$, consider the space ${\displaystyle W_{0}^{k,p}(X)}$, ${\displaystyle kp=n}$. There exist constants ${\displaystyle C_{1},C_{2}>0}$ such that

${\displaystyle \int _{X}\exp \left(\left({\frac {|u(x)|}{C_{1}\|\mathrm {D} ^{k}u\|_{L^{p}(X)}}}\right)^{p/(p-1)}\right)\,\mathrm {d} x\leq C_{2}|X|.}$

## Orlicz norm of a random variable

Similarly, the Orlicz norm of a random variable characterizes it as follows:

${\displaystyle \|X\|_{\Psi }\triangleq \inf \left\{k\in (0,\infty )\mid \operatorname {E} [\Psi (|X|/k)]\leq 1\right\}.}$

This norm is homogeneous and is defined only when this set is non-empty.

When ${\displaystyle \Psi (x)=x^{p}}$, this coincides with the p-th moment of the random variable. Other special cases in the exponential family are taken with respect to the functions ${\displaystyle \Psi _{q}(x)=\exp(x^{q})-1}$ (for ${\displaystyle q\geq 1}$). A random variable with finite ${\displaystyle \Psi _{2}}$ norm is said to be "sub-Gaussian" and a random variable with finite ${\displaystyle \Psi _{1}}$ norm is said to be "sub-exponential". Indeed, the boundedness of the ${\displaystyle \Psi _{p}}$ norm characterizes the limiting behavior of the probability density function:

${\displaystyle \|X\|_{\Psi _{p}}=c\rightarrow \lim _{x\rightarrow \infty }f_{X}(x)\exp(|x/c|^{p})=0,}$

so that the tail of this probability density function asymptotically resembles, and is bounded above by ${\displaystyle \exp(-|x/c|^{p})}$.

The ${\displaystyle \Psi _{1}}$ norm may be easily computed from a strictly monotonic moment-generating function. For example, the moment-generating function of a chi-squared random variable X with K degrees of freedom is ${\displaystyle M_{X}(t)=(1-2t)^{-K/2}}$, so that the inverse of the ${\displaystyle \Psi _{1}}$ norm is related to the functional inverse of the moment-generating function:

${\displaystyle \|X\|_{\Psi _{1}}^{-1}=M_{X}^{-1}(2)=(1-4^{-1/K})/2.}$

## References

• Birnbaum, Z. W.; Orlicz, W. (1931), "Über die Verallgemeinerung des Begriffes der zueinander Konjugierten Potenzen", Studia Mathematica, 3: 1–67 PDF.
• Bund, Iracema (1975), "Birnbaum–Orlicz spaces of functions on groups", Pacific Mathematics Journal, 58 (2): 351–359.
• Hewitt, Edwin; Stromberg, Karl, Real and abstract analysis, Springer-Verlag.
• Krasnosel'skii, M.A.; Rutickii, Ya.B. (1961), Convex Functions and Orlicz Spaces, Groningen: P.Noordhoff Ltd
• Rao, M.M.; Ren, Z.D. (1991), Theory of Orlicz Spaces, Pure and Applied Mathematics, Marcel Dekker, ISBN 0-8247-8478-2.
• Zygmund, Antoni, "Chapter IV: Classes of functions and Fourier series", Trigonometric series, Volume 1 (3rd ed.), Cambridge University Press.
• Ledoux, Michel; Talagrand, Michel, Probability in Banach Spaces, Springer-Verlag.