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 L^p spaces. Like the L^p spaces, they are Banach spaces. The spaces are named for Władysław Orlicz and Zygmunt William Birnbaum, who first defined them in 1931.

Besides the L^p 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 : X → R such that the integral

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

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

This might not be a vector space (i.e., it might 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 for products 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), called the Luxemburg norm, is defined on L_Φ by

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

and likewise L_Φ(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 L^p spaces (for ${\displaystyle 1<p<\infty }$) 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 reciprocal 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.