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

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

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

$\frac{\mathrm{\Phi }(x)}{x}\to \mathrm{\infty },\text{as }x\to \mathrm{\infty },$

$\frac{\mathrm{\Phi }(x)}{x}\to 0,\text{as }x\to 0.$

Let $L_{\mathrm{\Phi }}^{†}$ be the set of measurable functions f : X → R such that the integral

$\underset{X}{\int }\mathrm{\Phi }(|f|)d\mu$

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 $L_{\mathrm{\Phi }}^{†}$ is the Birnbaum–Orlicz space, denoted $L_{\mathrm{\Phi }}$.

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

$\mathrm{\Psi }(x)=\underset{0}{\overset{x}{\int }}({\mathrm{\Phi }}^{\prime }{)}^{-1}(t)dt.$

Note that Young's inequality for products holds:

$ab\le \mathrm{\Phi }(a)+\mathrm{\Psi }(b).$

The norm is then given by

$\Vert f{\Vert }_{\mathrm{\Phi }}=\sup \{\Vert fg{\Vert }_1\mid \int \mathrm{\Psi }\circ |g|d\mu \le 1\}.$

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

An equivalent norm (Rao Ren), called the Luxemburg norm, is defined on L_Φ by

$\Vert f\Vert {\text{'}}_{\mathrm{\Phi }}=\inf \{k\in (0,\mathrm{\infty })\mid \underset{X}{\int }\mathrm{\Phi }(|f|/k)d\mu \le 1\},$

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

Example

Here is an example where $L_{\mathrm{\Phi }}^{†}$ is not a vector space and is strictly smaller than $L_{\mathrm{\Phi }}$. 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 $L_{\mathrm{\Phi }}$ but is only in the set $L_{\mathrm{\Phi }}^{†}$ if |a| < 1.

Properties

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

Relations to Sobolev spaces

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

$W_0^{1,p}(X)\subseteq L^{\phi }(X)$

for

$\phi (t):=\exp (|t{|}^{p/(p-1)})-1.$

This is the analytical content of the Trudinger inequality: For $X\subseteq {\mathbb{ℝ}}^n$ open and bounded with Lipschitz boundary $\partial X$, consider the space $W_0^{k,p}(X)$, $kp=n$. There exist constants $C_1,C_2>0$ such that

$\underset{X}{\int }\exp \left({\left(\frac{|u(x)|}{C_1\Vert {\mathrm{D}}^{k}u{\Vert }_{L^p(X)}}\right)}^{p/(p-1)}\right)\mathrm{d}x\le C_2|X|.$

Orlicz norm of a random variable

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

$\Vert X{\Vert }_{\mathrm{\Psi }}\triangleq \inf \{k\in (0,\mathrm{\infty })\mid \mathrm{E}[\mathrm{\Psi }(|X|/k)]\le 1\}.$

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

When $\mathrm{\Psi }(x)=x^p$, 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 ${\mathrm{\Psi }}_q(x)=\exp (x^q)-1$ (for $q\ge 1$). A random variable with finite ${\mathrm{\Psi }}_2$ norm is said to be "sub-Gaussian" and a random variable with finite ${\mathrm{\Psi }}_1$ norm is said to be "sub-exponential". Indeed, the boundedness of the ${\mathrm{\Psi }}_p$ norm characterizes the limiting behavior of the probability density function:

$\Vert X{\Vert }_{{\mathrm{\Psi }}_p}=c\to \underset{x\to \mathrm{\infty }}{\lim }{f}_X(x)\exp (|x/c{|}^p)=0,$

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

The ${\mathrm{\Psi }}_1$ 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 $M_X(t)=(1-2t{)}^{-K/2}$, so that the reciprocal of the ${\mathrm{\Psi }}_1$ norm is related to the functional inverse of the moment-generating function:

$\Vert X{\Vert }_{{\mathrm{\Psi }}_1}^{-1}=M_X^{-1}(2)=(1-4^{-1/K})/2.$

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 .