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

[math]\displaystyle{ \int_{\mathbb{R}^n} |f(x)|\log^+ |f(x)|\,dx \lt \infty. }[/math]

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

[math]\displaystyle{ \frac{\Phi(x)} x \to \infty,\quad\text{as }x\to \infty, }[/math]

[math]\displaystyle{ \frac{\Phi(x)} x \to 0,\quad\text{as }x\to 0. }[/math]

Let [math]\displaystyle{ L^\dagger_\Phi }[/math] be the set of measurable functions f : X → R such that the integral

[math]\displaystyle{ \int_X \Phi(|f|)\, d\mu }[/math]

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 [math]\displaystyle{ L^\dagger_\Phi }[/math] is the Birnbaum–Orlicz space, denoted [math]\displaystyle{ L_\Phi }[/math].

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

[math]\displaystyle{ \Psi(x) = \int_0^x (\Phi')^{-1}(t)\, dt. }[/math]

Note that Young's inequality for products holds:

[math]\displaystyle{ ab\le \Phi(a) + \Psi(b). }[/math]

The norm is then given by

[math]\displaystyle{ \|f\|_\Phi = \sup\left\{\|fg\|_1\mid \int \Psi\circ |g|\, d\mu \le 1\right\}. }[/math]

Furthermore, the space [math]\displaystyle{ L_\Phi }[/math] 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

[math]\displaystyle{ \|f\|'_\Phi = \inf\left\{k\in (0,\infty)\mid\int_X \Phi(|f|/k)\,d\mu\le 1 \right\}, }[/math]

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

Example

Here is an example where [math]\displaystyle{ L^\dagger_\Phi }[/math] is not a vector space and is strictly smaller than [math]\displaystyle{ L_\Phi }[/math]. 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 [math]\displaystyle{ L_\Phi }[/math] but is only in the set [math]\displaystyle{ L^\dagger_\Phi }[/math] if |a| < 1.

Properties

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

Relations to Sobolev spaces

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

[math]\displaystyle{ W_0^{1, p} (X) \subseteq L^\varphi (X) }[/math]

for

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

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

[math]\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 |. }[/math]

Orlicz norm of a random variable

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

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

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

When [math]\displaystyle{ \Psi(x) = x^p }[/math], 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 [math]\displaystyle{ \Psi_q(x) = \exp(x^q)-1 }[/math] (for [math]\displaystyle{ q \geq 1 }[/math]). A random variable with finite [math]\displaystyle{ \Psi_2 }[/math] norm is said to be "sub-Gaussian" and a random variable with finite [math]\displaystyle{ \Psi_1 }[/math] norm is said to be "sub-exponential". Indeed, the boundedness of the [math]\displaystyle{ \Psi_p }[/math] norm characterizes the limiting behavior of the probability density function:

[math]\displaystyle{ \|X\|_{\Psi_p} = c \rightarrow \lim_{x \rightarrow \infty} f_X(x) \exp(|x/c|^p) = 0, }[/math]

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

The [math]\displaystyle{ \Psi_1 }[/math] 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 [math]\displaystyle{ M_X(t) = (1-2t)^{-K/2} }[/math], so that the reciprocal of the [math]\displaystyle{ \Psi_1 }[/math] norm is related to the functional inverse of the moment-generating function:

[math]\displaystyle{ \|X\|_{\Psi_1} ^{-1} = M_X^{-1}(2) = (1-4^{-1/K})/2. }[/math]

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 .