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

\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

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

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

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

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

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

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

Note that Young's inequality holds:

ab\leq \Phi (a)+\Psi (b).

The norm is then given by

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

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

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

\|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 $L_{\Phi }^{\dagger }$ is not a vector space and is strictly smaller than $L_{\Phi }$ . 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 $L_{\Phi }$ but is only in the set $L_{\Phi }^{\dagger }$ if |a|<1.

Properties

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

Relations to Sobolev spaces

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

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

for

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

\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:

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

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

When $\Psi (x)=x^{p}$ , 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 $\Psi _{q}(x)=\exp(x^{q})-1$ (for $q\geq 1$ ). A random variable with finite $\Psi _{2}$ norm is said to be "sub-Gaussian" and a random variable with finite $\Psi _{1}$ norm is said to be "sub-exponential". Indeed, the boundedness of the $\Psi _{p}$ norm characterizes the limiting behavior of the probability density function:

\|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 $\exp(-|x/c|^{p})$ .

The $\Psi _{1}$ 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 inverse of the $\Psi _{1}$ norm is related to the functional inverse of the moment-generating function:

\|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 .

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