Birnbaum–Orlicz space

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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

\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[edit]

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^\dagger_\Phi be the set of measurable functions f : XR 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^\dagger_\Phi 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\le \Phi(a) + \Psi(b).

The norm is then given by

\|f\|_\Phi = \sup\left\{\|fg\|_1\mid \int \Psi\circ |g|\, d\mu \le 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 LΦ by

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

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

Example[edit]

Here is an example where L^\dagger_\Phi 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^\dagger_\Phi if |a|<1.

Properties[edit]

Relations to Sobolev spaces[edit]

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), k p = 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[edit]

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)] \le 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[edit]

  • 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 .