Orlicz space

In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an 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, who was the first to define them in 1932.

Besides the Lp spaces, a variety of function spaces arising naturally in analysis are 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


 * $$\int_{\mathbb{R}^n} |f(x)|\log^+ |f(x)|\,dx < \infty. $$

Here log+ is the positive part of the logarithm. Also included in the class of Orlicz spaces are many of the most important Sobolev spaces. In addition, the Orlicz sequence spaces are examples of Orlicz spaces.

Terminology
These spaces are called Orlicz spaces by an overwhelming majority of mathematicians and by all monographies studying them, because Władysław Orlicz was the first who introduced them, in 1932. Some mathematicians, including Wojbor Woyczyński, Edwin Hewitt and Vladimir Mazya, include the name of Zygmunt Birnbaum as well, referring to his earlier joint work with Władysław Orlicz. However in the Birnbaum–Orlicz paper the Orlicz space is not introduced, neither explicitly nor implicitly, hence the name Orlicz space is preferred. By the same reasons this convention has been also openly criticized by another mathematician (and an expert in the history of Orlicz spaces), Lech Maligranda. Orlicz was confirmed as the person who introduced Orlicz spaces already by Stefan Banach in his 1932 monograph.

Setup
μ is a σ-finite measure on a set X,

$$\Phi: [0, \infty) \to [0, \infty]$$, is a Young function, i.e. convex, lower semicontinuous, and non-trivial, in the sense that it is not the zero function $$x \mapsto 0$$, and it is not the convex dual of the zero function $$x \mapsto \begin{cases} 0 \text{ if } x = 0, \\ +\infty \text{ else.}\end{cases}$$

Orlicz spaces
Let $$L^\dagger_\Phi$$ 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 might not be a vector space (i.e., it might fail to be closed under scalar multiplication). The vector space of functions spanned by $$L^\dagger_\Phi$$ is the Orlicz space, denoted $$L_\Phi$$. In other words, it is the smallest linear space containing $$L^\dagger_\Phi$$. In other words,$$L_{\Phi} = \left\{f \Big| \exists k > 0, \int_X \Phi(k|f|)\, d\mu < \infty \right\}$$There is another Orlicz space (the "small" Orlicz space) defined by$$M_{\Phi} = \left\{f \Big| \forall k > 0, \int_X \Phi(k|f|)\, d\mu < \infty \right\}$$In other words, it is the largest linear space contained in $$L^\dagger_\Phi$$.

Norm
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 for products holds:


 * $$ab\le \Phi(a) + \Psi(b).$$

The norm is then given by


 * $$\|f\|_\Phi = \sup\left\{\|fg\|_1\mid \int \Psi(|g|)\, d\mu \le 1\right\}.$$

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

An equivalent norm, called the Luxemburg norm, 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 $$ L_\Phi(\mu) $$ is the space of all measurable functions for which this norm is finite.

Proposition.


 * The two norms are equivalent in the sense that $$\| f \|_{\Phi}' \leq \| f \|_{\Phi} \leq 2 \| f \|_{\Phi}'$$ for all measurable $$f$$.
 * By monotone convergence theorem, if $$0 < \|f\|_\Phi' < \infty$$, then $$\int_X \Phi(|f|/\|f\|_{\Phi}')\,d\mu\le 1 $$.

Examples
For any $$p \in [1, \infty]$$, the $$L^p$$ space is the Orlicz space with Orlicz function $$\Phi (t) = t^p$$. Here $$t^\infty = \begin{cases} 0 &\text{ if } t \in [0, 1], \\ +\infty &\text{ else.} \end{cases}$$

When $$1 < p < \infty$$, the small and the large Orlicz spaces for $$\Phi(x) = x^p$$ are equal: $$M_{\Phi} \simeq L_{\Phi}$$.

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
Proposition. The Orlicz norm is a norm.

Proof. Since $$\Phi(x) > 0$$ for some $$x > 0$$, we have $$\|f \|_{\Phi} = 0 \to f = 0$$ a.e.. That $$\|kf\|_{\Phi} = |k| \|f\|_{\Phi}$$ is obvious by definition. For triangular inequality, we have:$$\begin{aligned} & \int_{\mathcal{X}} \Phi\left(\frac{f+g}{\|f\|_\Phi+\|g\|_\Phi}\right) d \mu \\ = & \int_{\mathcal{X}} \Phi\left(\frac{\|f\|_\Phi}{\|f\|_\Phi+\|g\|_\Phi} \frac{f}{\|f\|_\Phi}+\frac{\|g\|_\Phi}{\|f\|_\Phi+\|g\|_\Phi} \frac{g}{\|g\|_\Phi}\right) d \mu \\ \leq & \frac{\|f\|_\Phi}{\|f\|_\Phi+\|g\|_\Phi} \int_{\mathcal{X}} \Phi\left(\frac{f}{\|f\|_\Phi}\right) d \mu+\frac{\|g\|_\Phi}{\|f\|_\Phi+\|g\|_\Phi} \int_{\mathcal{X}} \Phi\left(\frac{g}{\|g\|_\Phi}\right) d \mu \\ \leq & 1 \end{aligned}$$Theorem. The Orlicz space $$L^\varphi (X)$$ is a Banach space &mdash; a complete normed vector space.

'''Theorem. ''' $$M_\Phi, L_{\Phi^*}$$ are topological dual Banach spaces.

In particular, if $$M_{\Phi} = L_{\Phi}$$, then $$L_{\Phi^*}, L_{\Phi}$$ are topological dual spaces. In particular, $$L^p, L^q$$ are dual Banach spaces when $$1/p + 1/q = 1$$ and $$1 < p < \infty$$.

Relations to Sobolev spaces
Certain Sobolev spaces are embedded in Orlicz spaces: for $$ n>1$$ and $$X \subseteq \mathbb{R}^{n}$$ open and bounded with Lipschitz boundary $$\partial X$$, we have


 * $$W_0^{1, n} (X) \subseteq L^\varphi (X)$$

for


 * $$\varphi (t) := \exp \left( | t |^{n / (n - 1)} \right) - 1.$$

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 $$W_0^{k, p} (X)$$ with $$k p = n$$ and $$p > 1$$. Then there exist constants $$C_1, C_2 > 0$$ such that


 * $$\int_X \exp \left( \left( \frac{| u(x) |}{C_1 \| \mathrm{D}^k u \|_{L^p (X)}} \right)^{n / (n - k)} \right) \, \mathrm{d} x \leq C_2 | X |.$$

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


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

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 distribution function:


 * $$\|X\|_{\Psi_p} <\infty \iff {\Bbb P}(|X|\ge x)\le Ke^{-K' x^p}\qquad {\rm for\ some\  constants\ } K, K'>0,$$

so that the tail of the probability distribution function is bounded above by $$O(e^{-K' x^p})$$.

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 reciprocal of the $$\Psi_1$$ norm is related to the functional inverse of the moment-generating function:


 * $$\|X\|_{\Psi_1} ^{-1} = M_X^{-1}(2) = (1-4^{-1/K})/2.$$