Orlicz space

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.[1] A small minority of 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 this naming convention is incorrect. 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.[2] Orlicz was confirmed as the person who introduced Orlicz spaces already by Stefan Banach in his 1932 monography.[3]

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

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

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

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

${\displaystyle \int _{X}\Phi (|f|)\,d\mu }$

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

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

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

Note that Young's inequality for products holds:

${\displaystyle ab\leq \Phi (a)+\Psi (b).}$

The norm is then given by

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

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

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

${\displaystyle \|f\|'_{\Phi }=\inf \left\{k\in (0,\infty )\mid \int _{X}\Phi (|f|/k)\,d\mu \leq 1\right\},}$

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

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

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

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

for

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

This is the analytical content of the Trudinger inequality: For ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ open and bounded with Lipschitz boundary ${\displaystyle \partial X}$, consider the space ${\displaystyle W_{0}^{k,p}(X)}$, ${\displaystyle kp=n}$. There exist constants ${\displaystyle C_{1},C_{2}>0}$ such that

${\displaystyle \int _{X}\exp \left(\left({\frac {|u(x)|}{C_{1}\|\mathrm {D} ^{k}u\|_{L^{p}(X)}}}\right)^{p/(p-1)}\right)\,\mathrm {d} x\leq C_{2}|X|.}$

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

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

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

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

${\displaystyle \|X\|_{\Psi _{p}}=c\rightarrow \lim _{x\rightarrow \infty }f_{X}(x)\exp(|x/c|^{p})=0,}$

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

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

${\displaystyle \|X\|_{\Psi _{1}}^{-1}=M_{X}^{-1}(2)=(1-4^{-1/K})/2.}$

• 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.

• "Orlicz space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]