In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the L spaces. Like the L spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.
Besides the L 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
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.
Definition
Setup
μ is a σ-finite measure on a set X,
, is a Young function, i.e. convex, lower semicontinuous, and non-trivial, in the sense that it is not the zero function , and it is not the convex dual of the zero function
Orlicz spaces
Let be the set of measurable functions f : X → R such that the integral
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 is the Orlicz space, denoted . In other words, it is the smallest linear space containing . In other words,There is another Orlicz space (the "small" Orlicz space) defined byIn other words, it is the largest linear space contained in .
Norm
To define a norm on , let Ψ be the Young complement of Φ; that is,
Note that Young's inequality for products holds:
The norm is then given by
Furthermore, the space 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
and likewise is the space of all measurable functions for which this norm is finite.
Proposition.
- The two norms are equivalent in the sense that for all measurable .
- By monotone convergence theorem, if , then .
Examples
For any , the space is the Orlicz space with Orlicz function . Here
When , the small and the large Orlicz spaces for are equal: .
Example where is not a vector space, and is strictly smaller than . 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 but is only in the set if |a| < 1.
Properties
Proposition. The Orlicz norm is a norm.
Proof. Since for some , we have a.e.. That is obvious by definition. For triangular inequality, we have:Theorem. The Orlicz space is a Banach space — a complete normed vector space.
Theorem. are topological dual Banach spaces.
In particular, if , then are topological dual spaces. In particular, are dual Banach spaces when and .
Relations to Sobolev spaces
Certain Sobolev spaces are embedded in Orlicz spaces: for and open and bounded with Lipschitz boundary , we have
for
This is the analytical content of the Trudinger inequality: For open and bounded with Lipschitz boundary , consider the space with and . Then there exist constants such that
Orlicz norm of a random variable
Similarly, the Orlicz norm of a random variable characterizes it as follows:
This norm is homogeneous and is defined only when this set is non-empty.
When , 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 (for ). A random variable with finite norm is said to be "sub-Gaussian" and a random variable with finite norm is said to be "sub-exponential". Indeed, the boundedness of the norm characterizes the limiting behavior of the probability distribution function:
so that the tail of the probability distribution function is bounded above by .
The 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 , so that the reciprocal of the norm is related to the functional inverse of the moment-generating function:
References
- Über eine gewisse Klasse von Räumen vom Typus B, Bull. Internat. Acad. Polon. Sci. Lett., Class. Sci. Math. Natur.: Sér. A, Sci. Math. 1932:8/9, 207–220.
- Lech Maligranda, Osiągnięcia polskich matematyków w teorii interpolacji operatorów: 1910–1960, 2015, „Wiadomości matematyczne”, 51, 239-281 (in Polish).
- Stefan Banach, 1932, Théorie des opérations linéaires, Warszawa (p.202)
- Rao, M.M.; Ren, Z.D. (1991). Theory of Orlicz Spaces. Pure and Applied Mathematics. Marcel Dekker. ISBN 0-8247-8478-2.
- ^ Léonard, Christian. "Orlicz spaces." (2007).
Further reading
- Krasnosel'skii, M.A.; Rutickii, Ya B. (1961-01-01). Convex Functions and Orlicz Spaces (1 ed.). Gordon & Breach. ISBN 978-0-677-20210-5. Contains most commonly used properties of Orlicz spaces over with the Lebesgue measure.
- Rao, M.M.; Ren, Z.D. (1991). Theory of Orlicz Spaces. Pure and Applied Mathematics. Marcel Dekker. ISBN 0-8247-8478-2. Contains properties of Orlicz spaces over general spaces with general measures, including many pathological examples.
- Rubshtein, Ben-Zion A.; Grabarnik, Genady Ya; Muratov, Mustafa A.; Pashkova, Yulia S. (2016-12-20). Foundations of Symmetric Spaces of Measurable Functions: Lorentz, Marcinkiewicz and Orlicz Spaces (1st ed.). New York, NY: Springer. ISBN 978-3-319-42756-0.
- Birnbaum, Z. W.; Orlicz, W. (1931), "Über die Verallgemeinerung des Begriffes der zueinander Konjugierten Potenzen" (PDF), Studia Mathematica, 3: 1–67, archived from the original (PDF) on 27 Sep 2011. The original paper.
- 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.
- 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.
External links
Functional analysis (topics – glossary) | |||||
---|---|---|---|---|---|
Spaces |
| ||||
Theorems | |||||
Operators | |||||
Algebras | |||||
Open problems | |||||
Applications | |||||
Advanced topics | |||||