Orlicz space

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In mathematical analysis, and especially in real and harmonic 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.

Contents

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 integral

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.

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. [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 monograph. [3]

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

Let be the set of measurable functions f : XR 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 .

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

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

Example

Here is an 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

Relations to Sobolev spaces

Certain Sobolev spaces are embedded in Orlicz spaces: for open and bounded with Lipschitz boundary ,

for

This is the analytical content of the Trudinger inequality: For open and bounded with Lipschitz boundary , consider the space , . 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 density function:

so that the tail of this probability density function asymptotically resembles, and 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:

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References

  1. Über eine gewisse Klasse von Räumen vom Typus B, Bull. Internat. Acad. Polon. Sci. Lett., Class. Sci. Math. Natur.: S\'{e}r. A, Sci. Math. 1932:8/9, 207--220.
  2. Lech Maligranda, Osiągnięcia polskich matematyków w teorii interpolacji operatorów: 1910–1960, 2015, „Wiadomości matematyczne”, 51 , 239-281 (in Polish).
  3. Stefan Banach, 1932, Théorie des opérations linéaires, Warszawa (p.202)

Further reading