Orlicz space

In mathematical analysis, and especially in real and harmonic analysis, an Orlicz space is a type of function space which generalizes the "L""p" spaces. Like the "L""p" spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.
Besides the "L""p" 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. 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. Orlicz was confirmed as the person who introduced Orlicz spaces already by Stefan Banach in his 1932 monograph.
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 formula_4 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 formula_4 is the Orlicz space, denoted formula_7.
To define a norm on formula_7, 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 formula_7 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 LΦ(μ) is the space of all measurable functions for which this norm is finite.
Example.
Here is an example where formula_4 is not a vector space and is strictly smaller than formula_7.
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 formula_7 but is only in the set formula_4 if |"a"| < 1.
Relations to Sobolev spaces.
Certain Sobolev spaces are embedded in Orlicz spaces: for formula_22 open and bounded with Lipschitz boundary formula_23,
for
This is the analytical content of the Trudinger inequality: For formula_22 open and bounded with Lipschitz boundary formula_23, consider the space formula_28, formula_29. There exist constants formula_30 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 formula_33, 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 formula_34 (for formula_35). A random variable with finite formula_36 norm is said to be "sub-Gaussian" and a random variable with finite formula_37 norm is said to be "sub-exponential". Indeed, the boundedness of the formula_38 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 formula_40.
The formula_37 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 formula_42, so that the reciprocal of the formula_37 norm is related to the functional inverse of the moment-generating function: