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
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.[1] 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.[2] Orlicz was confirmed as the person who introduced Orlicz spaces already by
Stefan Banach in his 1932 monograph.[3]
, 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 by
In 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,
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:
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:
^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.
ISBN0-8247-8478-2.
Krasnosel'skii, M.A.; Rutickii, Ya B. (1961-01-01).
Convex Functions and Orlicz Spaces (1 ed.). Gordon & Breach.
ISBN978-0-677-20210-5. 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.
ISBN0-8247-8478-2. Contains properties of Orlicz spaces over general spaces with general measures, including many pathological examples.