In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–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 and Zygmunt William Birnbaum, who first defined them in 1931.
Contents
- Formal definition
- Example
- Properties
- Relations to Sobolev spaces
- Orlicz Norm of a Random Variable
- References
Besides the Lp spaces, a variety of function spaces arising naturally in analysis are Birnbaum–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 Birnbaum–Orlicz spaces are many of the most important Sobolev spaces.
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
is finite, where, as usual, functions that agree almost everywhere are identified.
This may not be a vector space (it may fail to be closed under scalar multiplication). The vector space of functions spanned by
To define a norm on
Note that Young's inequality holds:
The norm is then given by
Furthermore, the space
An equivalent norm (Rao & Ren 1991, §3.3) 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
Properties
Relations to Sobolev spaces
Certain Sobolev spaces are embedded in Orlicz spaces: for
for
This is the analytical content of the Trudinger inequality: For
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
so that the tail of this probability density function asymptotically resembles, and is bounded above by
The