In mathematical analysis, and especially in real, harmonic analysis and functional 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.
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
\int | |
Rn |
|f(x)|log+|f(x)|dx<infty.
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.
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]
μ is a σ-finite measure on a set X,
\Phi:[0,infty)\to[0,infty]
x\mapsto0
x\mapsto\begin{cases}0ifx=0,\ +inftyelse.\end{cases}
Let
\dagger | |
L | |
\Phi |
\intX\Phi(|f|)d\mu
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
\dagger | |
L | |
\Phi |
L\Phi
\dagger | |
L | |
\Phi |
\dagger | |
L | |
\Phi |
To define a norm on
L\Phi
\Psi(x)=
x | |
\int | |
0 |
(\Phi')-1(t)dt.
Note that Young's inequality for products holds:
ab\le\Phi(a)+\Psi(b).
The norm is then given by
\|f\|\Phi=\sup\left\{\|fg\|1\mid\int\Psi(|g|)d\mu\le1\right\}.
Furthermore, the space
L\Phi
An equivalent norm,[4] called the Luxemburg norm, is defined on LΦ by
\|f\|'\Phi=inf\left\{k\in(0,infty)\mid\intX\Phi(|f|/k)d\mu\le1\right\},
and likewise
L\Phi(\mu)
Proposition.[5]
\|f\|\Phi'\leq\|f\|\Phi\leq2\|f\|\Phi'
f
0<\|f\|\Phi'<infty
\intX\Phi(|f|/\|f\|\Phi')d\mu\le1
For any
p\in[1,infty]
Lp
\Phi(t)=tp
tinfty=\begin{cases}0&ift\in[0,1],\ +infty&else.\end{cases}
When
1<p<infty
\Phi(x)=xp
M\Phi\simeqL\Phi
Example where
\dagger | |
L | |
\Phi |
L\Phi
L\Phi
\dagger | |
L | |
\Phi |
Proposition. The Orlicz norm is a norm.
Proof. Since
\Phi(x)>0
x>0
\|f\|\Phi=0\tof=0
\|kf\|\Phi=|k|\|f\|\Phi
L\varphi(X)
Theorem.
M\Phi,
L | |
\Phi* |
In particular, if
M\Phi=L\Phi
L | |
\Phi* |
,L\Phi
Lp,Lq
1/p+1/q=1
1<p<infty
Certain Sobolev spaces are embedded in Orlicz spaces: for
n>1
X\subseteqRn
\partialX
1,n | |
W | |
0 |
(X)\subseteqL\varphi(X)
for
\varphi(t):=\exp\left(|t|n\right)-1.
This is the analytical content of the Trudinger inequality: For
X\subseteqRn
\partialX
k,p | |
W | |
0 |
(X)
kp=n
p>1
C1,C2>0
\intX\exp\left(\left(
|u(x)| | ||||||||||
|
\right)n\right)dx\leqC2|X|.
Similarly, the Orlicz norm of a random variable characterizes it as follows:
\|X\|\Psi\triangleqinf\left\{k\in(0,infty)\mid\operatorname{E}[\Psi(|X|/k)]\le1\right\}.
This norm is homogeneous and is defined only when this set is non-empty.
When
\Psi(x)=xp
\Psiq(x)=\exp(xq)-1
q\geq1
\Psi2
\Psi1
\Psip
\|X\| | |
\Psip |
<infty\iff{\BbbP}(|X|\gex)\le
-K'xp | |
Ke |
{\rmfor some constants }K,K'>0,
so that the tail of the probability distribution function is bounded above by
-K'xp | |
O(e |
)
The
\Psi1
MX(t)=(1-2t)-K/2
\Psi1
\|X\| | |
\Psi1 |
-1=
-1 | |
M | |
X |
(2)=(1-4-1/K)/2.
\Rn