Inmathematics, and especially inharmonic analysis andfunctional analysis, anOrlicz space is a type of function space which generalizesL^p spaces. Like${\displaystyle L^p}$ spaces, they areBanach spaces. The spaces are named forWładysław Orlicz, who was the first to define them in 1932.

Besides${\displaystyle L^p}$ spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space is${\displaystyle L{\log }^{+}L}$, which arises in the study ofHardy–Littlewood maximal functions, consisting of measurable functions${\displaystyle f}$ such that

Here${\displaystyle {\log }^{+}}$ is thepositive part of the logarithm. Also included in the class of Orlicz spaces are many of the most importantSobolev spaces. In addition, theOrlicz sequence spaces are examples of Orlicz spaces.

These spaces are called Orlicz spaces becauseWładysław Orlicz was the first who introduced them, in 1932.^[1] Some mathematicians, including Wojbor Woyczyński,Edwin Hewitt andVladimir Mazya, include the name ofZygmunt Birnbaum as well, referring to his earlier joint work withWładysław 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 byStefan Banach in his 1932 monograph.^[3]

Let${\displaystyle \mu }$ be aσ-finite measure on a set${\displaystyle X}$, and${\displaystyle \mathrm{\Phi }:[0,\mathrm{\infty })\to [0,\mathrm{\infty }]}$ aYoung function; i.e., aconvex,lower semicontinuous, and non-trivial function. Non-trivial in the sense that it is neither the zero function${\displaystyle x\mapsto 0}$ nor the convex dual of the zero function

Now let${\displaystyle L_{\mathrm{\Phi }}^{†}}$ be the set of measurable functions${\displaystyle f:X\to \mathbb{R}}$ such that the integral

${\displaystyle {\int }_X\mathrm{\Phi }(|f|)d\mu }$

is finite, where, as usual, functions that agreealmost everywhere are identified.

This is not necessarily avector space (for example, it might fail to be closed under scalar multiplication). TheOrlicz space, denoted${\displaystyle L_{\mathrm{\Phi }}}$, is the vector space of functions spanned by${\displaystyle L_{\mathrm{\Phi }}^{†}}$; that is, the smallest linear space containing${\displaystyle L_{\mathrm{\Phi }}^{†}}$. Formally,

There is another Orlicz space, thesmall Orlicz space, defined by

In other words, it is the largest linear space contained in${\displaystyle L_{\mathrm{\Phi }}^{†}}$.

To define a norm on${\displaystyle L_{\mathrm{\Phi }}}$, let${\displaystyle \mathrm{\Psi }}$ be the complementary Young function to${\displaystyle \mathrm{\Phi }}$; i.e.,

${\displaystyle \mathrm{\Psi }(x)={\int }_0^x({\mathrm{\Phi }}^{\prime }{)}^{-1}(t)dt.}$

Note thatYoung's inequality for products holds:

${\displaystyle ab\le \mathrm{\Phi }(a)+\mathrm{\Psi }(b).}$

The norm is then given by

${\displaystyle \Vert f{\Vert }_{\mathrm{\Phi }}=sup\left\{\Vert fg{\Vert }_1|\int \mathrm{\Psi }(|g|)d\mu \le 1\right\}.}$

Furthermore, the space${\displaystyle L_{\mathrm{\Phi }}}$ is precisely the space of measurable functions for which this norm is finite.

An equivalent norm,^[4]: §3.3 called the Luxemburg norm, is defined on${\displaystyle L_{\mathrm{\Phi }}}$ by

${\displaystyle \Vert f{\Vert }_{\mathrm{\Phi }}^{\prime }=inf\left\{k\in (0,\mathrm{\infty })|{\int }_X\mathrm{\Phi }(|f|/k)d\mu \le 1\right\},}$

and likewise${\displaystyle L_{\mathrm{\Phi }}(\mu )}$ is the space of all measurable functions for which this norm is finite.

The two norms are equivalent in the sense that${\displaystyle \Vert f{\Vert }_{\mathrm{\Phi }}^{\prime }\le \Vert f{\Vert }_{\mathrm{\Phi }}\le 2\Vert f{\Vert }_{\mathrm{\Phi }}^{\prime }}$ for all measurable${\displaystyle f}$.^[5]

Note that by themonotone convergence theorem, if, then

${\displaystyle {\int }_X\mathrm{\Phi }(|f|/\Vert f{\Vert }_{\mathrm{\Phi }}^{\prime })d\mu \le 1}$.

For any${\displaystyle p\in [1,\mathrm{\infty }]}$,${\displaystyle L^p}$ space is an Orlicz space with Orlicz function${\displaystyle \mathrm{\Phi }(t)=t^p}$. Here

When, the small and the large Orlicz spaces for${\displaystyle \mathrm{\Phi }(t)=t^p}$ are equal:${\displaystyle M_{\mathrm{\Phi }}\simeq L_{\mathrm{\Phi }}}$.

For an example where${\displaystyle L_{\mathrm{\Phi }}^{†}}$ is not a vector space, and is strictly smaller than${\displaystyle L_{\mathrm{\Phi }}}$, let${\displaystyle X}$ be the open unit interval${\displaystyle (0,1)}$,${\displaystyle \mathrm{\Phi }(t)=e^t-1-t}$, and${\displaystyle f(t)=\log (t)}$. Then${\displaystyle af}$ is in the space${\displaystyle L_{\mathrm{\Phi }}}$ for all${\displaystyle a\in \mathbb{R}}$ but is only in${\displaystyle L_{\mathrm{\Phi }}^{†}}$ if.

Proposition. The Orlicz norm is anorm.

Proof. Since${\displaystyle \mathrm{\Phi }(x)>0}$ for some${\displaystyle x>0}$, we have${\displaystyle \Vert f{\Vert }_{\mathrm{\Phi }}=0\to f=0}$ a.e.. That${\displaystyle \Vert kf{\Vert }_{\mathrm{\Phi }}=|k|\Vert f{\Vert }_{\mathrm{\Phi }}}$ is obvious by definition. Fortriangular inequality, we have:Theorem. The Orlicz space${\displaystyle L^{\phi }(X)}$ is aBanach space — acompletenormedvector space.

Theorem.^[5]${\displaystyle M_{\mathrm{\Phi }},L_{{\mathrm{\Phi }}^{\ast }}}$ aretopological dual Banach spaces.

In particular, if${\displaystyle M_{\mathrm{\Phi }}=L_{\mathrm{\Phi }}}$, then${\displaystyle L_{{\mathrm{\Phi }}^{\ast }},L_{\mathrm{\Phi }}}$ are topological dual spaces. In particular,${\displaystyle L^p,L^q}$ are dual Banach spaces when${\displaystyle 1/p+1/q=1}$ and.

CertainSobolev spaces are embedded in Orlicz spaces: for${\displaystyle n>1}$ and${\displaystyle X\subseteq {\mathbb{R}}^n}$open andbounded withLipschitz boundary${\displaystyle \mathrm{\partial }X}$, we have

${\displaystyle W_0^{1,n}(X)\subseteq L^{\phi }(X)}$

for

${\displaystyle \phi (t):=\exp \left(|t{|}^{n/(n-1)}\right)-1.}$

This is the analytical content of theTrudinger inequality: For${\displaystyle X\subseteq {\mathbb{R}}^n}$ open and bounded with Lipschitz boundary${\displaystyle \mathrm{\partial }X}$, consider the space${\displaystyle W_0^{k,p}(X)}$ with${\displaystyle kp=n}$ and${\displaystyle p>1}$. Then there exist constants${\displaystyle C_1,C_2>0}$ such that

${\displaystyle {\int }_X\exp \left({\left(\frac{|u(x)|}{C_1\Vert {\mathrm{D}}^{k}u{\Vert }_{L^p(X)}}\right)}^{n/(n-k)}\right)\mathrm{d}x\le C_2|X|.}$

Similarly, the Orlicz norm of arandom variable characterizes it as follows:

${\displaystyle \Vert X{\Vert }_{\mathrm{\Psi }}\triangleq inf\left\{k\in (0,\mathrm{\infty })\mid \mathrm{E}[\mathrm{\Psi }(|X|/k)]\le 1\right\}.}$

This norm ishomogeneous and is defined only when this set is non-empty.

When${\displaystyle \mathrm{\Psi }(x)=x^p}$, this coincides with thep-thmoment of the random variable. Other special cases in the exponential family are taken with respect to the functions${\displaystyle {\mathrm{\Psi }}_q(x)=\exp (x^q)-1}$ (for${\displaystyle q\ge 1}$). A random variable with finite${\displaystyle {\mathrm{\Psi }}_2}$ norm is said to be "sub-Gaussian" and a random variable with finite${\displaystyle {\mathrm{\Psi }}_1}$ norm is said to be "sub-exponential". Indeed, the boundedness of the${\displaystyle {\mathrm{\Psi }}_p}$ norm characterizes the limiting behavior of the probability distribution function:

so that the tail of the probability distribution function is bounded above by${\displaystyle O(e^{-K^{\prime }{x}^p})}$.

The${\displaystyle {\mathrm{\Psi }}_1}$ norm may be easily computed from a strictly monotonicmoment-generating function. For example, the moment-generating function of achi-squared random variable X with K degrees of freedom is${\displaystyle M_X(t)=(1-2t{)}^{-K/2}}$, so that the reciprocal of the${\displaystyle {\mathrm{\Psi }}_1}$ norm is related to the functional inverse of the moment-generating function:

${\displaystyle \Vert X{\Vert }_{{\mathrm{\Psi }}_1}^{-1}=M_X^{-1}(2)=(1-4^{-1/K})/2.}$

• Krasnosel'skii, M.A.; Rutickii, Ya B. (1961-01-01).Convex Functions and Orlicz Spaces (1 ed.). Gordon & Breach.ISBN 978-0-677-20210-5.{{cite book}}:ISBN / Date incompatibility (help) Contains most commonly used properties of Orlicz spaces over${\displaystyle {\mathbb{R}}^n}$ with the Lebesgue measure.
• Rao, M.M.; Ren, Z.D. (1991).Theory of Orlicz Spaces. Pure and Applied Mathematics. Marcel Dekker.ISBN 0-8247-8478-2. Contains properties of Orlicz spaces over general spaces with general measures, including many pathological examples.
• Rubshtein, Ben-Zion A.; Grabarnik, Genady Ya; Muratov, Mustafa A.; Pashkova, Yulia S. (2016-12-20).Foundations of Symmetric Spaces of Measurable Functions: Lorentz, Marcinkiewicz and Orlicz Spaces (1st ed.). New York, NY: Springer.ISBN 978-3-319-42756-0.
• Birnbaum, Z. W.; Orlicz, W. (1931),"Über die Verallgemeinerung des Begriffes der zueinander Konjugierten Potenzen"(PDF),Studia Mathematica,3:1–67,doi:10.4064/sm-3-1-1-67, archived fromthe original(PDF) on 27 Sep 2011. The original paper.
• Bund, Iracema (1975), "Birnbaum–Orlicz spaces of functions on groups",Pacific Journal of Mathematics,58 (2):351–359,doi:10.2140/pjm.1975.58.351.
• Hewitt, Edwin; Stromberg, Karl,Real and abstract analysis, Springer-Verlag.
• Zygmund, Antoni, "Chapter IV: Classes of functions and Fourier series",Trigonometric Series, Volume 1 (3rd ed.), Cambridge University Press.
• Ledoux, Michel; Talagrand, Michel,Probability in Banach Spaces, Springer-Verlag.

• "Orlicz space",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Harmonic analysis
• Real analysis
• Banach spaces

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