Orlicz space

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In mathematics, Orlicz spaces are a generalization of L^p spaces. They are named after the Polish mathematician Władysław Orlicz.

1 Definition
2 Properties
3 Orlicz-Sobolev spaces
4 References

[edit] Definition

Let $X \subseteq \mathbb{R}^{n}$ be an open set and let $\varphi : [0, + \infty) \to [0, + \infty)$ be an increasing convex function. Then the Orlicz space $L^{\varphi} (X)$ is defined to be the space of all integrable functions $u : X \to \mathbb{R}$ such that the Orlicz norm

$\| u \|_{L^{\varphi} (X)} := \inf \left\{ \lambda > 0 \left| \int_{X} \varphi \left( \frac{| u(x) |}{\lambda} \right) \, \mathrm{d} x \leq 1 \right. \right\} < + \infty.$

[edit] Properties

Olicz spaces generalize L^p spaces in the sense that if $\varphi (t) = | t |^{p}$ , then $\| u \|_{L^{\varphi} (X)} = \| u \|_{L^{p} (X)}$ , so $L^{\varphi} (X) = L^{p} (X)$ .
The Orlicz space $L^{\varphi} (X)$ is a Banach space — a complete normed vector space.

[edit] Orlicz-Sobolev spaces

Certain Sobolev spaces are embedded in Orlicz spaces: for $X \subseteq \mathbb{R}^{n}$ open and bounded with Lipschitz boundary $\partial X$ ,

$W_{0}^{1, p} (X) \subseteq L^{\varphi} (X)$

for

$\varphi (t) := \exp \left( | t |^{p / (p - 1)} \right) - 1.$

More generally, for $X \subseteq \mathbb{R}^{n}$ open and bounded with Lipschitz boundary $\partial X$ , consider the space $W_{0}^{k, p} (X)$ , $k p = n$ . There there exists constants $C 1, C 2 > 0$ such that

$\int_{X} \exp \left( \left( \frac{| u(x) |}{C_{1} \| \mathrm{D}^{k} u \|_{L^{p} (X)}} \right)^{p / (p - 1)} \right) \, \mathrm{d} x \leq C_{2} | X |.$