Orlicz space
From Wikipedia, the free encyclopedia
In mathematics, Orlicz spaces are a generalization of Lp spaces. They are named after the Polish mathematician Władysław Orlicz.
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[edit] Definition
Let 
be an open set and let 
be an increasing convex function. Then the Orlicz space 
is defined to be the space of all integrable functions 
such that the Orlicz norm
[edit] Properties
- Olicz spaces generalize Lp spaces in the sense that if
, then 
, so 
. - The Orlicz space is a Banach space — a complete normed vector space.
[edit] Orlicz-Sobolev spaces
Certain Sobolev spaces are embedded in Orlicz spaces: for open and bounded with Lipschitz boundary 
,
for
More generally, for open and bounded with Lipschitz boundary , consider the space 
, kp = n. There there exists constants C1,C2 > 0 such that
[edit] References
- Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.
