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Moser–Trudinger inequality ：ウィキペディア英語版

Trudinger's theorem
In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser).
It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function. The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem:
Let

\Omega

be a bounded domain in

\mathbb^n

satisfying the cone condition. Let

mp=n

and

p>1

. Set
:

A(t)=\exp\left( t^ \right)-1.

Then there exists the imbedding
:

W^(\Omega)\hookrightarrow L_A(\Omega)

where
:

L_A(\Omega)=\left\ \right)~dx\leq 1 \}<\infty \right\}.

The space
:

L_A(\Omega)

is an example of an Orlicz space.
==References==

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