Trudinger's theorem explained

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 R ^{n} satisfying the cone condition. Let mp = n and p> 1 . SetA(t) = \exp\left( t ^{n/(n-m)} \right) -1 .

Then there exists the embedding

W ^{m,p} (\Omega)\hookrightarrow L _{A(\Omega)}

where

L _{A(\Omega)=\left\{} u\in M _{f(\Omega):\|u\|} _{A,\Omega} =inf \{ k> 0 :\int _{\Omega} A\left( (|u(x)|)/(k) \right)~dx\leq 1 \}< inf ty \right\}.

The space

L _{A(\Omega)}

is an example of an Orlicz space.

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