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 $\mathbb{R}^n$ satisfying the cone condition. Let $mp=n$ and $p>1$. Set

$A(t)=\exp\left( t^{n/(n-m)} \right)-1.$

Then there exists the imbedding

$W^{m,p}(\Omega)\hookrightarrow L_A(\Omega)$

where

$L_A(\Omega)=\left\{ u\in M_f(\Omega):\|u\|_{A,\Omega}=\inf\{ k>0:\int_\Omega A\left( \frac{|u(x)|}{k} \right)~dx\leq 1 \}<\infty \right\}.$

The space

$L_A(\Omega)$

is an example of an Orlicz space.

References

• Moser, J. (1971), "A Sharp form of an Inequality by N. Trudinger", Indiana Univ. Math. 20: 1077–1092.
• Trudinger, N. S. (1967), "On imbeddings into Orlicz spaces and some applications", J. Math. Mech. 17: 473–483.