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 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( \frac{|u(x)|}{k} \right)~dx\leq 1 \}<\infty \right\}. $$

The space


 * $$L_A(\Omega)$$

is an example of an Orlicz space.