# 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 be a bounded domain in satisfying the cone condition. Let and . Set

Then there exists the imbedding

where

The space

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.

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