mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...

s. It is named after Neil Trudinger (and

Jürgen Moser Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations. Life Moser's mother Ilse Strehl ...

). It provides an inequality between a certain

norm and an

Orlicz space In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the ''L'p'' spaces. Like the ''L'p'' spaces, they are Banach spaces. The spaces are n ...

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 In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the ...

. Let

mp=n

and

p>1

. Set :

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

Then there exists the embedding :

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

where :

L_A(\Omega)=\left\.

The space :

L_A(\Omega)

is an example of an

References

*. *{{citation, last=Trudinger, first=N. S., authorlink=Neil Trudinger, title=On imbeddings into Orlicz spaces and some applications, journal=J. Math. Mech. , volume=17, year=1967, pages=473–483. Sobolev spaces Inequalities Theorems in analysis