In 1967, Neil S. Trudinger announced a result in *J. Math. Mech.* (now known as *Indiana Univ. Math. J.*) which can be seen as a limiting case of the Sobolev inequality [here] or [here].

It is well-known from the Sobolev embedding theorem that

for

.

The case is commonly referred to the limitting case. If , and we obtain

.

In general one cannot take the limits and , i.e.

.

A counter-example is given by

on the unit ball in . Instead, Trudinger proved exponential -integrability in the following sense

Theorem(Trudinger). Let be a bounded domain and with.

Then there exist universal constants , such that

.

We write

.

Observe that the assumption

implies that inequality

is equivalent to

for some universal constant .

The way to see it is the following

- For all one has
hence

.

This proves the claim for . For odd a simple use of the Holder inequality gives

.

- Now obviously
So if one chooses so small that

which according to the Stirling formula implies that the infinie series

is finite.

*Proof*. It suffices to prove

for some constant . By symmetric rearrangement

and

and scaling we may take . Furthermore, we may assume .

We can represent as

which after integration by parts leads to the estimate

using the Holder inequality.

Now

is finite since for any one has and then

.

Consequently

where we uses the Fubini theorem to obtain the last inequality. By the assumption we have

for some universal constant .

The subscript 0 in the space of functions can be dropped

Corollary. Let be a compact and closed manifold. Then there exist constants and such that for all withone has

.

The proof replies on making use the unity of partition. Assumption allows us to use the Poincare inequality.

There is one more corollary which is frequently used in the literature.

Corollary. For a compact and closed manifold there are constants and such that for eachfor all where

.

The proof relies on an elementary inequality, for ,

where is the constant appeared in the previous corollary.

It is worth noticing that the above inequality can be rewritten as

.

Let us consider a simple application of the Trudinger inequality.

**Example**. If in as and

then for each

as .

*Proof*. Using the simple estimate we can write

using the Holder inequality.

There are some improvement of the Trudinger inequality in the literature. The most important one is the so-called Moser-Trudinger inequality which is a sharp version of a limiting case of the Sobolev inequality. This work had been done by Jürgen Moser around 1970. We will consider this inequality later. We end this entry by stating a version fo the -dimension spaces, see also Chang and Yang *Comm. Pure Appl. Math.* 2003 [here].

Theorem(Trudinger). Let be a bounded domain and with.

Then there exist universal constants , such that

.

Source: S-Y.A. Chang, *Non-linear elliptic equations in conformal geometry*, EMS, 2004.

Hi, well done as usual.

Chang’s book is a very nice and more or less complete work on this subject. I love it!

Quickly reading, I have noted two misprints: 1) on the third embedding you have to put a cross, 2) implies that in equality turns in implies that inequality.

Comment by fab — February 10, 2012 @ 17:59

Thanks Fab. You’re welcome. I have corrects those misprints as suggested. Thanks for your interest in my post(s).

Comment by Ngô Quốc Anh — February 10, 2012 @ 18:05