Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.

Let us remind the Trudinger inequality

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

Then there exist universal constants , such that

.

The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant . In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows

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

Then there exist sharp constants , given by

such that

.

The constant is sharp in the sense that for all there is a sequence of functions satisfying

but the integral

grow without bound.

For general compact closed manifold the constant on the right hand side of the Moser-Trudinger inequality depends on the metric . Working on a sphere with a canonical metric allows us to control the constants.

Theorem(Moser). There is a universal constant such that for all withand

we have

.

Observe that

.

In the same way as we introduce in the entry concerning the Trudinger inequality one can show

Corollary. Forone has

for all .

Obviously, since . It turns out to determine the best constant . This had been done by Onofri known as the Onofri inequality [here].

Theorem(Onofri).Let then we havewith the equality iff

.

The proof of the Onofri inequality relies on a result due to Aubin

Theorem(Aubin). For all there exists a constant such thatfor any belonging to the following class

.

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