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
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 with
In the same way as we introduce in the entry concerning the Trudinger inequality one can show
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 have
with 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 that
for any belonging to the following class
Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.