This entry can be considered as a continued part to a recent entry where lots of significantly important inequalities (Hardy, Opial, Rellich, Serrin, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg-Sobolev, Horgan) have been considered.
Today we shall continue to list here more important inequalities in the literature. Given , , , define the number by
Gagliardo-Nirenberg’s inequality. For any one has
When and , Gagliardo-Nirenberg’s inequality then becomes the well-known Sobolev inequality.
Sobolev’s inequality. For any one has
where is the volume of the unit ball in and the gamma function.
When , , and , Gagliardo-Nirenberg’s inequality then becomes the well-known Nash inequality.
Nash’s inequality. For any one has
The best constant for the Nash inequality is given by
where is the first non-zero Neumann eigenvalue of the Laplacian operator in the unit ball. This come from a joint work between Carlen and Loss [here].
Another consequence of Gagliardo-Nirenberg’s inequality is the logarithmic Sobolev inequality.
logarithmic Sobolev’s inequality. For any one has
where also satisfies
In fact, it can be obtained as the limit case when , that is, and , . To see this, let us first notice the fact that the constant in Gagliardo-Nirenberg’s inequality is independent of . We can rewrite Gagliardo-Nirenberg’s inequality as
where . It then follows that
Thus when we get
where we have used the fact that the function
Therefore, replacing and writing , we obtain the logarithmic Sobolev’s inequality.
The best constant for the logarithmic Sobolev inequality is given by
We refer the reader to a book due to Hebey entitled “Nonlinear analysis on manifolds: Sobolev spaces and inequalities” for details.