In this note, we prove a very interesting inequality known as the Agmon type inequality in space dimension 1.
for any smooth function with compact support in .
The proof is standard and classical. The trick is to use the integral representation for functions that we have already discussed when we talk about the Poincare inequality.
Proof. Since has compact support, there exists some sufficiently large such that vanishes outside of . Then we can write
Adding both inequalities we obtain
Then using the Cauchy-Schwarz inequality we find the desired inequality.
I will show some application of this equality, precisely, I will derive a proof of the Ladyzhenskaya inequalities
which plays an important role in the theory of Navier-Stokes equations.