Agmon's inequality
In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space and the Sobolev spaces. It is useful in the study of partial differential equations.
Let where. Then Agmon's inequalities in 3D state that there exists a constant such that
and
In 2D, the first inequality still holds, but not the second: let where. Then Agmon's inequality in 2D states that there exists a constant such that
For the -dimensional case, choose and such that. Then, if and, the following inequality holds for any
