Agmon's inequality
# Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space $L^{\infty }$ and the Sobolev spaces $H^{s}$. It is useful in the study of partial differential equations.

Let $u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )$ where $\Omega \subset \mathbb {R} ^{3}$. Then Agmon's inequalities in 3D state that there exists a constant $C$ such that

    $\displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2},$

and

    $\displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/4}\|u\|_{H^{2}(\Omega )}^{3/4}.$

In 2D, the first inequality still holds, but not the second: let $u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )$ where $\Omega \subset \mathbb {R} ^{2}$. Then Agmon's inequality in 2D states that there exists a constant $C$ such that

    $\displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2}.$

For the $n$-dimensional case, choose $s_{1}$ and $s_{2}$ such that $s_{1}<{\tfrac {n}{2}}<s_{2}$. Then, if $0<\theta <1$ and ${\tfrac {n}{2}}=\theta s_{1}+(1-\theta )s_{2}$, the following inequality holds for any $u\in H^{s_{2}}(\Omega )$

    $\displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{s_{1}}(\Omega )}^{\theta }\|u\|_{H^{s_{2}}(\Omega )}^{1-\theta }$
