Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,[1] consist of two closely related interpolation inequalities between the Lebesgue space ${\displaystyle L^{\infty }}$ and the Sobolev spaces ${\displaystyle H^{s}}$. It is useful in the study of partial differential equations.

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

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

and

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

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

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

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

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