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 $$u\in H^2(\Omega)\cap H^1_0(\Omega)$$ 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 $$u\in H^2(\Omega)\cap H^1_0(\Omega)$$ 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 $$u\in H^{s_2}(\Omega)$$


 * $$\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^{s_1}(\Omega)}^{\theta} \|u\|_{H^{s_2}(\Omega)}^{1-\theta}$$