According to Wikipedia, Agmon's inequality provides a bound on the $L^\infty$ norm of a $H^2$ function on a (regular) subset of $\mathbb{R}^3$. In the book of JC Robinson et al. "The Three-Dimensional Navier-Stokes Equations: Classical Theory" in Thm. 1.20 the same statement holds on the whole of $\mathbb{R}^3$, i.e.
$$\|u\|_{L^\infty(\mathbb{R}^3)}\leq C \|u\|_{H^1(\mathbb{R}^3)}^{1/2} \|u\|_{H^2(\mathbb{R}^3)}^{1/2}.$$
My question is:
- where can I find a proof of this inequality?
- are there any other useful inequalities which allow to bound the $L^\infty$ norm on $\mathbb{R}^3$.