According to Wikipedia, [Agmon's inequality](https://en.wikipedia.org/wiki/Agmon%27s_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 $\mathbb{R}^3$.



Proof of Agmon's inequality in $\mathbb{R}^3$
We note that $u$ is Hölder continuous,
and we may write for any $M>0$ (to be chosen later),
$$
u(x)=\int\_{\vert \xi\vert\le M} e^{2π i x\xi}\hat u(\xi)(1+\vert \xi\vert) (1+\vert \xi\vert)^{-1}d\xi
+\int\_{\vert \xi\vert\ge M} e^{2π i x\xi}\hat u(\xi)\vert \xi\vert^2 
\vert \xi\vert^{-2}d\xi,
$$
so that using Cauchy-Schwarz inequality, we get
\begin{multline}
\Vert u\Vert\_{L^\infty}\le c\_n\Vert u\Vert\_{H^1}\left(\int\_0^M\frac{r^2}{1+r^2} dr\right)^{1/2}+
c\_n\Vert u\Vert\_{H^2}\left(\int\_M^{+\infty}{r^2}r^{-4} dr\right)^{1/2}
\\\le c\_n\Vert u\Vert\_{H^1} M^{1/2}
+c\_n\Vert u\Vert\_{H^2}M^{-1/2}.
\end{multline}
Choosing now $M=\Vert u\Vert\_{H^2}\Vert u\Vert\_{H^1}^{-1}$ gives the sought answer. It is easy to get generalizations of this inequality in $n$ dimensions following the same method with given $s\_0, s\_1$ such that
$s\_0<\frac n2< s\_1$, yielding
$$
\Vert u\Vert\_{L^\infty}\le C\_{n, s\_0,s\_1}\Vert u\Vert\_{H^{s\_0}}^{1-\theta}\Vert u\Vert\_{H^{s\_1}}^\theta, \quad \frac n2
=(1-\theta) s\_0+\theta s\_1.$$
Needless to say the above inequality is not true for $s\_0=n/2=s\_1$.


