Proof of Agmon's inequality in $\mathbb{R}^3$ 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 $\mathbb{R}^3$. 

 A: 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 $s_0=n/2=s_1$.
