Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|\_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|\_{H^k(\mathbb{R}^m)}<M$ 


Can we say that 


$$|\sum\limits\_{|\alpha| = k}\sum\limits\_{|\beta| = k}\int\_{\mathbb{R}^m}D^{\alpha} f D^{\beta}\phi| <N$$


for some $N \in \mathbb{R}, N>0$ 



A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?
This seems to be a consequence of the Cauchy–Schwarz inequality:
$$
\begin{split}
\bigg\vert
\sum\_{\vert \alpha \vert = k}
\sum\_{\vert \beta \vert = k}
\int\_{\mathbb{R}^m}
D^\alpha f
D^\beta \phi
\bigg\vert
&\le
\bigg(
\sum\_{\vert \alpha \vert = k}
\int\_{\mathbb{R}^m}
(D^\alpha f)^2
\bigg)^\frac{1}{2}
\bigg(
\sum\_{\vert \beta \vert = k}
\int\_{\mathbb{R}^m}
(D^\beta \phi)^2
\bigg)^\frac{1}{2}\\
&\le \Vert f \Vert\_{H^k (\mathbb{R}^m)}\Vert \phi \Vert\_{H^k (\mathbb{R}^m)} \\
&\le KM.
\end{split}
$$


