A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?

Question

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

Answer 1

This seems to be a consequence of the Cauchy–Schwarz inequality: $$ \begin{split} \bigg\vert \sum_{\vert \alpha \vert = k} \sum_{\vert \beta \vert = k} \int_{\mathbb{R}^m} D^\alpha f D^\beta \phi \bigg\vert &\le \bigg( \sum_{\vert \alpha \vert = k} \int_{\mathbb{R}^m} (D^\alpha f)^2 \bigg)^\frac{1}{2} \bigg( \sum_{\vert \beta \vert = k} \int_{\mathbb{R}^m} (D^\beta \phi)^2 \bigg)^\frac{1}{2}\\ &\le \Vert f \Vert_{H^k (\mathbb{R}^m)}\Vert \phi \Vert_{H^k (\mathbb{R}^m)} \\ &\le KM. \end{split} $$