A specific problem on : Does bounding of the Sobolev norm can cause bounding of a higher derivative? 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 $N \in \mathbb{R}, N>0$ 

For a function $f$ to belong to $H^k(\mathbb R^m)$ its order $k$ partial derivatives are square integrable. If $|\alpha| = k$ and $|\beta| = k$, then Cauchy-Schwarz gives you
$$\int_{\mathbb R^m} D^\alpha f D^\beta \phi \le \|D^\alpha f\|_{L^2(\mathbb R^m)} \|D^\beta \phi\|_{L^2(\mathbb R^m)}  \le \|f\|_{H^k(\mathbb R^m)}\|\phi\|_{H^k(\mathbb R^m)}.$$
The sum in question is bounded by $$K\cdot M\cdot(\text{number of multi-indices of length $k$})^2$$
which I have no ambition to calculate.
