Morrey's Identity:
Define the Torus: $\pi^{d}=\frac{\mathbb{R}^{d}}{2 \pi \mathbb{Z}^{d}}$
Let $m \in \mathbb{N} $ and $s\in \mathbb{R}$ such that $s > m + \frac{d}{2}$ . Prove that there exists $ c > 0$ such that for all $u \in H^{s}(\pi ^{d})$ (Sobolev space) we have: $\|u\|_{C^{m}(\pi ^{d})} \leq c\|u\|_{H^{s}(\pi ^{d})} $ , for some constant $c$.
Remark that : $H^{s}(\pi ^{d}) \subset C^{m}(\pi^{d})$ for $s >m +\frac{d}{2}$.
Solution :
For $m=0$, $s>\frac{d}{2}$ we have to prove that there exist for all $u \in$ $H^{s}(\pi ^{d}$) we have : $ \|u\|_{C^{0}}$ $ \leqslant$ c$\|u\|_{H^{s}(\pi^{d})}$
Let $u \in H^{s}(\pi^{d})$
$\begin{aligned} | u ( x ) | & = \left| \sum _ { k \in \mathbb{Z} ^ { d } } \hat { u } ( k ) e_{ k} \right| \\ & \leqslant \sum_ { k \in \mathbb { Z } ^ { d } } | \hat { u } ( k ) | \langle k \rangle ^ { s } \langle k \rangle ^ { - s } \\ & \leqslant \left( \sum_ { k \in \mathbb{Z} ^ { d } } | \hat { u } ( k ) | ^ { 2 } < k > ^ { 2 s } \right) ^ { \frac { 1 } { 2 } } \left( \sum _ { \mu \in x ^ { d } } \langle k \rangle ^ { - 2 s } \right) ^ { \frac{1}{2} } \\ & \leqslant \| u \| _ { H ^ { s }(\pi ^ {d})} \left( \sum _ { k \in \mathbb{Z} ^ { d } } \frac { 1 } { \|k\| ^ { 2s } } \right) ^ { \frac { 1 } { 2 } } \\ & \leqslant c \|u\| _{H^{s}(\pi^{d})} \end{aligned}$
where $<k>^{s}$ = $(1+\|k\|^{2})^{\frac{s}{2}}$ and k$\in$ $\mathbb{Z}^{d}$
