Let $r>0$. I need to show that there is a constant $C>0$ such that $$||u- u_r||_{k} \leq \frac{C}{r}||u||_{k+1}$$ for all $u$ in the Sobolev space $H^{k+1}(\mathbb{R}^m)$, where $$u_r:=\mathcal{F}^{-1}[M_r \mathcal{F}[u]],$$ $M_r$ is the operator of multiplication by the characteristic function of the ball $B_r(0)$, $\mathcal{F}$ is the Fourier Transform and $||\cdot||_{k}$ is the norm in $H^k(\mathbb{R}^m)$ .
I tried to derive $u_r$, but I am stuck. I don't understand what is the relation between $u$ and $u_r$.
Can you help me?
Keep in mind that one way to define the $H^k$ norm is by $$||u||_{H^k}= ||\langle\xi \rangle^{k} \hat{u}(\xi)||_{L^2}$$ Where $\langle \xi \rangle = \sqrt{1 + |\xi|^2}$. Making use of this we write $$\begin{align}||u-u_r||_{H^k}^2 &= ||\langle \xi\rangle^k[\hat{u - u_r}]||_{L^2}^2\\ &= \int_{\mathbb{R}^m} \langle \xi \rangle ^{2k}(1 - \chi_{B_r(0)})\hat{u}^2(\xi)d\xi\\ &= \int _{\mathbb{R}^m\setminus B_r(0)} \langle \xi \rangle^{2k}\hat{u}(\xi)^2 d\xi\\ &= \int _{\mathbb{R}^m\setminus B_r(0)} \frac{\langle \xi \rangle^{2k+2}}{\langle \xi \rangle^{2}} \hat{u}(\xi)^2 d\xi \\ &\leq \frac{1}{r^2}||u||_{H^{k+1}}^2 \end{align}$$