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Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function with the following properties.

  1. $\phi(x) = 1$ if $|x| \leq 1$

  2. $\phi(x) = 0$ if $|x| \geq 2$

  3. $0 \leq \phi \leq 1$

  4. $\phi$ is radial.

Let $\phi_k(x) = \phi(\frac{x}{k})$. How do I show that for each multiindex $\alpha$, there exists a constant $C_\alpha$ such that $|D^\alpha \phi_k| \leq \frac{C_\alpha}{k^{\textrm{deg }\alpha}}$ uniformly in $k$?

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For $\alpha\in\mathbb N^n$ and $x\in\mathbb{R}^n$, we have $D^{\alpha}\phi_k(x) = \dfrac{x^{\alpha}}{k^{\mathrm{deg}\alpha}}D^{\alpha}\phi \left(\frac xk\right)$. Since the support of $\phi$ is contained in the ball $\bar{B}(0,2)$, we have $\lVert D^{\alpha}\phi_k\rVert \leq \dfrac{2^{\mathrm{deg}\alpha}}{k^{\mathrm{deg}\alpha}}\cdot \lVert D^{\alpha}\phi\rVert$, where \displaystyle\lVert f\rVert =\sup_{x\in \mathbb R} |f(x)| . We get the result with $C_{\alpha} = 2^{\mathrm{deg}\alpha}\cdot \lVert D^{\alpha}\phi\rVert$.