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Let $\phi$ a smooth function with compact support on $\mathbb R^n$ such that $\phi=1$ on $B(0;1)$, $\phi=0$ on $\mathbb R^n\setminus B(0;2)$ and $0\leq \phi\leq 1$ on $B(0;2)\setminus B(0;1)$. Define $\phi_{m}=\phi\left(\dfrac{x}{m}\right)$ where $\alpha$ is a multi index.

Then what is $||D^\alpha\phi_{m}||_{\infty}$ for all $|\alpha|\leq k$ where $k\in \mathbb N$ also what is $supp(D^\alpha\phi)$?

I am trying but I am stuck.

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  • $\begingroup$ What did you tried and what exactly is the difficulty you are having? $\endgroup$ – mucciolo Jan 2 '18 at 11:18
  • $\begingroup$ Without knowing more about your $\phi$ where is no way to calculate. $\endgroup$ – user99914 Jan 2 '18 at 11:20
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First try to do the exercise in dimension one. You will find that if $\alpha$ is a positive integer, then $D^\alpha f(x)=m^{-\alpha} f^{(\alpha)}\left(x /m\right)$ (prove it by induction). The best you can do it to express everything in terms of the supremum norm/support of the derivatives of $f$, $m$ and $\alpha$.

Then in dimension $n$, you will find that $$ D^\alpha f(x)=m^{-\left\lvert \alpha\right\rvert } D^{\alpha} f\left(\frac 1m x\right). $$

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