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the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$: $$ \|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| \leq N}} |\,x^\beta(\partial^\alpha_x \phi)(x)\,| $$ defined for each non-negative integer $N$, where the multi - index notation is used and $\phi$ is a $C^\infty$ function.

in particular, this is done in the book by E. Stein (et al), "Functional Analysis" (Ch. 3). There it is also stated (in the proof of Proposition 1.5 in Ch.3, Sect. 1.5) that for any compactly supported $C^\infty$ function $\psi$ and any $N$, if $\psi^\backsim_x := \psi(x - y)$ then we have the estimate $$ \|\psi^\backsim_x\| \leq c(1 + |x|)^N\|\psi\|_N \,, $$ and more generally, $$ \|\partial^\alpha_x \psi^\backsim_x\| \leq c(1 + |x|)^N\|\psi\|_{N + |\alpha|} \,, $$

this confuses me and clearly shows that I don't understand the notation of the semi-norm well enough.

here is what I struggle with: since $\psi^\backsim_x$ denotes translation by $x$ and this is done before I take the norm, I would have thought that this operation has no impact on the size of the norm, i.e. just plugging in the translated function in the norm I'd have $$ \|\psi^\backsim_x\|_N := \sup_{\substack{(x-y)\, \in \, \mathbb{R}^n \\ |\alpha|\,,|\beta| \leq N}} |\,(x - y)^\beta(\partial^\alpha_{(x - y)} \psi)(x - y)\,| = \|\psi\|_N $$

Why is this not the correct way to measure $\psi^\backsim_x$ with respect to the family $\|\cdot\|_N$ ?

thanks a lot for clarification!

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Could someone explain to me how these two estimates for $ \|\psi^\backsim_x\| $ and for $\|\partial^\alpha_x \psi^\backsim_x\|$ follows from Julian's answer? I understand his answer but I can't see how to procceed, to obtain the desired estimate. I think I am missing something obvious... – passenger Nov 30 '13 at 13:00
up vote 2 down vote accepted

The expression for $\|\psi_{\tilde x}\|_N$ is incorrect; $\psi_{\tilde x}$ does not depend on $x$. It should be $$ \|\psi_{\tilde x}\|_N=\sup_{\substack{y \in \mathbb{R}^n \\ |\alpha|,\,|\beta| \le N}} |y^\beta(\partial^\alpha_y \psi)(y-x)|= \sup_{\substack{y \in \mathbb{R}^n \\ |\alpha|,\,|\beta| \le N}} |(y+x)^\beta(\partial^\alpha_y \psi)(y)|. $$

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