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Is the following true?

A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following: For every $\alpha$ there is a $N_\alpha>0$ such that $|\partial^\alpha m(x)|\le (1+|x|)^{N_\alpha}$ for all $x\in\mathbb R^n$?

If this is true, is there existed reference?

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    $\begingroup$ This is true (up to a multiplicative constant). It is in Laurent Schwartz's famous book Théorie des distributions. He called this multiplier space $\mathscr O_M(\mathbb R^n)$. $\endgroup$
    – Jochen Wengenroth
    Commented Mar 30 at 6:50
  • $\begingroup$ @JochenWengenroth Thanks! I didn't locate the index of theorem for which characterize $\mathscr O_M$ in his book (since I don't know French:( ). Could you help pointing that? $\endgroup$
    – Liding Yao
    Commented Mar 30 at 17:02
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    $\begingroup$ It is also in Horváth's book Topological Vector Spaces and Distribution as Proposition 4.11.5 on page 417 with a 1½ page long proof. $\endgroup$
    – TaQ
    Commented Mar 30 at 18:37

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