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I am currently working my way through Stein's Functional Analysis and at one point in the text he claims without proof that for all $\phi \in\mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of functions on $\mathbb{R}^d$, $$||\phi_y^{\widetilde{}}||_N \leq c_N(1 + |y|)^N ||\phi||_N$$ where $$\phi_y^{\widetilde{}}(x) = \phi(y-x)$$ and $$||\phi(x)||_N = \sup_{x \in \mathbb{R}^d, |\alpha|, |\beta|\leq N}|x^\beta (\partial_x^\alpha \phi)(x)|$$ Notation: $\partial_x^\alpha = (\partial/\partial x)^\alpha = (\partial/\partial x_1)^{\alpha_1 }\cdots (\partial/\partial x_d)^{\alpha_d}$ and $|\alpha| = \alpha_1 + \cdots + \alpha_d$.

I cannot seem to figure out why the above inequality is true. It must be somewhat simple for the author to not provide a proof. Any guidance on this would be greatly appreciated!

This seems like a very relevant post, but I can't figure out how to adapt to my case. here

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First note that $||\cdot||_N$ and $$|||f|||_N := \sup_{x\in \mathbb{R}^d, |\alpha|\leq N} |(1+|x|^2)^{N/2}\partial_x^\alpha f(x)|$$ are equivalent, i.e. it exist $c_N,C_N \in \mathbb{R}$ with $$||.||_N \leq C_N|||.|||_N \leq C_Nc_N||.||_N$$ Then use Peetre's lemma: for $x,y \in \mathbb{R}^d$ and $m \in \mathbb{R}$ it holds $$(1+|x+y|^2)^{m/2} \leq 2^{|m|}(1+|x|^2)^{|m|/2}(1+|y|^2)^{m/2}$$ It follows $$||\tilde\phi_y||_N \leq C_N|||\tilde\phi_y|||_N = C_N\sup_{x\in \mathbb{R}^d, |\alpha|\leq N} |(1+|x|^2)^{N/2}\partial_x^\alpha \phi(y-x)| = C_N\sup_{x\in \mathbb{R}^d, |\alpha|\leq N} |(1+|y-x|^2)^{N/2}\partial_x^\alpha \phi(x)| \leq C_N2^N(1+|y|^2)^{N/2}\sup_{x\in \mathbb{R}^d, |\alpha|\leq N} |(1+|x|^2)^{N/2}\partial_x^\alpha \phi(x)| \leq C_N2^N(1+|y|)^N|||\phi|||_N \leq c_NC_N2^N(1+|y|)^N||\phi||_N$$

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  • $\begingroup$ This looks correct. Thank you so much! $\endgroup$
    – slowspider
    Apr 17 at 22:26

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