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I'm reading about the Littlewood-Paley decomposition, but there is a definition I can't understand, it says:

We denote by $S'h(\mathbb{R}^d)$ the space of tempered distributions $u$ such that $\displaystyle\lim_{\lambda\to\infty}\|\theta(\lambda D)u\|_{L^{\infty}} = 0$ for any $\theta$ in $D(\mathbb{R}^d)$.

What does it mean?

The remark below the definion says the distribution $u$ belongs to $S'h$ iff one can find some smooth compactly supported function $\theta$ satisfying the above equality and such that $\theta(0)$ cannot be $0$. Why?

Thanks a lot.

Gerry Myerson
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user38448-po
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  • Let me know if I haven't correctly formatted your post. – Michael Albanese Jan 03 '13 at 02:25
  • To check if a distribution lies in $S'h(\mathbb{R}^d)$ using the definition, one must verify the appropriate limit for all $\theta \in D(\mathbb{R}^d)$. I suspect that the remark points out that this is equivalent to finding a single element of $D(\mathbb{R}^d)$ for which the limit holds. Obviously it is not enough to just consider the zero distribution, which is why we require at least one point of support. – Adam Saltz Jan 03 '13 at 05:53

1 Answers1

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To make the definition clearer, the word "any" should be replaced with the word "every."

The remark then states that it suffices to have the limit property for some nowhere vanishing smooth compactly supported function.

Rasmus
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