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Hello,

It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like

$$ \dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \mathcal{B}'(R),\quad D_{L^p}(R),\quad D_{L^p}'(R) $$ etc.

I have never seen any English references on these spaces. Does anyone know one?

EDIT:

For example, $f\in D_{L^p}(R)$ if and only if $f$ is a smooth function such that it, together with all its derivatives, belongs to $L^p(R)$. $D_{L^p}'(R)$ is its dual.

$\mathcal{B}(R)=D_{L^\infty}(R)$, and $\mathcal{B}'(R)$ is its dual.

Thank you very much!

Anand

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    $\begingroup$ Perhaps you or somebody else should at least give the definition, since notation changes often ... $\endgroup$
    – Marc Palm
    Jul 26, 2011 at 19:56
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    $\begingroup$ @pm, these notations are from Schwartz's book. I didn't find them at any other places. There is no wonder many people may not familiar with these spaces. $\endgroup$
    – Anand
    Jul 26, 2011 at 20:00
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    $\begingroup$ Do you need these specific names or do you just need the same space but perhaps denoted differently? There are plenty of modern books on distributions, including ones by Treves, Taylor, Hormander, and others. $\endgroup$
    – Deane Yang
    Feb 9, 2012 at 13:07
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    $\begingroup$ Anand, you're right. Based on your explanation of what these spaces are, they are not often used, except maybe when $p = 2$. Why do you need them? $\endgroup$
    – Deane Yang
    Feb 9, 2012 at 14:49
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    $\begingroup$ When the order of the distributions is zero, they are measures. These spaces then give a way to describe the tail behaviors of measures. That's what I need in my work. :-) $\endgroup$
    – Anand
    Feb 10, 2012 at 21:51

1 Answer 1

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As far as I remember, the book "Topological vector spaces and distributions" by Juan Horvath discusses at least the $\mathcal{D}_{L^1}$ and the $\dot{\mathcal{B}}$-spaces.

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  • $\begingroup$ Dear VoBo, I checked this book, especially Chapter 4. I didn't find these spaces you mentioned. However, I find two spaces akin to the above mentioned the spaces: (1) the space of integrable distributions which corresponds to $\mathcal{D}_{L^1}'$; (2) the space of smooth functions with bounded derivatives which corresponds to $\mathcal{D}_{L^\infty}$ in my case. Thank you for the reference. $\endgroup$
    – Anand
    Feb 22, 2012 at 14:41
  • $\begingroup$ Oh, I am sorry. I checked my old thesis, $\dot{\mathcal{B}}$ in Schwartz' book ist the subspace of $\mathcal{B}$ of functions with all their derivates vanishing at infinity. – VoBo 0 secs ago $\endgroup$
    – VoBo
    Feb 22, 2012 at 21:03

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