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I'm reading a paper about PDE, in which the author mentioned the space $W^{-1,1}(\Omega)$, does this mean the dual space $W^{1,\infty}(\Omega)^{*}$ ? I hope not.

I only know the Sobolev dual space when $p>1$, so who can give some suggestions about its definition and norm. Thank you very much!

Kira Yamato
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    The following links might help you: http://math.stackexchange.com/questions/445428/the-dual-of-the-sobolev-space-wk-p https://en.wikipedia.org/wiki/User:Igny/Sobolev_space – Tomas Aug 20 '15 at 07:00
  • I think $W^{-1,1}(\Omega)$ is the dual of $W^{1,1}(\Omega)$. – Ellya Aug 21 '15 at 07:49
  • @ellya No. The point of using the Hölder conjugate exponent is so that by formal integration by parts, $W^{-1,p}$ can comprise second derivatives of the functions in $W^{1,p}$. A search for "negative order Sobolev spaces" will bring up examples such as http://mathoverflow.net/a/166141 –  Aug 22 '15 at 06:31

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