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!

  • 1
    $\begingroup$ The following links might help you: math.stackexchange.com/questions/445428/… en.wikipedia.org/wiki/User:Igny/Sobolev_space $\endgroup$ – Tomas Aug 20 '15 at 7:00
  • $\begingroup$ I think $W^{-1,1}(\Omega)$ is the dual of $W^{1,1}(\Omega)$. $\endgroup$ – Ellya Aug 21 '15 at 7:49
  • $\begingroup$ @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 mathoverflow.net/a/166141 $\endgroup$ – user147263 Aug 22 '15 at 6:31

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