# About the dual of Sobolev spaces

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!

• I think $W^{-1,1}(\Omega)$ is the dual of $W^{1,1}(\Omega)$. – Ellya Aug 21 '15 at 7:49
• @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 mathoverflow.net/a/166141 – user147263 Aug 22 '15 at 6:31