# One of conjectures of De Giorgi

conjecture: If $\exp(tw), \exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight.

$w$ is regular if weighted Sobolev space $W^l_p(\Omega,w)$ is equal to the completion of $C^{\infty}$ with respect to the weighted norm $\|\cdot \mid W^l_p(\Omega,w)\|$.

Question: Is there some progress with this conjecture?

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Do you have a reference for this conjecture? –  begeistzwerst Nov 23 '12 at 18:08
I've read about it here spm.uem.br/bspm/pdf/vol26/Art11.pdf –  nikita2 Nov 24 '12 at 4:53