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Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$, \begin{align*} \exp \left(t \rho\right), \exp \left(t \rho^{-1}\right) \in L^1_{loc}. \end{align*} Then, a conjecture of De Giorgi asserts that the Meyers-Serrin theorem holds for the weighted Sobolev spaces associated with $\mu$, namely $H=W$. The only reference I know speaking of this conjecture is this paper

https://iopscience.iop.org/article/10.1070/SM1998v189n08ABEH000344/meta

Thus, I would like to know if there have been recent advances in proving or disproving this conjecture.

Thanks in advance.

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    $\begingroup$ It might make sense making the title for specific with something like “A conjecture of De Giorgi on weighted Sobolev spaces”. $\endgroup$ – Jean Van Schaftingen Jun 10 '20 at 11:31
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On De Giorgi’s conjecture: Recent progress and open problems by Chan and Wei reviews the status of the problem in 2017.

A second attempt to locate this elusive conjecture: De Giorgi gave a talk in Lecce (Italy) in 1995 entitled "Congetture sulla continuità delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati" (Conjectures on the continuity of solutions of selfadjoint elliptic linear equations with unbounded coefficients"). The talk was typed out but never published.$^\ast$ Progress on these conjectures is discussed in

$^\ast$ De Giorgi's Bibliography says: "De Giorgi, in particular in the last years of his life, used to circulated his writings among friends and colleagues, asking for opinions. We plan in the future to collect and to make available all these unpublished writings." I tried to contact professor Zhong for a copy, but the email bounced.

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    $\begingroup$ It appears De Giorgi has more than a single conjecture. $\endgroup$ – Mr Pie Jun 10 '20 at 6:49
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I did some diggings and some readings and found out that the conjecture has been solved here

https://link.springer.com/article/10.1134/S1064562413060173

and extended recently to a wider context in

https://www.degruyter.com/view/journals/crll/2019/746/article-p39.xml

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