Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions for compactness of this embedding?
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1$\begingroup$ There are several definitions of Sobolev spaces on metric-measure spaces. Which one are you using? There are many results about compactness of the embedding, but until I know more about what $W^{1,2}$ is, I cannot answer the question. $\endgroup$– Piotr HajlaszCommented Nov 30, 2021 at 17:24
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$\begingroup$ I know that you did some work in analysis on graphs. Are you interested in Sobolev embeddings on graphs? $\endgroup$– Piotr HajlaszCommented Nov 30, 2021 at 17:28
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$\begingroup$ @PiotrHajlasz Yes, this would be my most immediate interest. In the case of finite graphs, Sobolev spaces are defined in a pretty canonical way. Infinite graphs and fractals would be the next step, I guess. $\endgroup$– Delio MugnoloCommented Nov 30, 2021 at 21:25
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$\begingroup$ @PiotrHajlasz So, the different definitions of Sobolev spaces on m-m-s are diverging? In this case, I'd probably pick what works best in this context. $\endgroup$– Delio MugnoloCommented Nov 30, 2021 at 21:26
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