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I'm reading a paper: https://arxiv.org/pdf/1003.2074.pdf

In (4.1) they use the space $H_0^{1,2}$. I know that $H_0^1=W_0^{1,2}$ is the closure of $C_c^\infty$ under the $W^{1,2}$ norm. What is $H_0^{1,2}$? I'm assuming that they're mixing notations and that $H_0^{1,2}=H_0^1=W_0^{1,2}$. Is this correct? I am confused because in other places they use $H_0^1$ or $H^1$.

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  • $\begingroup$ I suspect it could mean $H^1_0 \cap H^2$. But that's just a gut feeling. $\endgroup$ Oct 27 '17 at 20:43
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Read $H^{1,2}_0$ as $H^1_0$; it's a common notational mishap from mixing $H^1$ and $W^{1,2}$, especially in papers with multiple authors. They also alternate between $\operatorname{arctan}$ and $\operatorname{arctg}$ notation.

By the way, the published version of this paper is somewhat different, with section 4 expanded. (It still has $H^{1,2}_0$, though.)

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  • $\begingroup$ I was just reading it as $H_0^1$. Thanks. $\endgroup$
    – user223391
    Oct 28 '17 at 0:37

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