# What does the $2$ mean in the space $H_0^{1,2}$?

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 $H^1$.

• I suspect it could mean $H^1_0 \cap H^2$. But that's just a gut feeling. Oct 27 '17 at 20:43

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.)
• I was just reading it as $H_0^1$. Thanks.