I read some books and papers and I am confused about some norms with a variety of subscripts that are not uniform. I mean, some of the subscripts are sets while others are spaces.
In particular, can you help me to understand the specific definitions of the following norms I see in the books and papers?
\begin{align} &\|x\|_\Sigma \\ &\|x\|_{L^\infty(\Sigma)} \\ &\|x\|_{L^p(\Sigma)} \\ &\|x\|_{W_\infty^2(\Sigma)} \\ &\|x\|_{H^1(\Sigma)} \\ &\|x\|_{H^2 \;\text{without}\; (\Sigma)} \end{align}
where $x$ may be a function defined on $\Omega \supset \Sigma$.
In addition, I also see some subscripts are numbers or $\infty$, such as $\|x\|_2$ and $\|x\|_\infty$. In these cases I think $x$ must be a sequence rather than a function, and they are abbreviations for $\|x\|_{l_2}$ and $\|x\|_{l_\infty}$?
i checked the paper so far. The norms are defined as above: For $\Sigma\subset\mathbb{R}^n$ and $v:\Sigma\to\mathbb{R}$ we have
$||v||_{\Sigma}=\left(\int_{\Sigma}v(t)^2dt\right)^{1/2}$, the "square mean value".
For $1\leq p < \infty$ we have
$||v||_{L^p(\Sigma)}=\left(\int_{\Sigma}|v(t)|^p dt\right)^{1/p}$, the $L^p$ norm on $\Sigma$,
for the case $p=\infty$ and so for the $L^{\infty}$-norm i refer to https://en.wikipedia.org/wiki/Lp_space and the construction of the essential supremum.
Then for $s\in\mathbb{N}_0$ and $1\leq p\leq \infty$ we have
$||v||_{W^s_p(\Sigma)}=\sum_{i=1}^{s}||\partial_t^{i}v||_{L^p(\Sigma)},$ the Sobolev norm,
for a first introduction see https://en.wikipedia.org/wiki/Sobolev_space.
If we choose $p=2$ in the Sobolev setting we use for shorten the notion
$W^s_2(\Sigma)=H^s(\Sigma)$
and so we have
$||v||_{H^1(\Sigma)}=||v||_{W^1_2(\Sigma)}, \quad ||v||_{H^2(\Sigma)}=||v||_{W^2_2(\Sigma)}.$