Subscript Notation on Real Numbers

Question

I was reading a wiki page on oscillatory integrals and I came across two notations I am unfamiliar with.

$$\Bbb R_{x}^{n}\ \times \ R_{\zeta}^{N}$$

$$S_{1,0}^{m}(\Bbb R_{x}^{n}\ \times \ R_{\zeta}^{N})$$

Firstly, what do the subscripts for the real numbers mean? Also what is $S_{1,0}^{m}$ and what does the second notation mean? There is no explanation on the page and trying to search around has not gotten me far.

Answer 1

Subscripts appear be simply the names of the variables. For the definition of $S^m_{1,0} = S^m$, quote from Pseudo-Differential Operators and Symmetries:

Definition (Symbol Classes $S^m(\Bbb R^n\times\Bbb R^n)$). We will say that $a\in S^m(\Bbb R^n\times\Bbb R^n)$ if $a = a(x,\xi)$ is smooth in $\Bbb R^n\times\Bbb R^n$ and if the estimates $$|\partial^\beta_x\partial^\alpha_\xi a(x,\xi)|\le A_{\alpha,\beta}(1 + |\xi|)^{m - |\alpha|}$$ hold for all $\alpha,\beta$ [multi-indexes] and $x,\xi\in\Bbb R^n$. Constants $A_{\alpha,\beta}$ may depend on $a,\alpha,\beta$, but no on $x,\xi$.

Answer 2

At the bottom of the linked Wikipedia page, there is a set of references:

The second reference is to an article which is freely available online. The notation $S_{\varrho,\delta}^{m}(X\times\mathbb{R}^N)$ is defined on page 83:

Definition 1.1.1. Let $m$, $\varrho$, $\delta$ be real numbers with $0\le \varrho \le 1$, $0\le \delta \le 1$. Then we denote by $S_{\varrho,\delta}^{m}(X\times\mathbb{R}^N)$ the set of all $a \in C^{\infty}(X\times\mathbb{R}^N)$ such that... [technical conditions follow].

The technical conditions basically give bounds on how large the derivatives can be on compact subsets of $X$, where $X$ is an open subset of $\mathbb{R}^n$ (note that the first coordinate comes from $X \subset \mathbb{R}^n$, while the second coordinate comes from $\mathbb{R}^N$—it appears that $n$ and $N$ may be distinct). The sets defined here are the symbols (of order $m$ and type $\varrho, \delta$). $S^{\infty}_{\varrho,\delta}$ is the union of all symbols, taken over the order $m$.

Regarding the meaning of $\mathbb{R}_x^n$ and $R_\xi^N$, I have two comments:

1. It appears that there is a typo in the Wikipedia article, and that $R_{\xi}^N$ should really be $\mathbb{R}_{\xi}^N$. However, I don't know a whole lot about oscillattory integrals, nor the notational conventions of the field, so this is only a guess.
2. From context, it appears that the subscripts denote the names of the variables which are taken from the given coordinate. For example, if $$ a \in C^{\infty}(\mathbb{R}_x^n \times \mathbb{R}_{\xi}^N),$$ then $a$ is function with domain $\mathbb{R}^n \times \mathbb{R}^N$, where $a$ takes as input an ordered pair $(x,\xi)$, where $x \in \mathbb{R}^n$ and $\xi \in \mathbb{R}^N$. This appears to be a shorthand notation used to remind the reader where the variables come from.