From Terence Tao's Nonlinear dispersive equations: local and global analysis, Definition 2.7:
The Bourgain space $X^{s,b}(\mathbb R\times\mathbb R^n)$ is defined to be the closure of the set of Schwartz functions $\mathcal S_{t,x}(\mathbb R\times\mathbb R^n)$ under the norm
$$\|u\|_{X^{s,b}}:=\|\langle\xi\rangle^s\langle\tau-h(\xi)\rangle^b\hat u(\tau,\xi)\|_{L_{\tau,\xi}^2}$$
for $\{s,b\}\subset\mathbb R$ and a given continuous function $h:\mathbb R^n\to\mathbb R$.
In the definition above I would assume there is a reason it's defined to be the closure of $\mathcal S_{t,x}$ rather than the set of all function such that $\|u\|_{X_{\tau=h(\xi)}^{s,b}}<\infty$. What kinds of functions have finite $X^{s,b}$-norms but are not limits of Schwartz functions? And what kinds of problems, if any, can we encounter with the $\|u\|_{X_{\tau=h(\xi)}^{s,b}}<\infty$ version of the space?
Best Answer
I think I figured out why. The Bourgain norm is defined in terms of an $L^2$-norm, and we know the Schwartz function space $\mathcal S$ is dense in $L^2$. Also $\langle\xi\rangle^s\langle\tau-h(\xi)\rangle^b$ is virtually more or less a polynomial* so if $\widehat u\in\mathcal S$ then $\langle\xi\rangle^s\langle\tau-h(\xi)\rangle^b\widehat u\in\mathcal S$ and vice versa. Therefore in fact both versions of definitions are the same. There is no function $u$ such that $\|u\|_{X^{s,b}}<\infty$ as well as $u\notin X^{s,b}$. The reason we think of the closure of $\mathcal S$ is that it tells us what qualifies as a function in a given space; we know $\mathcal S$ is dense in $L^p$, but an $L^p$ function does not have to be the kind of function that we learnt at high school. An $L^p$ function can take $\infty$ as its value at countably many points, and the Sobolev space $H^{-1}(\mathbb R)$ even contains the Dirac delta function $\delta$ which is, in fact, not a function. Just like we can define the sine or cosine function reversely with its Taylor series, we can define the $L^p$ space as the closure of $\mathcal S$, and its variants likewise, such as the Bourgain space $X^{s,b}$, the Sobolev space $W^{k,p}$, etc. Such a definition naturally also defines what a function is in the given context.
* For $b<0$ with $h(\xi)$ growing faster than polynomials, things could go wrong, but so far I've never seen the $X^{s,b}$ space used in such a setting, in which we would need something more than tempered distributions if we were to adopt the finite-norm definition. This is maybe pointless in practice.