(Reposted from MSE after no responses) Introduce the following weighted Sobolev space norm on $\mathbb{R}^n$ (common in the study of hyperbolic PDE): $$ \|u\|_{H_{k,\delta}}^2 = \sum_{0 \leq i \leq k} \int_{\mathbb{R}^n} \langle x \rangle^{2(\delta + i)}|\nabla^i u|^2 \, dx. $$ This is found (for instance; I'm sure they're found in other places) in Christodoulou & Klainerman's stability of Minkowski spacetime, and in Choquet-Bruhat's book on general relativity and the Einstein equations. Here, $\langle x \rangle = (1 + |x|^2)^{1/2}$.
They behave nicely and have their own set of embedding theorems (see Choquet-Bruhat, General Relativity and the Einstein Equations, Appendix I, Theorem 3.4). The idea of their construction seems to be that $|u|$ has a certain (integrated) decay rate (related to $\delta$), and each derivative of $u$ behaves one power of decay better.
Let us focus on the case $\delta = 0$, so $u \in H_{k, 0}$. This means, in particular, that $\langle x \rangle^\delta u \in L^2$. (However, this is not the same as the usual Sobolev space $H^k$, since recall each derivative of $u$ gains a power of decay in our case. This does not happen with classical Sobolev spaces.). Now, from learning about power functions, integration, and the $p$-test in calculus class, one might expect this (heuristically) to imply that $|u| \lesssim \langle x \rangle^{-n/2}$, as this is the "borderline" case of integrability. Of course, this is in general false. But the weighted Sobolev embeddings tell us that if $k > n/2$, then in fact $$ \sup_{x \in \mathbb{R}^n} \langle x \rangle^{n/2} |u| \lesssim \|{u}\|_{H_{k,\delta}}. $$ So we recover the decay one might heuristically have expected!
My question: It seems to me like the derivative estimates (i.e. the fact that $\langle x \rangle^{i}\nabla^i u \in L^2$) did not come into the estimate here: in fact, if $u \in H_{100, 0}$ versus $u \in H_{100000,0}$, one cannot improve the decay of $u$ just by having "more derivatives". (For instance let $u(x) = \langle x \rangle^{-q}$.) It seemed like once you had $u \in H_{k,0}$ with $k > n/2$, that was all you need, and any more derivatives are "useless." Does this seem right?
And on a related note, the decay could be guessed at from the zeroth-order behavior, i.e. just from $u \in L^2$ one expects $|u| \lesssim \langle x \rangle^{-n/2}$. The higher derivatives seemed like they were just there to rule out highly irregular behavior, like narrow/tall spikes that might spoil this. So, my second question: is the same embedding true if one just assumes $u \in L^2$ and $u \in H^k$ with $k > n/2$, in the sense of classical Sobolev spaces? That is, can the Sobolev embeddings for classical Sobolev spaces (without weights) be improved to give decay?
