Weighted $L^2$ space where $\int_{\mathbb{R}^n}\langle x\rangle ^\delta |u|^2\, dx<\infty $, and the Laplacian

Question

Let $\Delta$ denote the Laplacian on $\mathbb{R}^n,$ and define the weighted $L^2$ space $$L^{2,\delta}=\left\lbrace u:\mathbb{R}^n\rightarrow\mathbb{C}: \int\limits_{\mathbb{R}^n}\langle x\rangle ^\delta |u|^2\, dx<\infty\right\rbrace,$$ where $\langle x\rangle=(1+|x|^2)^{1/2}.$ In some sense, they're sort of like Sobolev spaces, but they explicitly track spatial growth/decay instead of frequency growth/decay. I am wondering about mapping properties $\Delta:L^{2,\delta_1}\rightarrow L^{2,\delta_2}.$ My guess is that for any $\delta,$ $\Delta:L^{2,\delta}\rightarrow L^{2,\delta+2}.$ This guess is due to the fact that the Laplacian is a second order elliptic operator, so it satisfies elliptic regularity hypotheses, but one would replace the regularity changes with the analogous weightings. However, I have not been able to show it. My intuition is that one can utilize some argument rooted in combining elliptic regularity with the asymptotics of the fundamental solution of Poisson's equation, but I have not made much progress on that end.

Answer 1

As in the comments, the "only natural" definition of $\Delta f$ for some continuous functions takes you out of the class of functions; e.g. $\frac{d^2}{dx^2}\max(x,0)= \delta_0$.

Even if you assume that $\Delta f$ is a function, you have to deal with large oscillations, and/or the fact that you are not controlling spatial growth at small scales (really, another manifestation of large frequencies).

Example due to no spatial control: well known fact- $f(x) =\frac1{|x|^{1/4}\langle x\rangle^{100}} \in L^2(\mathbb R) = L^{2,0}(\mathbb R)$. Differentiate twice and the singularity at zero is now $\sim \frac1{|x|^{2+1/4}}$ which is not in any $L^{2,\delta}$, $\delta\in\mathbb R$.

Smooth example due to no frequency control: $f(x) = \sin( e^{x})\in C^\infty\cap L^\infty \subset L^{2,-2}(\mathbb R)$, with $$ \sin(e^x)''=e^x\Big(\cos(e^x) -e^x \sin(e^x)\Big)$$ and when $|\sin(e^x)| > 1/2$, $|\cos(e^x)|<1/2$, so $$ |\sin(e^x)''|^2 \ge e^{2x}\left(e^{x}|\sin(e^x)|-\frac12\right)^2$$ Further restricting to e.g. $x>10$, $$e^x|\sin e^x|-\frac12>e^{x/2}2 |\sin e^x|-\frac12 > e^{x/2}-1 > 1$$

Since $$|\sin(e^x)| > 1/2 \iff e^x \in \left(\frac\pi6 + 2\pi n,\ \frac{5\pi}6+2\pi n\right),\ n\ge 0 \\ \iff x \in \left(\log\Big(\frac\pi6 + 2\pi n\Big),\ \log\Big(\frac{5\pi}6+2\pi n\Big)\right),\ n\ge 0.$$ Each $n$th interval is a region of size (mean value theorem) $$ \log\Big(\frac{5\pi}6+2\pi n\Big) - \log\Big(\frac\pi6 + 2\pi n\Big) \ge \frac{2\pi}3 \frac1{\frac{5\pi}6+2\pi n}\ge \frac1{1000n}$$ on which $$ \langle x\rangle^{\delta}|\sin(e^x)''|^2 \ge \langle \log(\frac\pi6+2\pi n)\rangle^\delta e^{2\log(\frac\pi6+2\pi n)}\ge C_\delta \langle \log n \rangle^\delta n^2$$ It easily follows $\sin(e^x)'' \notin L^{2,\delta}$ for any $\delta\in\mathbb R$.