Density of Schwartz test restrictions in the scalar shell space

In the chart $\mathbf{k}\mapsto p_{\mu }(\mathbf{k})$, write $w_{\mu }(\mathbf{k})=[2(2\pi {)}^3{\omega }_{\mu }(\mathbf{k}){]}^{-1}$. By [F3] and the fixed invertible reflection and rescaling in Scalar future mass shell, invariant measure and test restriction, ${\mathcal{F}}_{+}f$ is Schwartz for every $f\in \mathcal{S}({\mathbb{R}}^4)$. Thus $q_2(f):={\sup }_{q\in {\mathbb{R}}^4}(1+\mathrm{\mid }q\mathrm{\mid }{)}^2\mathrm{\mid }{\mathcal{F}}_{+}f(q)\mathrm{\mid }<\mathrm{\infty }\mathrm{.}$ On the shell, $\mathrm{\mid }{\mathcal{F}}_{+}f({\omega }_{\mu }(\mathbf{k}),\mathbf{k})\mathrm{\mid }\le q_2(f)(1+\mathrm{\mid }\mathbf{k}\mathrm{\mid }{)}^{-2}$, so $\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_{H_{\mu }}^2\le \frac{q_2(f{)}^2}{2(2\pi {)}^3}{\int }_{{\mathbb{R}}^3}\frac{(1+\mathrm{\mid }\mathbf{k}\mathrm{\mid }{)}^{-4}}{{\omega }_{\mu }(\mathbf{k})}{d}^3\mathbf{k}<\mathrm{\infty }\mathrm{.}$ The radial integrand is $O(r)$ near zero if $\mu =0$, $O(r^2)$ near zero if $\mu >0$, and $O(r^{-3})$ at infinity. Therefore every $K_{\mu }f$ lies in $H_{\mu }$, so the claimed range is a subset of the Hilbert space. By [F1], real-valued $L^2({\mathbb{R}}^3)$ has a countable dense subset $S_{\mathbb{R}}$. For complex-valued $h=u+iv\in L^2({\mathbb{R}}^3;\mathbb{C})$, its real and imaginary parts lie in real-valued $L^2$. For any $\epsilon >0$, choose $s,t\in S_{\mathbb{R}}$ with $\mathrm{\parallel }u-s{\mathrm{\parallel }}_2,\mathrm{\parallel }v-t{\mathrm{\parallel }}_2<\epsilon \mathrm{/}\sqrt{2}$. Then $\mathrm{\parallel }h-(s+it){\mathrm{\parallel }}_2^2=\mathrm{\parallel }u-s{\mathrm{\parallel }}_2^2+\mathrm{\parallel }v-t{\mathrm{\parallel }}_2^2<{\epsilon }^2$. Enumerate $S_{\mathbb{R}}$ by $(s_j{)}_{j\in {\mathbb{N}}_0}$, repeating terms if it is finite. The Cantor pairing map $(j,k)\mapsto (j+k)(j+k+1)\mathrm{/}2+k$ enumerates ${\mathbb{N}}_0^2$, so $S_{\mathbb{R}}+i{S}_{\mathbb{R}}$ is countable and dense in complex-valued $L^2({\mathbb{R}}^3)$. Multiplication by $\sqrt{w_{\mu }}$ is an isometry from $H_{\mu }$ onto $L^2({\mathbb{R}}^3\setminus \{0\},d^3\mathbf{k};\mathbb{C})$ (for $\mu =0$) or $L^2({\mathbb{R}}^3,d^3\mathbf{k};\mathbb{C})$ (for $\mu >0$), with inverse division by $\sqrt{w_{\mu }}$. In the massless case, extend any function on the punctured space by zero at the null point, approximate it by the countable dense family just constructed, and restrict those approximants back to the punctured space; restriction does not increase the norm. Thus both chart spaces, and therefore $H_{\mu }$, are separable.

Let $\mathrm{\Omega }={\mathbb{R}}^3$ for $\mu >0$ and $\mathrm{\Omega }={\mathbb{R}}^3\setminus \{0\}$ for $\mu =0$. For ${\mathrm{\Omega }}_n=\{1\mathrm{/}n<\mathrm{\mid }\mathbf{k}\mathrm{\mid }<n\}$, these sets increase to $\mathrm{\Omega }\setminus \{0\}$; its complement in $\mathrm{\Omega }$ is null for the shell measure. For $h\in L^2(\mathrm{\Omega },w_{\mu }{d}^3\mathbf{k})$, monotone convergence [F5] gives ${\int }_{{\mathrm{\Omega }}_n}\mathrm{\mid }h{\mathrm{\mid }}^2{w}_{\mu }{d}^3\mathbf{k}⟶{\int }_{\mathrm{\Omega }}\mathrm{\mid }h{\mathrm{\mid }}^2{w}_{\mu }{d}^3\mathbf{k}<\mathrm{\infty }\mathrm{.}$ Subtracting these finite integrals from the total shows $\mathrm{\parallel }h{\mathbf{1}}_{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_n}{\mathrm{\parallel }}_{L^2(w_{\mu })}\to 0$. On each ${\mathrm{\Omega }}_n$, $w_{\mu }$ is smooth and bounded above and below by positive constants. Approximate $h{\mathrm{\mid }}_{{\mathrm{\Omega }}_n}$ in ordinary $L^2({\mathrm{\Omega }}_n)$ by a function in $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_n)$ using [F2]; equivalence of the weighted and ordinary norms on ${\mathrm{\Omega }}_n$ gives weighted approximation. Thus $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is dense in the shell space.

Fix $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Choose $\rho \in C_c^{\mathrm{\infty }}(\mathbb{R})$ with $\rho (0)=1$, using [F4], and put $G(q^0,\mathbf{q})=\psi (\mathbf{q})\rho (q^0-{\omega }_{\mu }(\mathbf{q}))\mathrm{.}$ Because $\psi$ has compact support away from the massless apex, ${\omega }_{\mu }$ is smooth on a neighborhood of its support, so this is a smooth compactly supported function on ${\mathbb{R}}^4$. By the Fourier automorphism [F3] there is $f\in \mathcal{S}({\mathbb{R}}^4)$ with ${\mathcal{F}}_{+}f(q^0,\mathbf{q})=G(q^0,\mathbf{q})$ in the convention of Scalar future mass shell, invariant measure and test restriction. On the shell, $K_{\mu }f(p_{\mu }(\mathbf{k}))=G({\omega }_{\mu }(\mathbf{k}),\mathbf{k})=\psi (\mathbf{k})$. Hence the test restriction range contains $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, which is dense by 2.1.