Mellin incoming and outgoing propagation

The change of variables $y=e^{-s}x$ gives $\mathrm{\parallel }U(s)\psi {\mathrm{\parallel }}_2=\mathrm{\parallel }\psi {\mathrm{\parallel }}_2$, and $U(s)U(r)=U(s+r)$, $U(s{)}^{-1}=U(-s)$. Polar coordinates give $dx=e^{3\rho }d\rho d\sigma (\omega )$, hence $T_x$ and $T_p$ are unitary, with inverse $u\mapsto [\psi (r\omega )=r^{-3\mathrm{/}2}u(\log r,\omega )]$ for $r>0$; the omitted origin has measure zero. Direct substitution gives $T_{x}U(s)T_x^{\ast }u(\rho ,\omega )=u(\rho -s,\omega )$ and $\mathcal{G}{T}_{x}U(s)T_x^{\ast }{\mathcal{G}}^{\ast }h=e^{-is\eta }h$. Dominated convergence for $\mathrm{\mid }(e^{-is\eta }-1)h{\mathrm{\mid }}^2\le 4\mathrm{\mid }h{\mathrm{\mid }}^2$ proves strong continuity. Multiplication by the real coordinate $\eta$ is self-adjoint on $\{h:\eta h\in L^2\}$: if $h$ belongs to its adjoint domain with adjoint value $k$, testing against arbitrary $L^2$ functions supported where $\mathrm{\mid }\eta \mathrm{\mid }\le m$ gives $k=\eta h$ there, and exhaustion over integers $m$ gives the exact multiplier domain. For $\eta h\in L^2$, dominated convergence using $\mathrm{\mid }(e^{-is\eta }-1)\mathrm{/}s\mathrm{\mid }\le \mathrm{\mid }\eta \mathrm{\mid }$ gives the derivative $-i\eta h$ at zero. Conversely, if the difference quotients converge in $L^2$, their norms are bounded; Fatou applied to their pointwise limit $-i\eta h$ gives $\eta h\in L^2$, so this is the exact generator domain. Thus $D=T_x^{\ast }{\mathcal{G}}^{\ast }{M}_{\eta }\mathcal{G}{T}_x$ is the generator in the declared convention. Fourier differentiation identifies its log domain with $\{u:{\mathrm{\partial }}_{\rho }u\in L^2\}$ and its action with $-i{\mathrm{\partial }}_{\rho }$, where the derivative is weak.

Smooth functions compactly supported in $\rho$, with smooth angular dependence, are dense for the norm $\mathrm{\parallel }u{\mathrm{\parallel }}_2+\mathrm{\parallel }{\mathrm{\partial }}_{\rho }u{\mathrm{\parallel }}_2$: first multiply by $\chi (\rho \mathrm{/}m)$ for a fixed smooth cutoff $\chi =1$ near zero; the derivative error from the cutoff is bounded by $\mathrm{\parallel }{\chi }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{/}m$, and the two tail errors tend to zero. Next convolve in $\rho$ with a compactly supported smooth approximate identity, which converges in $L^2$ for both $u$ and its weak derivative; finally smooth the angular variable in a finite smooth coordinate cover of the sphere with a partition of unity. This angular smoothing converges in both norms because it acts only in $\omega$ and commutes with ${\mathrm{\partial }}_{\rho }$. Each resulting inverse $T_x$ image belongs to $C_c^{\mathrm{\infty }}({\mathbb{R}}^3\setminus \{0\})$. These images therefore form a graph core. Every $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ also has $T_x\psi$ and ${\mathrm{\partial }}_{\rho }{T}_x\psi$ in $L^2$, by the polar change of variables, and differentiation gives $T_x^{\ast }(-i{\mathrm{\partial }}_{\rho })T_x\psi =-i(x\cdot \mathrm{\nabla }+3\mathrm{/}2)\psi$. The larger space $C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ is consequently a core as well. Countable approximation sequences suffice for these density arguments.

Changing variables in the momentum Fourier integral gives ${\mathcal{F}}_{p}U(s){\mathcal{F}}_p^{\ast }\varphi (p)=e^{3s\mathrm{/}2}\varphi (e^{s}p)$. Its momentum-log action is $u(\rho ,\omega )\mapsto u(\rho +s,\omega )$, so $WU(s)W^{\ast }=M_{e^{is\eta }}$ and $WD{W}^{\ast }=M_{-\eta }$, with the multiplier domain stated in (a). Equivalently Fourier conjugation gives $+i(p\cdot {\mathrm{\nabla }}_p+3\mathrm{/}2)$ on smooth vectors, while the position action in step 2.1 has the opposite sign. By the multiplication PVM, $W{P}_{+}{W}^{\ast }=M_{{\mathbf{1}}_{\eta <0}}$ and $W{P}_{-}{W}^{\ast }=M_{{\mathbf{1}}_{\eta >0}}$. The set $\{0\}\times S^2$ has product measure zero, so the zero projection vanishes and $P_{+}+P_{-}=I$.

For $t\ge 1$ set ${\chi }_t={\mathbf{1}}_{\{\mathrm{\mid }x\mathrm{\mid }<2at\}}$. On the dense set of $h(\eta ,\omega )$ smooth in both variables and compactly supported in $\eta$, substitute ${\mathcal{G}}^{\ast }h$ into $T_p^{\ast }$ and then into the inverse momentum Fourier integral. The polar Jacobian $e^{3\rho }$ combines with $e^{-3\rho \mathrm{/}2}$ from $T_p^{\ast }$ to give the amplitude $A(\rho )=f(e^{2\rho })e^{3\rho \mathrm{/}2}$. All integrals are absolutely convergent on this dense set because $A$ is compactly supported and the input has compact $\eta$ support. Thus ${\chi }_t{e}^{-it{H}_0}f(H_0)P_{+}{W}^{\ast }$ has the kernel $K_t(x;\eta ,\omega )=\frac{{\chi }_t(x){\mathbf{1}}_{\eta <0}}{(2\pi {)}^2}{\int }_{\mathbb{R}}{e}^{i{\mathrm{\Phi }}_t(\rho ;x,\eta ,\omega )}A(\rho )d\rho ,{\mathrm{\Phi }}_t=e^{\rho }\omega \cdot x-t{e}^{2\rho }+\eta \rho \mathrm{.}$ The factor $(2\pi {)}^{-2}$ is exactly $(2\pi {)}^{-3\mathrm{/}2}(2\pi {)}^{-1\mathrm{/}2}$. There is no input-dependent amplitude in this kernel.

If $f=0$ every remaining claim is immediate; otherwise choose a compact interval $J$ containing $\mathrm{supp}A$ in its interior with $e^{\rho }\in [r_0,r_1]\subset (a,b)$ throughout $J$. For $\mathrm{\mid }x\mathrm{\mid }<2at$ and $\eta <0$, ${\mathrm{\partial }}_{\rho }{\mathrm{\Phi }}_t=e^{\rho }\omega \cdot x-2t{e}^{2\rho }+\eta \le -2t{r}_0(r_0-a)-\mathrm{\mid }\eta \mathrm{\mid }\mathrm{.}$ Writing $q=t+\mathrm{\mid }\eta \mathrm{\mid }$, this implies $\mathrm{\mid }{\mathrm{\partial }}_{\rho }{\mathrm{\Phi }}_t\mathrm{\mid }\ge cq$ with $c=\min \{2r_0(r_0-a),1\}>0$. For every $j\ge 1$, $\mathrm{\mid }{\mathrm{\partial }}_{\rho }^{j+1}{\mathrm{\Phi }}_t\mathrm{\mid }\le C_{j}t\le C_{j}q$ on $J$, uniformly in $x,\omega ,\eta ,t$ in this region. Repeated quotient differentiation therefore gives $\mathrm{\mid }{\mathrm{\partial }}_{\rho }^j(1\mathrm{/}{\mathrm{\partial }}_{\rho }{\mathrm{\Phi }}_t)\mathrm{\mid }\le C_j^{\mathrm{\prime }}{q}^{-1}$, including $j=0$: each term is a product of higher phase derivatives divided by a power of the first derivative, with one more denominator factor than numerator factors.

Define $B_0=A$ and $B_{m+1}=-{\mathrm{\partial }}_{\rho }(B_m\mathrm{/}(i{\mathrm{\partial }}_{\rho }{\mathrm{\Phi }}_t))$. Induction using step 5.1 and the product rule shows, for every fixed $m,j$, ${\sup }_J\mathrm{\mid }{\mathrm{\partial }}_{\rho }^j{B}_m\mathrm{\mid }\le C_{m,j}{q}^{-m}$. Each $B_m$ is supported in the same fixed compact subset of the interior of $J$. Since $e^{i{\mathrm{\Phi }}_t}=(i{\mathrm{\partial }}_{\rho }{\mathrm{\Phi }}_t{)}^{-1}{\mathrm{\partial }}_{\rho }{e}^{i{\mathrm{\Phi }}_t}$, integrating by parts $m$ times has no boundary terms and yields ${\int }_J{e}^{i{\mathrm{\Phi }}_t}A={\int }_J{e}^{i{\mathrm{\Phi }}_t}{B}_m$. Hence $\mathrm{\mid }{K}_t(x;\eta ,\omega )\mathrm{\mid }\le C_m{\chi }_t(x){\mathbf{1}}_{\eta <0}(t+\mathrm{\mid }\eta \mathrm{\mid }{)}^{-m}$. For $m\ge 1$ its squared integral is at most $C_m{t}^3{\int }_{-\mathrm{\infty }}^0(t+\mathrm{\mid }\eta \mathrm{\mid }{)}^{-2m}d\eta =C_m^{\mathrm{\prime }}{t}^{4-2m},$ where the finite area of $S^2$ and volume of the radius-$2at$ ball have been included in the constant. Cauchy–Schwarz in $(\eta ,\omega )$ bounds the operator norm by the square root of this integral, $C_m^{\mathrm{\prime }\mathrm{\prime }}{t}^{2-m}$; the resulting bounded integral operator agrees on the dense set of step 4.1 and hence everywhere with the original localized operator. Choose $m=N+1$ to obtain $C_m^{\mathrm{\prime }\mathrm{\prime }}{t}^{1-N}\le 2^N{C}_m^{\mathrm{\prime }\mathrm{\prime }}t(1+t{)}^{-N}$ for $t\ge 1$. This proves the entire outgoing norm estimate.

For $t=-s\le -1$ the same kernel construction with $P_{-}$ has $\eta >0$ and phase $e^{\rho }\omega \cdot x+s{e}^{2\rho }+\eta \rho$. In $\mathrm{\mid }x\mathrm{\mid }<2as$ its first derivative is at least $2s{r}_0(r_0-a)+\eta$, and its higher derivatives are bounded by $C_{j}s$. The recursion and squared integral of steps 5.1–6.1, with $q=s+\eta$, give $\mathrm{\parallel }{\chi }_s{e}^{is{H}_0}f(H_0)P_{-}\mathrm{\parallel }\le C_{N}s(1+s{)}^{-N}$. Taking the larger of the outgoing and incoming constants supplies the common $C_N$ in (c). If $N<0$, boundedness by $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ suffices because $s(1+s{)}^{-N}\ge 1$.

Apply step 6.1 to the cutoff $\bar{f}$, and take adjoints, to obtain $\mathrm{\parallel }{P}_{+}f(H_0)e^{is{H}_0}{\chi }_s\mathrm{\parallel }\to 0$ as $s\to +\mathrm{\infty }$; here $\bar{f}(H_0{)}^{\ast }=f(H_0)$, and the free group commutes with its spectral cutoff. For a fixed $\psi \in L^2$, $\mathrm{\parallel }{P}_{+}f(H_0)e^{is{H}_0}\psi \mathrm{\parallel }\le \mathrm{\parallel }{P}_{+}f(H_0)e^{is{H}_0}{\chi }_s\mathrm{\parallel }\mathrm{\parallel }\psi \mathrm{\parallel }+\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }(1-{\chi }_s)\psi \mathrm{\parallel }⟶0.$ The first term vanishes by the norm estimate with, for example, $N=3$, and the second by integrability of $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ on the expanding balls. Similarly, the adjoint of step 7.1 with cutoff $\bar{f}$ gives $\mathrm{\parallel }{P}_{-}f(H_0)e^{-is{H}_0}{\chi }_s\mathrm{\parallel }\to 0$ and the same tail split proves the incoming strong limit. Setting $t=-s$ in the first limit gives exactly the opposite-time limits stated in (c). This completes (a)–(c).