[proofplan]
We choose a nonnegative cutoff $\chi$ supported in the measured set $V$ and nonzero at a data point realizing the visibility of $(x_0,\xi_0)$. The Radon transform is a Fourier integral operator whose canonical relation records the incidence relation $s=x\cdot\omega$ together with the normal covector direction $\xi=\lambda\omega$. Composing this canonical relation with its transpose gives the diagonal relation near the visible branch, so $R^*\chi R$ is microlocally a pseudodifferential operator of order $-1$. Its principal symbol is a nonzero constant times $|\xi|^{-1}$ multiplied by the sum of the two antipodal cutoff values, hence is elliptic near $(x_0,\xi_0)$. The elliptic parametrix theorem then gives an order $1$ inverse modulo smoothing terms, and the Sobolev wave front equivalence follows from the mapping and elliptic regularity properties of pseudodifferential operators.
[/proofplan]
[step:Choose a measured cutoff that is nonzero on a visible branch]
Because $(x_0,\xi_0)$ is visible from $V$, there exist $\omega_0\in S^1$ and $\lambda_0\in\mathbb{R}\setminus\{0\}$ such that $s_0:=x_0\cdot\omega_0$ satisfies
\begin{align*}
(\omega_0,s_0)\in V
\end{align*}
and
\begin{align*}
\xi_0=\lambda_0\omega_0.
\end{align*}
Since $V$ is open and $(\omega_0,s_0)\ne(-\omega_0,-s_0)$, choose an open neighbourhood $W\subset V$ of $(\omega_0,s_0)$ whose closure in $S^1\times\mathbb{R}$ is disjoint from a neighbourhood of the antipodal data point $(-\omega_0,-s_0)$. Choose a function
\begin{align*}
\chi:S^1\times\mathbb{R}\to[0,\infty)
\end{align*}
with $\chi\in C_c^\infty(V)$, $\operatorname{supp}\chi\subset W$, and $\chi(\omega_0,s_0)>0$. This is possible by the smooth bump function construction on the smooth manifold $S^1\times\mathbb{R}$.
The nonnegativity is used to prevent cancellation between the two oriented parametrizations of the same unoriented line. Since $\operatorname{supp}\chi\subset V$, the distribution $\chi\,Rf$ depends only on the known restriction of $Rf$ to $V$, and hence so does $N_\chi f=R^*(\chi\,Rf)$.
[/step]
[step:Record the canonical relation of the Radon transform]
Let $\mathcal{L}^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}$, and let $\mathcal{L}^2$ denote two-dimensional Lebesgue measure on $\mathbb{R}^2$. For $f\in C_c^\infty(\mathbb{R}^2)$, the Radon transform can be represented microlocally by the oscillatory integral
\begin{align*}
Rf(\omega,s)=\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}^2}e^{i\sigma(s-y\cdot\omega)}f(y)\,d\mathcal{L}^2(y)\,d\mathcal{L}^1(\sigma).
\end{align*}
Here the phase function
\begin{align*}
\phi:S^1\times\mathbb{R}\times\mathbb{R}^2\times(\mathbb{R}\setminus\{0\})\to\mathbb{R}
\end{align*}
is defined by
\begin{align*}
\phi(\omega,s,y,\sigma)=\sigma(s-y\cdot\omega).
\end{align*}
The critical equation in the frequency variable is
\begin{align*}
\partial_\sigma\phi(\omega,s,y,\sigma)=s-y\cdot\omega=0.
\end{align*}
Thus the canonical relation $C_R\subset T^*(S^1\times\mathbb{R})\setminus 0\times T^*\mathbb{R}^2\setminus 0$ is
\begin{align*}
C_R=\{((\omega,s;\eta,\sigma),(y,\xi)):s=y\cdot\omega,\ \xi=-\sigma\omega,\ \eta(v)=-\sigma\,y\cdot v\text{ for every }v\in T_\omega S^1,\ \sigma\ne0\}.
\end{align*}
Changing the parameter $\sigma$ to $-\sigma$ gives the equivalent convention $\xi=\sigma\omega$. We use this latter sign convention below; the sign has no effect on ellipticity because $\sigma$ ranges over $\mathbb{R}\setminus\{0\}$.
By the standard microlocal Radon-transform calculus assumed in the theorem statement, the preceding phase representation is a Fourier integral representation of $R$. The phase is nondegenerate on the incidence hypersurface because $\partial_\sigma\phi=s-y\cdot\omega$ and the differential of $s-y\cdot\omega$ in the $s$ variable is $1$. The single nonzero oscillatory parameter $\sigma$ parametrizes the conormal direction to this hypersurface, and the standard order convention for this Radon kernel gives order $-1/2$. Hence $R$ is a Fourier integral operator of order $-1/2$ associated to $C_R$, and its adjoint $R^*$ is a Fourier integral operator of order $-1/2$ associated to the transposed canonical relation $C_R^t$.
[guided]
We first need to know which singularities of $f$ can contribute to singularities of $Rf$. The oscillatory representation gives this information directly. For $f\in C_c^\infty(\mathbb{R}^2)$, write
\begin{align*}
Rf(\omega,s)=\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}^2}e^{i\sigma(s-y\cdot\omega)}f(y)\,d\mathcal{L}^2(y)\,d\mathcal{L}^1(\sigma).
\end{align*}
This is just the Fourier representation of the delta distribution imposing the equation $s=y\cdot\omega$. The phase function is the smooth map
\begin{align*}
\phi:S^1\times\mathbb{R}\times\mathbb{R}^2\times(\mathbb{R}\setminus\{0\})\to\mathbb{R}
\end{align*}
defined by
\begin{align*}
\phi(\omega,s,y,\sigma)=\sigma(s-y\cdot\omega).
\end{align*}
The critical equation in the auxiliary variable $\sigma$ is
\begin{align*}
\partial_\sigma\phi(\omega,s,y,\sigma)=s-y\cdot\omega=0.
\end{align*}
Therefore the incidence relation is exactly the condition that $y$ lies on the line with normal $\omega$ and signed distance $s$.
The covectors are obtained by differentiating the phase in the base variables. The derivative in $s$ is $\sigma$. If $v\in T_\omega S^1$ is a tangent vector at $\omega$, the derivative in the $\omega$ variable is $-\sigma\,y\cdot v$. The derivative in the spatial variable $y$ is $-\sigma\omega$. Hence the canonical relation is
\begin{align*}
C_R=\{((\omega,s;\eta,\sigma),(y,\xi)):s=y\cdot\omega,\ \xi=-\sigma\omega,\ \eta(v)=-\sigma\,y\cdot v\text{ for every }v\in T_\omega S^1,\ \sigma\ne0\}.
\end{align*}
Because $\sigma$ runs through all nonzero [real numbers](/page/Real%20Numbers), replacing $\sigma$ by $-\sigma$ gives the equivalent convention $\xi=\sigma\omega$. This convention matches the visibility condition $\xi=\lambda\omega$.
The standard Radon-transform Fourier integral operator theorem, included as an assumption in the theorem statement, applies to this phase. Its nondegeneracy is visible directly: the defining incidence function $s-y\cdot\omega$ has derivative $1$ in the $s$ variable, so the incidence hypersurface is smooth. The nonzero parameter $\sigma$ parametrizes the conormal line to that hypersurface, and the standard Radon kernel order convention gives order $-1/2$. Therefore this oscillatory representation makes $R$ a Fourier integral operator of order $-1/2$ associated to $C_R$, and $R^*$ is a Fourier integral operator of order $-1/2$ associated to $C_R^t$. This is the microlocal mechanism behind visibility: spatial covectors normal to measured lines are precisely the covectors that enter the canonical relation.
[/guided]
[/step]
[step:Compose the Radon canonical relation with its transpose near the visible covector]
Let $\Omega\subset T^*\mathbb{R}^2\setminus0$ be a sufficiently small conic neighbourhood of $(x_0,\xi_0)$. Define the oriented-normal maps
\begin{align*}
\omega_+:\Omega\to S^1
\end{align*}
and
\begin{align*}
\omega_-:\Omega\to S^1
\end{align*}
by
\begin{align*}
\omega_+(x,\xi)=\frac{\xi}{|\xi|}
\end{align*}
and
\begin{align*}
\omega_-(x,\xi)=-\frac{\xi}{|\xi|}.
\end{align*}
The corresponding oriented line-parameter maps are
\begin{align*}
\ell_+:\Omega\to S^1\times\mathbb{R}
\end{align*}
and
\begin{align*}
\ell_-:\Omega\to S^1\times\mathbb{R}
\end{align*}
defined by
\begin{align*}
\ell_+(x,\xi)=(\omega_+(x,\xi),x\cdot\omega_+(x,\xi))
\end{align*}
and
\begin{align*}
\ell_-(x,\xi)=(\omega_-(x,\xi),x\cdot\omega_-(x,\xi)).
\end{align*}
Shrink $\Omega$ so that the branch realizing visibility stays in $W$ and the nonselected antipodal branch avoids $\operatorname{supp}\chi$. More explicitly, if $\lambda_0>0$, shrink $\Omega$ so that $\ell_+(\Omega)\subset W$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi=\varnothing$; if $\lambda_0<0$, shrink $\Omega$ so that $\ell_-(\Omega)\subset W$ and $\ell_+(\Omega)\cap\operatorname{supp}\chi=\varnothing$. This shrinking is possible because $\ell_+$ and $\ell_-$ are continuous, $\operatorname{supp}\chi$ is a closed subset of $W$, and $W$ was chosen away from the antipodal data point $(-\omega_0,-s_0)$. Consequently each set $\ell_+(\Omega)\cap\operatorname{supp}\chi$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi$ is either empty or contained in a single localized oriented branch. This is exactly the branch-isolation hypothesis in the Radon normal-operator calculus assumed in the theorem statement.
Suppose two nearby spatial covectors $(x,\xi)$ and $(y,\eta)$ are connected through the same oriented Radon covector in the localized composition. Then the canonical relation gives a common $\omega\in S^1$ and nonzero parameters $\sigma,\tau\in\mathbb{R}\setminus\{0\}$ such that $\xi=\sigma\omega$, $\eta=\tau\omega$, $x\cdot\omega=y\cdot\omega$, and the full data covectors agree. Equality of the data covectors includes equality of the $s$-covector components, so $\sigma=\tau$. Writing $v\in T_\omega S^1$ for the unit tangent vector, equality of the $S^1$-covector components gives $\sigma x\cdot v=\sigma y\cdot v$. Since $\sigma\ne0$, this gives $x\cdot v=y\cdot v$. Since $\{\omega,v\}$ is a basis of $\mathbb{R}^2$, the two scalar equalities $x\cdot\omega=y\cdot\omega$ and $x\cdot v=y\cdot v$ imply $x=y$, and then $\xi=\eta$. Thus the localized composed relation has diagonal output near the selected visible branch.
We now use the Radon normal-operator composition statement assumed in the theorem. Its hypotheses are met on the chosen conic neighbourhood: multiplication by $\chi$ is a pseudodifferential operator of order $0$ on $S^1\times\mathbb{R}$, each supported oriented branch is isolated in the explicit sense that $\ell_+(\Omega)\cap\operatorname{supp}\chi$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi$ are separately localized or empty, and the preceding calculation identifies the localized composed relation with the diagonal in $T^*\mathbb{R}^2\setminus0$. Therefore the assumed clean composition result applies directly and gives that $N_\chi=R^*\chi R$ is a classical pseudodifferential operator of order $-1$ microlocally near $(x_0,\xi_0)$.
[/step]
[step:Compute the principal symbol and prove ellipticity at the visible covector]
By the positive Radon normal-symbol formula assumed in the theorem statement, the principal symbol of the localized Radon normal operator has the form
\begin{align*}
\sigma_{-1}(N_\chi)(x,\xi)=c_R|\xi|^{-1}\left[\chi\left(\frac{\xi}{|\xi|},x\cdot\frac{\xi}{|\xi|}\right)+\chi\left(-\frac{\xi}{|\xi|},-x\cdot\frac{\xi}{|\xi|}\right)\right],
\end{align*}
where $c_R>0$ is the positive density constant obtained from the stationary-phase evaluation of the one-dimensional clean fiber in the Radon normal composition. The positivity follows because the normal operator is formed as $R^*\chi R$ and the principal density contribution is the positive fiber density multiplied by the nonnegative cutoff values. This is the principal symbol formula for the Radon normal operator assumed in the theorem statement.
Evaluate this formula at $(x_0,\xi_0)$. If $\lambda_0>0$, then
\begin{align*}
\frac{\xi_0}{|\xi_0|}=\omega_0,
\end{align*}
and the first term equals $\chi(\omega_0,s_0)>0$. If $\lambda_0<0$, then
\begin{align*}
-\frac{\xi_0}{|\xi_0|}=\omega_0,
\end{align*}
and the second term equals $\chi(\omega_0,s_0)>0$. Since $\chi\ge0$, the sum cannot cancel. Hence
\begin{align*}
\sigma_{-1}(N_\chi)(x_0,\xi_0)\ne0.
\end{align*}
By continuity and homogeneity of the principal symbol, there exists a conic neighbourhood $U_0\subset T^*\mathbb{R}^2\setminus0$ of $(x_0,\xi_0)$ and a constant $a_0>0$ such that
\begin{align*}
|\sigma_{-1}(N_\chi)(x,\xi)|\ge a_0|\xi|^{-1}
\end{align*}
for every $(x,\xi)\in U_0$. Thus $N_\chi$ is elliptic of order $-1$ on $U_0$.
[/step]
[step:Construct a microlocal parametrix for the elliptic normal operator]
By the [elliptic parametrix theorem for classical pseudodifferential operators](/theorems/7708), applied to the order $-1$ operator $N_\chi$ on the elliptic conic neighbourhood $U_0$, there exist a conic neighbourhood $U\subset U_0$ of $(x_0,\xi_0)$ and a classical pseudodifferential operator
\begin{align*}
Q:\mathcal{D}'(\mathbb{R}^2)\to\mathcal{D}'(\mathbb{R}^2)
\end{align*}
of order $1$ such that, with $I:\mathcal{D}'(\mathbb{R}^2)\to\mathcal{D}'(\mathbb{R}^2)$ denoting the identity operator,
\begin{align*}
QN_\chi-I=S
\end{align*}
microlocally on $U$, where $S$ is microlocally smoothing on $U$. Therefore, for every $f\in\mathcal{E}'(\mathbb{R}^2)$,
\begin{align*}
QN_\chi f-f=Sf
\end{align*}
is smooth microlocally on $U$.
[/step]
[step:Translate elliptic inversion into Sobolev wave front equivalence]
Fix $t\in\mathbb{R}$ and $(x,\xi)\in U$. Since $N_\chi$ is a pseudodifferential operator of order $-1$, the Sobolev mapping property for pseudodifferential operators gives
\begin{align*}
(x,\xi)\notin WF^t(f)\quad\Longrightarrow\quad (x,\xi)\notin WF^{t+1}(N_\chi f).
\end{align*}
Indeed, an operator of order $-1$ raises local Sobolev regularity by one.
Conversely, suppose $(x,\xi)\notin WF^{t+1}(N_\chi f)$. Since $Q$ has order $1$, the same mapping property gives
\begin{align*}
(x,\xi)\notin WF^t(QN_\chi f).
\end{align*}
The parametrix identity gives $f=QN_\chi f-Sf$ microlocally on $U$, and $Sf$ is smooth microlocally on $U$. Smooth microlocal regularity implies absence from every Sobolev wave front set, so
\begin{align*}
(x,\xi)\notin WF^t(f).
\end{align*}
This proves
\begin{align*}
(x,\xi)\notin WF^t(f)\quad\Longleftrightarrow\quad (x,\xi)\notin WF^{t+1}(N_\chi f)
\end{align*}
for every $t\in\mathbb{R}$ and every $(x,\xi)\in U$.
The only data used to form $N_\chi f$ are $\chi Rf$ with $\operatorname{supp}\chi\subset V$. Hence the measured Radon data on $V$ determine the Sobolev wave front presence and Sobolev order of $f$ at the visible covectors in $U$, modulo the microlocally smooth remainder produced by the elliptic parametrix.
[/step]
