This questions is slightly related to Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems, in which I asked for some references. Now I having troubles with the last part (pag. 21-24) of the paper mentioned there: http://www.math.washington.edu/~gunther/publications/Papers/magneticlocal15.pdf (this is freely available in the web, so I guess is OK if I link it here).

They end up having, for the scalar potential: $$\int_{P \cap \Omega} q \hspace{1mm} dm_{P}=0, $$ for any -they say- 2-plane $P$ such that $d\left( (0,e_{1}, T(P) \right) < \delta.$ I don't understand this notation, but I guess that in the three dimensional case it just means (as is natural from the rest of stuff there) that that holds for any $P$ with director vector in a small conic neighborhood of $e_{1}$ and whose distance to the origin is in a interval $(-\epsilon, \epsilon).$

The next step is to combine theorems 5.3 and 5.4 on that paper (pag. 21), which are the microlocal versions of Helgason's support theorem and of Holmgren theorem. For any three-dimensional subspace $H$ such that $d\left( (0,e_{1}, T(H) \right) < \delta$ (again, this notation?) It is immediate that it holds: $$N^{\ast}(P) \cap N(\text{supp}\hspace{0.5mm}q \hspace{0.5mm}\vert _{H})=0.$$ And then they conclude that $q=0$ on $H.$ How does one proof that conclusion? And further, how does one conclude from here that $q$ vanishes in all $\Omega$,? I don't even understand the three-dimensional case: they would end with $q=0$ only on a subspace of $\Omega$ which is not too big.

I guess one of the problems is that I am not visualizing the set of 2-planes for which we have that integral identity. How is that set?

This injectivity result is in contrast with one of Boman and Quinto which I had studied previously, which says that if we have a neighborhood $V$ of $\theta \in S^{n-1}$ and $Rf(s,\theta)=0$ for all $\theta \in V$ and all $s > s_{0},$ then $f$ vanishes in the half-space $\langle x , \theta_{0} \rangle >s_{0}.$ This is from "Support theorems for real-analytic Radon transform", and I have simplified slightly the statement.

I guess this result cannot be applied because the condition $s > s_{0}$ fails, right? The planes they end up having in the paper "Determining a magnetic..." cannot separate arbitrarily far way from the observation/illumination point $x_{0}.$

In short, I am asking:

1. For a clarification of how is the family of planes they can plug into their Integral identity

2)For an explanation of how one derive the injectivity from microlocal Helgason + Holmgren

3)And.. cannot one employ the mentioned result of Boman and Quinto somehow to simplify the last part of the proof?

Thank you very much