Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems

Question

I would like to find good references for the theorems mentioned above in the title. I am reading chapter VIII of Hörmander's classic, but I wonder whether there is something more up-to-date.

My motivations are the uniqueness results in Calderon problem for partial data and the linearized problem. These results were first derived by Dos Santos, Kenig, Sjöstrand and Uhlmann, and in the end they end up having to study the injectivity of a certain limited-angle Radon transform. To this end, they need the mentioned theorems.

Thanks.

Answer 1

This answer only concerns support theorems.

First, some non-microlocal support theorems (to put more readers on the map):

• Helgason's support theorem: If a compactly supported continuous function in $\mathbb R^2$ integrates to zero over every line that does not meet a compact convex set $K$, then the function vanishes in $\mathbb R^2\setminus K$.
• A local support theorem near the boundary of a strictly convex real analytic Riemannian surface: A Support Theorem for the Geodesic Ray Transform of Functions, Krishnan
• A local support theorem in higher dimensional manifolds without real-analyticity: The inverse problem for the local geodesic ray transform, Uhlmann and Vasy

The last two of these results use microlocal techniques, so reading the papers can be illuminating for the microlocal perspectives as well. For microlocal aspects, I suggest taking a look at The geodesic ray transform on Riemannian surfaces with conjugate points and references therein. In the absence of conjugate points, if the X-ray transform for geodesics in a neighborhood of a geodesic $\gamma$ vanishes, the wavefront set of the unknown function doesn't meet the conormal bundle of $\gamma$. (This is true in all dimensions. This talk I heard is a nice summary of what is known.) The X-ray transform fails to an elliptic Fourier integral operator if there are conjugate points, but the attenuated X-ray transform still recovers singularities if there are no more than two conjugate points per geodesic.

I haven't found a good reference for the fact that one can recover $N^*\Gamma\cap WF(f)$ from the knowledge of $Xf(\gamma)$ for $\gamma\in\Gamma$, where $\Gamma$ is an open set of lines in a Euclidean space and $N^*$ denotes the conormal bundle. This result and many others are covered in the book "Microlocal Analysis and Integral Geometry" by Stefanov and Uhlmann, but the book is still in progress. That book will certainly be an up-to-date reference material for microlocal techniques for ray transforms.