Revisiting the Anisotropic Fractional Calder\'on Problem Using the Caffarelli-Silvestre Extension

We revisit the source-to-solution anisotropic fractional Calder\'on problem introduced and analyzed in [FGKU21] and [F21]. Using the Caffarelli-Silvestre interpretation of the fractional Laplacian, we provide an alternative argument for the recovery of the heat and wave kernels from [FGKU21]. This shows that in the setting of the source-to-solution anisotropic fractional Calder\'on problem the heat and Caffarelli-Silvestre approach give rise to equivalent perspectives and that each kernel can be recovered from the other. Moreover, we also discuss the Dirichlet-to-Neumann anisotropic source-to-solution problem and provide a direct link between the Dirichlet Poisson kernel and the wave kernel. This illustrates that it is also possible to argue completely on the level of the Poisson kernel, bypassing the recovery of the heat kernel as an additional auxiliary step. Last but not least, as in [CGRU23], we relate the local and nonlocal source-to-solution Calder\'on problems.