In this talk we consider the Dirichlet to Neumann map (D-N) for the unit sphere in $R^3$. When we are sufficiently far from the origin, the spectrum of such an operator consists of eigenvalue clusters around the natural numbers. The distribution of the corresponding scaled eigenvalue shifts has an asymptotic expansion when the label of the cluster goes to infinity. The asymptotic expansion consists of distributions called spectral invariants. By using the averaging method, asymptotics of the Berezin symbol of the D-N map and a suitable symbol calculus, we compute the first terms of such an expansion in terms of the Radon transform (averages along geodesis of the unit sphere) of derivatives of the function that encodes the conductivity properties of the media in the unit ball.

The Mizohata-Takeuchi conjecture is a weighted estimate for the Fourier extension operator over the sphere. For radial weights, this conjecture was proved by Barceló, Ruiz and Vega when they were studying weighted estimates for the resolvent of the Laplace operator with radial weights that have the X-ray transform bounded. In this talk, we will study the analogous problem for the resolvent of the Lamé operator that appears in linear elasticity. Estimates for this resolvent play an important role in the resolution of both direct and inverse problems in elastic media. The proof will require the spherical harmonics decomposition of the fundamental solution and a very deep study of Bessel and Hankel functions.