Nachman's reconstruction for the Calderon problem with discontinuous conductivities

We show that Nachman's integral equations for the Calder\'on problem, derived for conductivities in $W^{2,p}(\Omega)$, still hold for $L^\infty$ conductivities which are $1$ in a neighborhood of the boundary. We also prove convergence of scattering transforms for smooth approximations to the scattering transform of $L^\infty$ conductivities. We rely on Astala-P\"aiv\"arinta's formulation of the Calder\'on problem for a framework in which these convergence results make sense.