Anisotropic regularity of linearized compressible vortex sheets

We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[vcdot au]|>2sqrt{2},c$, and
the well-posedness was obtained in standard weighted Sobolev spaces.
The aim of the present paper is to improve this result, by showing the existence of the solution in function spaces with some additional weighted anisotropic regularity in the frequency space.