A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey–Stewartson equation and to the inverse boundary value problem of Calderón

Abstract

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey–Stewartson II equation. We then use it to prove global well-posedness and scattering in $$L^2$$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderon in dimension 2, for conductivities $$\sigma >0$$ with $$\log \sigma \in \dot{H}^1$$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $$L^2$$-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.