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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 $$L^2$$ L 2 for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension 2, for conductivities $$\sigma >0$$ σ > 0 with $$\log \sigma \in \dot{H}^1$$ log σ ∈ H ˙ 1 . The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $$L^2$$ L 2 -boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

inventiones mathematicae

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