  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 Calder\'on 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.

Analysis of PDEs