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- Title
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.
- Authors
Nachman, Adrian; Regev, Idan; Tataru, Daniel
- 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 Calderón in dimension 2, for conductivities σ > 0 with 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 -boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.
- Subjects
BOUNDARY value problems; PSEUDODIFFERENTIAL operators; NONLINEAR operators; FRACTIONAL integrals; EQUATIONS; NONLINEAR boundary value problems; FOURIER transforms
- Publication
Inventiones Mathematicae, 2020, Vol 220, Issue 2, p395
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-019-00930-0