A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calder\'on

Adrian I. Nachman; Idan Regev; Daniel I. Tataru

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arXiv · 1708.04759

A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calder\'on

Adrian I. Nachman·Idan Regev·Daniel I. Tataru

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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 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.

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Adrian I. Nachman, Idan Regev, Daniel I. Tataru. 2017-08-16. A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calder\'on. https://arxiv.org/abs/1708.04759

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