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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

  • Adrian NachmanEmail author
  • Idan Regev
  • Daniel Tataru
Article
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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 Calderón 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.

Notes

Acknowledgements

The authors would like to thank Alexandru Tamasan for many helpful discussions at the early stages of investigation. A. Nachman and D. Tataru are grateful to IHP for hospitality and support during the program on Inverse Problems in 2015, which allowed us to initiate this project. The authors are also grateful to Xian Liao, Peter Perry, Mihai Tohaneanu, Pavel Zorin-Kranich and the anonymous referees for carefully reading the manuscript and helping us correct a number of typos and inaccuracies. D. Tataru was partially supported by the NSF grant DMS-1266182 as well as by the Simons Investigator grant from the Simons Foundation. A. Nachman was partially supported by the NSERC Discovery Grant RGPIN-06329.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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