A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey-Stewartson equation and to the inverse boundary value problem of Calderon

• 成果类型: Article
• 署名作者: Nachman, Adrian; Regev, Idan; Tataru, Daniel
• 署名单位: University of Toronto; University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00930-0
• 发表日期: 2020
• 页码: 395-451
• 关键词: scattering transform; schrodinger-equation; conductivity problem; uniqueness; reconstruction; asymptotics

摘要

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 L2 for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderon in dimension 2, for conductivities sigma>0 with log sigma is an element of H1. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on L2-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.