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 $$L^2$$ 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 $$\sigma >0$$ σ > 0 with $$\log \sigma \in \dot{H}^1$$ 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 $$L^2$$ L 2 -boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

中文翻译：

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非线性 Plancherel 定理，适用于散焦 Davey-Stewartson 方程的全局适定性和 Calderón 的逆边值问题

我们证明了在散焦 Davey-Stewartson II 方程的逆散射方法中出现的二维非线性傅立叶变换的 Plancherel 定理。然后，我们使用它来证明散焦 DSII 中 $$L^2$$L 2 中的全局适定性和散射。Plancherel 定理也暗示了 Calderón 在第 2 维的逆边值问题中的全局唯一性，对于电导率 $$\sigma >0$$ σ > 0 with $$\log \sigma \in \dot{H}^1$$ log σ ∈ H ˙ 1 。非线性 Plancherel 定理的证明包括对经典分数积分的新估计，以及对具有非光滑符号的伪微分算子的 $$L^2$$L 2 有界的新结果，在所有维度上都有效。