On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$

Abstract

We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H_0^s(\mathbb{T},\mathbb{R})$, $s>-1/2$, to the scale of weighted ${\ell }^2-$sequence spaces, ${\mathfrak{h}}_{r,0}^{s+1/2}(\mathbb{N},\mathbb{C})$, $s>-1/2$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation ${\mathcal{S}}_0^t:H_0^s(\mathbb{T},\mathbb{R})\to H_0^s(\mathbb{T},\mathbb{R})$ is {\em nowhere locally uniformly continuous} in $H_0^s(\mathbb{T},\mathbb{R})$.

References in corpus (2)

• Sharp well-posedness results of the Benjamin-Ono equation in $H^s(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution
• On smoothing properties and Tao's gauge transform of the Benjamin-Ono equation on the torus