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

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 $H^{s}_{0}(\mathbb{T},\mathbb{R})$, $s > -1/2$, to the scale of weighted $\ell^2-$sequence spaces, $\mathfrak{h}^{s +1/2}_{r,0}(\mathbb{N},\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2

Keywords:

• Uniform continuity
• Benjamin–Ono equation
• nonlinear fourier transform
• Sobolev space
• scale
• Mathematical physics
• Diffeomorphism
• Physics

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