<p>We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{s}_{0}(\mathbb {T}, \mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$s > -1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, to the scale of weighted <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell ^2-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>sequence spaces, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq7.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {h}^{s +1/2}_{r,0}(\mathbb N, \mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="fraktur">h</mi> </mrow> <mrow> <mi>r</mi> <mo>,</mo> <mn>0</mn> </mrow> <mrow> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$s >-1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. As an application we show that for any <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$-1/2<s<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the flow map of the Benjamin-Ono equation <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="182" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {S}_0^t : H^{s}_{0}(\mathbb {T}, \mathbb {R})\rightarrow H^{s}_{0}(\mathbb {T}, \mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">S</mi> <mn>0</mn> <mi>t</mi> </msubsup> <mo>:</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is <i>nowhere locally uniformly continuous</i> in <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1053_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{s}_{0}(\mathbb {T}, \mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

期刊论文
发表日期：

2025年6月11日

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P. Gérard,
T. Kappeler,
P. Topalov

摘要

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

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

摘要