Unconditional deep-water limit of the intermediate long wave equation in low-regularity

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi> <mml:mi>s</mml:mi></mml:msup> </mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>s</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo><</mml:mo> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac><mml:mn>1</mml:mn> <mml:mn>4</mml:mn></mml:mfrac> </mml:mrow> </mml:math> on the line and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>s</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo><</mml:mo> <mml:mi>s</mml:mi> <mml:mo><</mml:mo> <mml:mfrac><mml:mn>1</mml:mn> <mml:mn>2</mml:mn></mml:mfrac> </mml:mrow> </mml:math> on the circle, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>s</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>-</mml:mo> <mml:msqrt><mml:mrow><mml:mn>33</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn></mml:mrow> </mml:msqrt> <mml:mo>≈</mml:mo> <mml:mn>0.1277</mml:mn></mml:mrow> </mml:math> . Here, we adapt the strategy of Moşincat-Pilod (Pure Appl Anal 5:285-322, 2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.