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 Hs when s0 < s ≤ ¼ on the line and s0 < s < ½ on the circle, where [Formula: see text] . 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.