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 $H^s$ when $s_0<s\le \frac{1}{4}$ on the line and $s_0<s<\frac{1}{2}$ on the circle, where $s_0=3-\sqrt{33\mathrm{/}4}\approx 0.1277$. Here, we adapt the strategy of Moşincat-Pilod (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.