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≤ 14 on the line and s₀<s< 12 on the circle, where s₀ = 3-√33/4≈ 0.1277. Here, we adapt the strategy of Mo{s}incat-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.