We prove uniform-in-time a priori \(H^s\) bounds for solutions to the intermediate long wave equation posed both on the line and on the circle, covering the range \(-\frac{1}{2}<s\le 0\) . Additionally, we prove that the set of orbits emanating from a bounded and equicontinuous set in \(H^s\) is also bounded and equicontinuous in \(H^s\) . Our proof is based on the identification of a suitable Lax pair formulation for the intermediate long wave equation.
