A priori bounds and equicontinuity of orbits for the intermediate long wave equation

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.