Improved low regularity theory for gravity-capillary waves

This article concerns the Cauchy problem for the gravity-capillary water waves system in general dimensions. We establish local well-posedness for initial data in $H^s$, with $s>\frac{d}{2}+2-\mu$, with $\mu =\frac{3}{14}$ and $\mu =\frac{3}{7}$ in the cases $d=1$ and $d\ge 2$ respectively. This represents an improvement over the the state-of-the-art low regularity theory in $d\ge 2$ dimensions.