Low regularity solutions for gravity water waves

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - \mu$,where$\mu = \frac{1}{10}$inthecase$d = 1$and$\mu = \frac{1}{5}$ in the case $d\ge 2$, extending previous results of Alazard-Burq-Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.