A sharp Cauchy theory for the 2D gravity-capillary waves

This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for data $\frac{1}{4}$ derivatives less regular than the energy threshold (obtained by Alazard-Burq-Zuily), which corresponds to the gain of Holder regularity of the semiclassical Strichartz estimate for the fully nonlinear system. To obtain this result, we establish global, quantitative results for the paracomposition theory of Alinhac.