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 1/4 derivatives less regular than the energy threshold (obtained by Alazard-Burq-Zuily), which corresponds to the gain of Hölder regularity of the semi-classical Strichartz estimate for the fully nonlinear system. To obtain this result, we establish global, quantitative results for the paracomposition theory of Alinhac