We prove an inequality of the type$$\parallel |x{|}^{r_f}{\parallel }_{L^p({\mathbf{R}}^n)}\le c(n,p,q,r)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{L^q({\mathbf{R}}^n)}.$$This is then used to derive the unique continuation property for the differential inequality $\left|\Delta f\right(x\left)\right|\le \left|v\right(x\left)\right|\left|f\right(x\left)\right|$ under suitable local integrability assumptions on the function $v$.

The purpose of this talk is to present some recent results about the Cauchy theory of the gravity water waves equations (without surface tension). In particular, we clarify the theory as well in terms of regularity indexes for the initial conditions as fin terms of smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developed in [1, 2], after suitable para-linearizations, the system can be arranged into...