Bilinear estimates and applications to 2d NLS

The three bilinearities $u v, \overline {uv},\overline {u}v$ for functions $u, v : \mathbb {R}^2 \times [0,T] \longmapsto \mathbb {C}$ are sharply estimated in function spaces $X_{s,b}$ associated to the Schrödinger operator $i \partial _t + \Delta$. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.