Nonlinear fourth-order Schrödinger equations with radial data

We prove the small data scattering and large data local well-posedness for the nonlinear fourth-order Schrödinger equation with radial data in critical Ḣsc space, and in particular, for some sc≤0sc≤0. The new ingredient is the improved Strichartz estimates for fourth-order Schrödinger equations with radial initial data.