Weighted fractional chain rule and nonlinear wave equations with minimal regularity

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: \begin{eqnarray*} &\Box u = a |\partial_t u|^2+b|\nabla_x u|^2,& \\ & u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}, \quad \partial_t u(0,x)=u_1(x)\in H^{s-1}_{\mathrm{rad}}.& \end{eqnarray*} It has been known that the problem is well-posed for $s\ge 2$ and ill-posed for $s3/2$ and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.