A Note on the Refined Strichartz Estimates and Maximal Extension Operator

There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to more general class of phase functions. The main methods Bourgain-Demeter's $l^2$ decoupling theorem induction on scales. second prove positive maximal extension operator hypersurfaces with principal curvatures. sharp $L^2$ estimates Du Zhang, bilinear method Wolff Tao.