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