Regularity of k-curvature equation

this is a note after reading the article”” of Cafferalli.

in his article,a large type of fully nonlinear elliptic equation has been particular,including the k-curvature equation.and use the continue method,we just need to establish a ingredient estimate,$latex C^2$ estimate in the interior and $latex C^2$ estimate near the boundary.we establish these estimate step by step,base on construct special flexible function and use the maximum principle to establish the first and second estimate,for the $latex C^2$ estimate near the boundary we need to investigate the influence of permutation on the boundary carefully.