Local smoothing and maximal estimates for average over surfaces of codimension 2 in $\mathbb R^4$

In this paper, we obtain local smoothing estimates for the averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We make use of multilinear restriction estimates and decoupling inequalities for a hypersurface in $\mathbb R^5$, a conical extension of a two-dimensional nondegenerate surface along two flat directions. We also establish sharp $L^p$--$L^q$ estimates for maximal averages over nondegenerate surfaces of half the ambient dimension in $\mathbb R^{2n}$ for even $n \ge 2$.