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 $n\ge 2$.