Improved curvature conditions on $L^2\times \cdots \times L^2\to L^{2\mathrm{/}m}$ bounds for multilinear maximal averages

In this article, we focus on $L^2({\mathbb{R}}^d)\times \cdots \times L^2({\mathbb{R}}^d)\to L^{2\mathrm{/}m}({\mathbb{R}}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear averages with $m\ge 3$, and represent a reproof of the recent result of T. Borges, B. Foster, and Y. Ou on the curvature conditions of the hypersurfaces required in establishing $L^2({\mathbb{R}}^d)\times L^2({\mathbb{R}}^d)\to L^1({\mathbb{R}}^d)$ estimates of bilinear maximal functions.