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

In this article, we focus on $L^{2}(\mathbb{R}^d)\times\cdots\times L^{2}(\mathbb{R}^d)\rightarrow L^{2/m}(\mathbb{R}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear averages with $m\geq3$, 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 $L^{2}(\mathbb{R}^d)\times L^{2}(\mathbb{R}^d)\rightarrow L^{1}(\mathbb{R}^d)$ estimates of bilinear maximal functions.