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

Abstract

In this article, we focus on $L^2({\mathbb{R}}^d)\times \cdots \times L^2({\mathbb{R}}^d)\to L^{2/m}({\mathbb{R}}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear averages with $m\ge 3$, 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.

14 pages. We changed Lemma 2.2 due to some errors, hence the condition in Theorem 1.2, $s>(m-2)d/2+1$ is replaced by $s>(m-1)d/2+1/2$