The purpose of this work is to provide an investigation of \(L^p\) and \(H^p\) estimates for N-linear Fourier multipliers with flag-type singularities. The involved flag symbols of these multipliers include \(m(\xi ):=\prod \nolimits _{S\subseteq \{1,\cdots ,N\}}m_S(\xi _S),\) which was introduced by Muscalu (Rev Mat Iberoam 23:705–742, 2007; Contemp. Math. 505, 131–151, 2010), as well as the ones that may have singularities along “diagonal planes”. The trilinear models of such operators have been widely studied in the literature, and we hope to show the general multilinear result. Our work is inspired by Muscalu (Rev Mat Iberoam 23:705–742, 2007; Contemp. Math. 505, 131–151, 2010) and Miyachi and Tomita (Math Z 282(1–2):577–613, 2016).
