L^p boundedness for maximal functions associated with multi-linear pseudo-differential operators

In this paper, we establish the L^p estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra dificulties in our proof from those in Grafakos, Honzfik and Seeger [15] and Honzfik [22] for maximal functions generated by multipliers. First, in order to eliminate the variable x in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in [26, 30]. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-λ inequality, we can derive the desired localized estimates.