# American Institute of Mathematical Sciences

May  2017, 16(3): 883-898. doi: 10.3934/cpaa.2017042

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

 1 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China 2 School of Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100191, China 3 Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

* Corresponding author: Guozhen Lu

Received  July 2016 Revised  December 2016 Published  February 2017

Fund Project: The first two authors were partly supported by the NNSF of China (No. 11371056), the second author was also supported by the NNSF of China (No. 11501021), the third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation

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 difficulties in our proof from those in Grafakos, Honzík and Seeger [15] and Honzík [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-$\lambda$ inequality, we can derive the desired localized estimates.

Citation: JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042
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